An Introduction to the Theory of Point Processes: General theory and structure

Probability and its Applications A Series of the Applied Probability Trust Editors: J. Gani, C.C. Heyde, T.G. Kurtz Sp...

Author: Daryl J. Daley | David Vere-Jones

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Probability and its Applications A Series of the Applied Probability Trust

Editors: J. Gani, C.C. Heyde, T.G. Kurtz

Springer New York Berlin Heidelberg Hong Kong London Milan Paris Tokyo

D.J. Daley

D. Vere-Jones

An Introduction to the Theory of Point Processes Volume I: Elementary Theory and Methods Second Edition

D.J. Daley Centre for Mathematics and its Applications Mathematical Sciences Institute Australian National University Canberra, ACT 0200, Australia [email protected]

Series Editors: J. Gani Stochastic Analysis Group, CMA Australian National University Canberra, ACT 0200 Australia

D. Vere-Jones School of Mathematical and Computing Sciences Victoria University of Wellington Wellington, New Zealand [email protected]

C.C. Heyde Stochastic Analysis Group, CMA Australian National University Canberra, ACT 0200 Australia

T.G. Kurtz Department of Mathematics University of Wisconsin 480 Lincoln Drive Madison, WI 53706 USA

Library of Congress Cataloging-in-Publication Data Daley, Daryl J. An introduction to the theory of point processes / D.J. Daley, D. Vere-Jones. p. cm. Includes bibliographical references and index. Contents: v. 1. Elementary theory and methods ISBN 0-387-95541-0 (alk. paper) 1. Point processes. I. Vere-Jones, D. (David) II. Title QA274.42.D35 2002 519.2´3—dc21 2002026666 ISBN 0-387-95541-0

Printed on acid-free paper.

© 2003, 1988 by the Applied Probability Trust. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. 9 8 7 6 5 4 3 2 1

SPIN 10885680

Typesetting: Photocomposed pages prepared by the authors using plain TeX files. www.springer-ny.com Springer-Verlag New York Berlin Heidelberg A member of BertelsmannSpringer Science+Business Media GmbH

◆ To Nola, and in memory of Mary ◆

Preface to the Second Edition

In preparing this second edition, we have taken the opportunity to reshape the book, partly in response to the further explosion of material on point processes that has occurred in the last decade but partly also in the hope of making some of the material in later chapters of the first edition more accessible to readers primarily interested in models and applications. Topics such as conditional intensities and spatial processes, which appeared relatively advanced and technically difficult at the time of the first edition, have now been so extensively used and developed that they warrant inclusion in the earlier introductory part of the text. Although the original aim of the book— to present an introduction to the theory in as broad a manner as we are able—has remained unchanged, it now seems to us best accomplished in two volumes, the first concentrating on introductory material and models and the second on structure and general theory. The major revisions in this volume, as well as the main new material, are to be found in Chapters 6–8. The rest of the book has been revised to take these changes into account, to correct errors in the first edition, and to bring in a range of new ideas and examples. Even at the time of the first edition, we were struggling to do justice to the variety of directions, applications and links with other material that the theory of point processes had acquired. The situation now is a great deal more daunting. The mathematical ideas, particularly the links to statistical mechanics and with regard to inference for point processes, have extended considerably. Simulation and related computational methods have developed even more rapidly, transforming the range and nature of the problems under active investigation and development. Applications to spatial point patterns, especially in connection with image analysis but also in many other scientific disciplines, have also exploded, frequently acquiring special language and techniques in the different fields of application. Marked point processes, which were clamouring for greater attention even at the time of the first edition, have acquired a central position in many of these new applications, influencing both the direction of growth and the centre of gravity of the theory. vii

viii

Preface to the Second Edition

We are sadly conscious of our inability to do justice to this wealth of new material. Even less than at the time of the first edition can the book claim to provide a comprehensive, up-to-the-minute treatment of the subject. Nor are we able to provide more than a sketch of how the ideas of the subject have evolved. Nevertheless, we hope that the attempt to provide an introduction to the main lines of development, backed by a succinct yet rigorous treatment of the theory, will prove of value to readers in both theoretical and applied fields and a possible starting point for the development of lecture courses on different facets of the subject. As with the first edition, we have endeavoured to make the material as self-contained as possible, with references to background mathematical concepts summarized in the appendices, which appear in this edition at the end of Volume I. We would like to express our gratitude to the readers who drew our attention to some of the major errors and omissions of the first edition and will be glad to receive similar notice of those that remain or have been newly introduced. Space precludes our listing these many helpers, but we would like to acknowledge our indebtedness to Rick Schoenberg, Robin Milne, Volker Schmidt, G¨ unter Last, Peter Glynn, Olav Kallenberg, Martin Kalinke, Jim Pitman, Tim Brown and Steve Evans for particular comments and careful reading of the original or revised texts (or both). Finally, it is a pleasure to thank John Kimmel of Springer-Verlag for his patience and encouragement, and especially Eileen Dallwitz for undertaking the painful task of rekeying the text of the first edition. The support of our two universities has been as unflagging for this endeavour as for the first edition; we would add thanks to host institutions of visits to the Technical University of Munich (supported by a Humboldt Foundation Award), University College London (supported by a grant from the Engineering and Physical Sciences Research Council) and the Institute of Mathematics and its Applications at the University of Minnesota. Daryl Daley Canberra, Australia

David Vere-Jones Wellington, New Zealand

Preface to the First Edition

This book has developed over many years—too many, as our colleagues and families would doubtless aver. It was conceived as a sequel to the review paper that we wrote for the Point Process Conference organized by Peter Lewis in 1971. Since that time the subject has kept running away from us faster than we could organize our attempts to set it down on paper. The last two decades have seen the rise and rapid development of martingale methods, the surge of interest in stochastic geometry following Rollo Davidson’s work, and the forging of close links between point processes and equilibrium problems in statistical mechanics. Our intention at the beginning was to write a text that would provide a survey of point process theory accessible to beginning graduate students and workers in applied fields. With this in mind we adopted a partly historical approach, starting with an informal introduction followed by a more detailed discussion of the most familiar and important examples, and then moving gradually into topics of increased abstraction and generality. This is still the basic pattern of the book. Chapters 1–4 provide historical background and treat fundamental special cases (Poisson processes, stationary processes on the line, and renewal processes). Chapter 5, on finite point processes, has a bridging character, while Chapters 6–14 develop aspects of the general theory. The main difficulty we had with this approach was to decide when and how far to introduce the abstract concepts of functional analysis. With some regret, we finally decided that it was idle to pretend that a general treatment of point processes could be developed without this background, mainly because the problems of existence and convergence lead inexorably to the theory of measures on metric spaces. This being so, one might as well take advantage of the metric space framework from the outset and let the point process itself be defined on a space of this character: at least this obviates the tedium of having continually to specify the dimensions of the Euclidean space, while in the context of completely separable metric spaces—and this is the greatest ix

x


generality we contemplate—intuitive spatial notions still provide a reasonable guide to basic properties. For these reasons the general results from Chapter 6 onward are couched in the language of this setting, although the examples continue to be drawn mainly from the one- or two-dimensional Euclidean spaces R1 and R2 . Two appendices collect together the main results we need from measure theory and the theory of measures on metric spaces. We hope that their inclusion will help to make the book more readily usable by applied workers who wish to understand the main ideas of the general theory without themselves becoming experts in these fields. Chapter 13, on the martingale approach, is a special case. Here the context is again the real line, but we added a third appendix that attempts to summarize the main ideas needed from martingale theory and the general theory of processes. Such special treatment seems to us warranted by the exceptional importance of these ideas in handling the problems of inference for point processes. In style, our guiding star has been the texts of Feller, however many lightyears we may be from achieving that goal. In particular, we have tried to follow his format of motivating and illustrating the general theory with a range of examples, sometimes didactical in character, but more often taken from real applications of importance. In this sense we have tried to strike a mean between the rigorous, abstract treatments of texts such as those by Matthes, Kerstan and Mecke (1974/1978/1982) and Kallenberg (1975, 1983), and practically motivated but informal treatments such as Cox and Lewis (1966) and Cox and Isham (1980). Numbering Conventions. Each chapter is divided into sections, with consecutive labelling within each of equations, statements (encompassing Definitions, Conditions, Lemmas, Propositions, Theorems), examples, and the exercises collected at the end of each section. Thus, in Section 1.2, (1.2.3) is the third equation, Statement 1.2.III is the third statement, Example 1.2(c) is the third example, and Exercise 1.2.3 is the third exercise. The exercises are varied in both content and intention and form a significant part of the text. Usually, they indicate extensions or applications (or both) of the theory and examples developed in the main text, elaborated by hints or references intended to help the reader seeking to make use of them. The symbol denotes the end of a proof. Instead of a name index, the listed references carry page number(s) where they are cited. A general outline of the notation used has been included before the main text. It remains to acknowledge our indebtedness to many persons and institutions. Any reader familiar with the development of point process theory over the last two decades will have no difficulty in appreciating our dependence on the fundamental monographs already noted by Matthes, Kerstan and Mecke in its three editions (our use of the abbreviation MKM for the 1978 English edition is as much a mark of respect as convenience) and Kallenberg in its two editions. We have been very conscious of their generous interest in our efforts from the outset and are grateful to Olav Kallenberg in particular for saving us from some major blunders. A number of other colleagues, notably


xi

David Brillinger, David Cox, Klaus Krickeberg, Robin Milne, Dietrich Stoyan, Mark Westcott, and Deng Yonglu, have also provided valuable comments and advice for which we are very grateful. Our two universities have responded generously with seemingly unending streams of requests to visit one another at various stages during more intensive periods of writing the manuscript. We also note visits to the University of California at Berkeley, to the Center for Stochastic Processes at the University of North Carolina at Chapel Hill, and to Zhongshan University at Guangzhou. For secretarial assistance we wish to thank particularly Beryl Cranston, Sue Watson, June Wilson, Ann Milligan, and Shelley Carlyle for their excellent and painstaking typing of difficult manuscript. Finally, we must acknowledge the long-enduring support of our families, and especially our wives, throughout: they are not alone in welcoming the speed and efficiency of Springer-Verlag in completing this project. Daryl Daley Canberra, Australia

David Vere-Jones Wellington, New Zealand

Contents

Preface to the Second Edition Preface to the First Edition

vii ix

Principal Notation Concordance of Statements from the First Edition 1

Early History

1

1.1 Life Tables and Renewal Theory 1.2 Counting Problems 1.3 Some More Recent Developments 2

Basic Properties of the Poisson Process

2.1 The Stationary Poisson Process 2.2 Characterizations of the Stationary Poisson Process: I. Complete Randomness 2.3 Characterizations of the Stationary Poisson Process: II. The Form of the Distribution 2.4 The General Poisson Process 3

Simple Results for Stationary Point Processes on the Line

3.1 3.2 3.3 3.4 3.5 3.6

xvii xxi

Specification of a Point Process on the Line Stationarity: Definitions Mean Density, Intensity, and Batch-Size Distribution Palm–Khinchin Equations Ergodicity and an Elementary Renewal Theorem Analogue Subadditive and Superadditive Functions xiii

1 8 13 19 19 26 31 34 41 41 44 46 53 60 64

xiv 4

Contents

Renewal Processes

4.1 4.2 4.3 4.4 4.5 4.6 5

Basic Properties Stationarity and Recurrence Times Operations and Characterizations Renewal Theorems Neighbours of the Renewal Process: Wold Processes Stieltjes-Integral Calculus and Hazard Measures

Finite Point Processes

5.1 An Elementary Example: Independently and Identically Distributed Clusters 5.2 Factorial Moments, Cumulants, and Generating Function Relations for Discrete Distributions 5.3 The General Finite Point Process: Definitions and Distributions 5.4 Moment Measures and Product Densities 5.5 Generating Functionals and Their Expansions 6

Models Constructed via Conditioning: Cox, Cluster, and Marked Point Processes

6.1 6.2 6.3 6.4 7

Conditional Intensities and Likelihoods

7.1 7.2 7.3 7.4 7.5 7.6 8

Infinite Point Families and Random Measures Cox (Doubly Stochastic Poisson) Processes Cluster Processes Marked Point Processes

Likelihoods and Janossy Densities Conditional Intensities, Likelihoods, and Compensators Conditional Intensities for Marked Point Processes Random Time Change and a Goodness-of-Fit Test Simulation and Prediction Algorithms Information Gain and Probability Forecasts

Second-Order Properties of Stationary Point Processes

8.1 8.2 8.3 8.4 8.5 8.6

Second-Moment and Covariance Measures The Bartlett Spectrum Multivariate and Marked Point Processes Spectral Representation Linear Filters and Prediction P.P.D. Measures

66 66 74 78 83 92 106 111 112 114 123 132 144

157 157 169 175 194 211 212 229 246 257 267 275 288 289 303 316 331 342 357

Contents

A1 A1.1 A1.2 A1.3 A1.4 A1.5 A1.6 A2 A2.1 A2.2 A2.3 A2.4 A2.5 A2.6 A2.3 A2.3 A3 A3.1 A3.2 A3.3 A3.4

A Review of Some Basic Concepts of Topology and Measure Theory Set Theory Topologies Finitely and Countably Additive Set Functions Measurable Functions and Integrals Product Spaces Dissecting Systems and Atomic Measures Measures on Metric Spaces

xv

368 368 369 372 374 377 382 384

Borel Sets and the Support of Measures Regular and Tight Measures Weak Convergence of Measures Compactness Criteria for Weak Convergence Metric Properties of the Space MX Boundedly Finite Measures and the Space M# X Measures on Topological Groups Fourier Transforms

384 386 390 394 398 402 407 411

Conditional Expectations, Stopping Times, and Martingales

414

Conditional Expectations Convergence Concepts Processes and Stopping Times Martingales

414 418 423 428

References with Index

432

Subject Index

452 Chapter Titles for Volume II

9 General Theory of Point Processes and Random Measures 10 Special Classes of Processes 11 Convergence Concepts and Limit Theorems 12 Ergodic Theory and Stationary Processes 13 Palm Theory 14 Evolutionary Processes and Predictability 15 Spatial Point Processes

Principal Notation

Very little of the general notation used in Appendices 1–3 is given below. Also, notation that is largely confined to one or two sections of the same chapter is mostly excluded, so that neither all the symbols used nor all the uses of the symbols shown are given. The repeated use of some symbols occurs as a result of point process theory embracing a variety of topics from the theory of stochastic processes. Where they are given, page numbers indicate the first or significant use of the notation. Generally, the particular interpretation of symbols with more than one use is clear from the context. Throughout the lists below, N denotes a point process and ξ denotes a random measure.

Spaces C Rd R = R1 R+ S Ud2α Z, Z+ X Ω ∅, ∅(·) E (Ω, E, P) X (n) X∪

complex numbers d-dimensional Euclidean space real line nonnegative numbers circle group and its representation as (0, 2π] d-dimensional cube of side length 2α and vertices (± α, . . . , ± α) integers of R, R+ state space of N or ξ; often X = Rd ; always X is c.s.m.s. (complete separable metric space) space of probability elements ω null set, null measure measurable sets in probability space basic probability space on which N and ξ are defined n-fold product space X × · · · × X = X (0) ∪ X (1) ∪ · · · xvii

158 123 129

xviii

Principal Notation

B(X )

Borel σ-field generated by open spheres of c.s.m.s. X 34 BX = B(X ), B = BR = B(R) 34, 374 (n) BX = B(X (n) ) product σ-field on product space X (n) 129 BM(X ) measurable functions of bounded support 161 BM+ (X ) measurable nonnegative functions of bounded support 161 K mark space for marked point process (MPP) 194 MX (NX ) totally finite (counting) measures on c.s.m.s. X 158, 398 boundedly finite measures on c.s.m.s. X 158, 398 M# X # NX boundedly finite counting measures on c.s.m.s. X 131 P+ p.p.d. (positive positive-definite) measures 359 S infinitely differentiable functions of rapid decay 357 U complex-valued Borel measurable functions on X of modulus ≤ 1 144 U ⊗V product topology on product space X × Y of topological spaces (X , U), (Y, V) 378 V = V(X ) [0, 1]-valued measurable functions h(x) with 1 − h(x) of bounded support in X 149, 152

General Unless otherwise specified, A ∈ BX , k and n ∈ Z+ , t and x ∈ R, h ∈ V(X ), and z ∈ C. ˜

˘ #

µ a.e. µ, µ-a.e. a.s., P-a.s. A(n) A Bu (Tu ) ck , c[k]

ν˜, F = Fourier–Stieltjes transforms of measure ν or d.f. F φ˜ = Fourier transform of Lebesgue integrable function φ for counting measures reduced (ordinary or factorial) (moment or cumulant) measure extension of concept from totally finite to boundedly finite measure space variation norm of measure µ almost everywhere with respect to measure µ almost sure, P-almost surely n-fold product set A × · · · × A family of sets generating B; semiring of bounded Borel sets generating BX backward (forward) recurrence time at u kth cumulant, kth factorial cumulant, of distribution {pn }

411–412 357 160 158 374 376 376 130 31, 368 58, 76 116

c(x) = c(y, y + x) covariance density of stationary mean square continuous process on Rd

160, 358

Principal Notation

factorial cumulant measure and density C[k] (·), c[k] (·) reduced covariance measure of stationary N or ξ C˘2 (·), c˘(·) c˘(·) reduced covariance density of stationary N or ξ δ(·) Dirac delta function δx (A) Dirac measure, = A δ(u − x) du = IA (x) ∆F (x) = F (x) − F (x−) jump at function F dx in right-continuous eλ (x) = ( 12 λ)d exp − λ i=1 |xi | two-sided exponential density in Rd F renewal process lifetime d.f. F n∗ n-fold convolution power of measure or d.f. F F (· ; ·) finite-dimensional (fidi) distribution F history Φ(·) characteristic functional G[h] probability generating functional (p.g.fl.) of N , G[h | x] member of measurable family of p.g.fl.s Gc [·], Gm [· | x] p.g.fl.s of cluster centre and cluster member processes Nc and Nm (· | x) G, GI expected information gain (per interval) of stationary N on R Γ(·), γ(·) Bartlett spectrum, its density when it exists H(P; µ) generalized entropy H, H∗ internal history of ξ on R+ , R IA (x) = δx (A) indicator function of element x in set A modified Bessel function of order n In (x) Jn (A1 × · · · × An ) Janossy measure jn (x1 , . . . , xn ) Janossy density local Janossy measure Jn (· | A) K compact set Kn (·), kn (·) Khinchin measure and density (·) Lebesgue measure in B(Rd ), Haar measure on σ-group Lu = Bu + Tu current lifetime of point process on R L[f ] (f ∈ BM+ (X )) Laplace functional of ξ Lξ [1 − h] p.g.fl. of Cox process directed by ξ L2 (ξ 0 ), L2 (Γ) Hilbert spaces of square integrable r.v.s ξ 0 , and of functions square integrable w.r.t. measure Γ LA (x1 , . . . , xn ), = jN (x1 , . . . , xN | A) likelihood, local Janossy density, N ≡ N (A) λ rate of N , especially intensity of stationary N λ∗ (t) conditional intensity function kth (factorial) moment of distribution {pn } mk (m[k] )

xix 147 292 160, 292 382 107 359 67 55 158–161 236, 240 15 15, 144 166 178 280, 285 304 277, 283 236 72 124 125 137 371 146 31 408–409 58, 76 161 170 332 22, 212 46 231 115

xx

Principal Notation

˘2 m ˘ 2, M

reduced second-order moment density, measure, of stationary N mg mean density of ground process Ng of MPP N N (A) number of points in A N (a, b] number of points in half-open interval (a, b], = N ((a, b]) N (t) = N (0, t] = N ((0, t]) Nc cluster centre process N (· | x) cluster member or component process {(pn , Πn )} elements of probability measure for finite point process P (z) probability generating function (p.g.f.) of distribution {pn } P (x, A) Markov transition kernel P0 (A) avoidance function Pjk set of j-partitions of {1, . . . , k} P probability measure of stationary N on R, probability measure of N or ξ on c.s.m.s. X {πk } batch-size distribution q(x) = f (x)/[1 − F (x)] hazard function for lifetime d.f. F Q(z) = − log P (z) Q(·), Q(t) hazard measure, integrated hazard function (IHF) ρ(x, y) metric for x, y in metric space {Sn } random walk, sequence of partial sums S(x) = 1 − F (x) survivor function of d.f. F Sr (x) sphere of radius r, centre x, in metric space X d t(x) = i=1 (1 − |xi |)+ triangular density in Rd Tu forward recurrence time at u T = {S1 (T ), . . . , Sj (T )} a j-partition of k T = {Tn } = {{Ani }} dissecting system of nested partitions U (A) = E[N (A)] renewal measure U (x) = U ([0, x]), expectation function, renewal function (U (x) = 1 + U0 (x)) V (A) = var N (A), variance function V (x) = V ((0, x]) variance function for stationary N or ξ on R {Xn } components of random walk {Sn }, intervals of Wold process

289 198, 323 42 19 42 42 176 176 123 10, 115 92 31, 135 121 53 158 28, 51 2, 106 27 109 370 66 2, 109 35, 371 359 58, 75 121 382 67 61 67 295 80, 301 66 92

Concordance of Statements from the First Edition

The table below lists the identifying number of formal statements of the first edition (1988) of this book and their identification in this volume. 1988 edition

this volume

1988 edition

this volume

2.2.I–III 2.3.III

2.2.I–III 2.3.I

2.4.I–II 2.4.V–VIII

2.4.I–II 2.4.III–VI

8.1.II 8.2.I 8.2.II 8.3.I–III

6.1.II, IV 6.3.I 6.3.II, (6.3.6) 6.3.III–V

3.2.I–II 3.3.I–IX 3.4.I–II 3.5.I–III

3.2.I–II 3.3.I–IX 3.4.I–II 3.5.I–III

8.5.I–III

6.2.II

3.6.I–V

3.6.I–V

4.2.I–II

4.2.I–II

11.1.I–V 11.2.I–II 11.3.I–VIII 11.4.I–IV 11.4.V–VI

8.6.I–V 8.2.I–II 8.4.I–VIII 8.5.I–IV 8.5.VI–VII

4.3.I–III 4.4.I–VI 4.5.I–VI 4.6.I–V

4.3.I–III 4.4.I–VI 4.5.I–VI 4.6.I–V

13.1.I–III 13.1.IV–VI 13.1.VII 13.4.III

7.1.I–III 7.2.I–III 7.1.IV 7.6.I

5.2.I–VII 5.3.I–III 5.4.I–III 5.4.IV–VI 5.5.I

5.2.I–VII 5.3.I–III 5.4.I–III 5.4.V–VII 5.5.I

A1.1.I–5.IV A2.1.I–III A2.1.IV A2.1.V–VI A2.2.I–7.III A3.1.I–4.IX

A1.1.I–5.IV A2.1.I–III A1.6.I A2.1.IV–V A2.2.I–7.III A3.1.I–4.IX

7.1.XII–XIII

6.4.I(a)–(b) xxi

CHAPTER 1

Early History

The ancient origins of the modern theory of point processes are not easy to trace, nor is it our aim to give here an account with claims to being definitive. But any retrospective survey of a subject must inevitably give some focus on those past activities that can be seen to embody concepts in common with the modern theory. Accordingly, this first chapter is a historical indulgence but with the added benefit of describing certain fundamental concepts informally and in a heuristic fashion prior to possibly obscuring them with a plethora of mathematical jargon and techniques. These essentially simple ideas appear to have emerged from four distinguishable strands of enquiry—although our division of material may sometimes be a little arbitrary. These are (i) (ii) (iii) (iv)

life tables and the theory of self-renewing aggregates; counting problems; particle physics and population processes; and communication engineering.

The first two of these strands could have been discerned in centuries past and are discussed in the first two sections. The remaining two essentially belong to the twentieth century, and our comments are briefer in the remaining section.

1.1. Life Tables and Renewal Theory Of all the threads that are woven into the modern theory of point processes, the one with the longest history is that associated with intervals between events. This includes, in particular, renewal theory, which could be defined in a narrow sense as the study of the sequence of intervals between successive replacements of a component that is liable to failure and is replaced by a new 1

2

1. Early History

component every time a failure occurs. As such, it is a subject that developed during the 1930s and reached a definitive stage with the work of Feller, Smith, and others in the period following World War II. But its roots extend back much further than this, through the study of ‘self-renewing aggregates’ to problems of statistical demography, insurance, and mortality tables—in short, to one of the founding impulses of probability theory itself. It is not easy to point with confidence to any intermediate stage in this chronicle that recommends itself as the natural starting point either of renewal theory or of point process theory more generally. Accordingly, we start from the beginning, with a brief discussion of life tables themselves. The connection with point processes may seem distant at first sight, but in fact the theory of life tables provides not only the source of much current terminology but also the setting for a range of problems concerning the evolution of populations in time and space, which, in their full complexity, are only now coming within the scope of current mathematical techniques. In its basic form, a life table consists of a list of the number of individuals, usually from an initial group of 1000 individuals so that the numbers are effectively proportions, who survive to a given age in a given population. The most important parameters are the number x surviving to age x, the number dx dying between the ages x and x + 1 (dx = x − x+1 ), and the number qx of those surviving to age x who die before reaching age x + 1 (qx = dx /x ). In practice, the tables are given for discrete ages, with the unit of time usually taken as 1 year. For our purposes, it is more appropriate to replace the discrete time parameter by a continuous one and to replace numbers by probabilities for a single individual. Corresponding to x we have then the survivor function S(x) = Pr{lifetime > x}. To dx corresponds f (x), the density of the lifetime distribution function, where f (x) dx = Pr{lifetime terminates between x and x + dx}, while to qx corresponds q(x), the hazard function, where q(x) dx = Pr{lifetime terminates between x and x + dx | it does not terminate before x.} Denoting the lifetime distribution function itself by F (x), we have the following important relations between the functions above: x ∞ S(x) = 1 − F (x) = f (y) dy = exp − q(y) dy , (1.1.1) x

0

dF dS f (x) = = , dx dx d d f (x) = [log S(x)] = − {log[1 − F (x)]}. q(x) = S(x) dx dx

(1.1.2) (1.1.3)

1.1.

Life Tables and Renewal Theory

3

The first life table appeared, in a rather crude form, in John Graunt’s (1662) Observations on the London Bills of Mortality. This work is a landmark in the early history of statistics, much as the famous correspondence between Pascal and Fermat, which took place in 1654 but was not published until 1679, is a landmark in the early history of formal probability. The coincidence in dates lends weight to the thesis (see e.g. Maistrov, 1967) that mathematical scholars studied games of chance not only for their own interest but for the opportunity they gave for clarifying the basic notions of chance, frequency, and expectation, already actively in use in mortality, insurance, and population movement contexts. An improved life table was constructed in 1693 by the astronomer Halley, using data from the smaller city of Breslau, which was not subject to the same problems of disease, immigration, and incomplete records with which Graunt struggled in the London data. Graunt’s table was also discussed by Huyghens (1629–1695), to whom the notion of expected length of life is due. A. de Moivre (1667–1754) suggested that for human populations the function S(x) could be taken to decrease with equal yearly decrements between the ages 22 and 86. This corresponds to a uniform density over this period and a hazard function that increases to infinity as x approaches 86. The analysis leading to (1.1.1) and (1.1.2), with further elaborations to take into account different sources of mortality, would appear to be due to Laplace (1747–1829). It is interesting that in A Philosophical Essay on Probabilities (1814), where the classical definition of probability based on equiprobable events is laid down, Laplace gave a discussion of mortality tables in terms of probabilities of a totally different kind. Euler (1707–1783) also studied a variety of problems of statistical demography. From the mathematical point of view, the paradigm distribution function for lifetimes is the exponential function, which has a constant hazard independent of age: for x > 0, we have f (x) = λe−λx ,

q(x) = λ,

S(x) = e−λx ,

F (x) = 1 − e−λx .

(1.1.4)

The usefulness of this distribution, particularly as an approximation for purposes of interpolation, was stressed by Gompertz (1779–1865), who also suggested, as a closer approximation, the distribution function corresponding to a power-law hazard of the form q(x) = Aeαx

(A > 0, α > 0, x > 0).

(1.1.5)

With the addition of a further constant [i.e. q(x) = B + Aeαx ], this is known in demography as the Gompertz–Makeham law and is possibly still the most widely used function for interpolating or graduating a life table. Other forms commonly used for modelling the lifetime distribution in different contexts are the Weibull, gamma, and log normal distributions, corresponding, respectively, to the formulae q(x) = βλxβ−1

with S(x) = exp(−λxβ )

(λ > 0, β > 0),

(1.1.6)

4

1. Early History

f (x) = λαxα−1 e−λx Γ(α), √ 2 f (x) = (σx 2π )−1 e−[(log x−µ)/σ] /2 .

(1.1.7) (1.1.8)

The Weibull distribution was introduced by Weibull (1939a, b) as a model for brittle fracture. Both this and the preceding distribution have an interpretation in terms of extreme value theory (see e.g. Exercise 1.1.2), but it should be emphasized that as a general rule the same distribution may arise from several models (see Exercise 1.1.3). The gamma distribution has a long history and arises in many different contexts. When α = 12 k and λ = 12 , it is nothing other than the chi-squared distribution with k degrees of freedom, with well-known applications in mathematical statistics. When α = 1, it reduces to the exponential distribution, and when α = 32 , it reduces to the Maxwell distribution for the distribution of energies of molecules in a perfect gas. The most important special cases in the context of life tables arise when α is a positive integer, say α = k. It then has an interpretation as the sum of k independent random variables, each having an exponential distribution. Although commonly known as the Erlang distribution, after the Danish engineer and mathematician who introduced it as a model for telephone service and intercall distributions in the 1920s, this special form and its derivation were known much earlier. One of the earliest derivations, if not the first, is due to the English mathematician R.C. Ellis (1817–1859) in a remarkable paper in 1844 that could well be hailed as one of the early landmarks in stochastic process theory, although in fact it is rarely quoted. In addition to establishing the above-mentioned result as a special case, Ellis studied a general renewal process and in that context established the asymptotic normality of the sum of a number of independent nonnegative random variables. It is particularly remarkable in that he used Fourier methods; in other words, essentially the modern characteristic function proof (with a few lacunae from a modern standpoint) of the central limit theorem. An equally interesting aspect of Ellis’ paper is the problem that inspired the study. This takes us back a century and a half to an even less familiar statistician in the guise of Sir Isaac Newton (1642–1728). For much of his later life, Newton’s spare time was devoted to theological problems, one of which was to reconcile the ancient Greek and Hebrew chronologies. In both chronologies, periods of unknown length are spanned by a list of successive rulers. Newton proposed to estimate such periods, and hence to relate the two chronologies, by supposing each ruler to reign for a standard period of 22 years. This figure was obtained by a judicious comparison of averages from a miscellany of historical data for which more or less reliable lengths of reigns were known. It is a statistical inference in the same sense as many of Graunt’s inferences from the London Bills of Mortality: a plausible value based on the best or only evidence available and supported by as many cross-checks as can be devised. How far it was explicitly present in Newton’s mind that he was dealing with a statistical problem and whether he made any attempts

1.1.


5

to assess the likely errors of his results himself are questions we have not been able to answer with any certainty. In an informal summary of his work, Newton (1728) wrote: “I do not pretend to be exact to a year: there may be errors of five or ten years, and sometimes twenty, and not much above.” However, it appears unlikely that these figures were obtained by any theory of compounding of errors. It is tempting to conjecture that he may have discussed the problems with such friends and Fellows of the Royal Society as Halley, whose paper to the Royal Society would have been presented while Newton was president, and de Moivre, who dedicated the first edition of The Doctrine of Chances to Newton, but if records of such discussions exist, we have not found them. Up until the middle of the nineteenth century, as will be clear even from the brief review presented above, mathematical problems deriving from life tables not only occupied a major place in the subject matter of probability and statistics but also attracted the attention of many leading mathematicians of the time. From the middle of the nineteenth century onward, however, actuarial mathematics (together, it may be added, with many other probabilistic notions), while important in providing employment for mathematicians, became somewhat disreputable mathematically, a situation from which it has not fully recovered. (How many elementary textbooks in statistics, for example, even mention life tables, let alone such useful descriptive tools as the hazard function?) The result was that when, as was inevitably the case, new applications arose that made use of the same basic concepts, the links with earlier work were lost or only partially recognized. Moreover, the new developments themselves often took place independently or with only a partial realization of the extent of common material. In the twentieth century, at least three such areas of application may be distinguished. The first, historically, was queueing theory, more specifically the theory of telephone trunking problems. Erlang’s (1909) first paper on this subject contains a derivation of the Poisson distribution for the number of calls in a fixed time interval. It is evident from his comments that even before that time the possibility of using probabilistic methods in that context was being considered by engineers in several countries. The work here appears to be quite independent of earlier contributions. In later work, the analysis was extended to cover queueing systems with more general input and service distributions. Mathematical interest in actuarial problems as such re-emerged in the 1910s and 1920s in connection with the differential and integral equations of population growth. Here at least there is a bridge between the classical theory of life tables on the one hand and the modern treatments of renewal processes on the other. It is provided by the theory of ‘self-renewing aggregates’ [to borrow a phrase from the review by Lotka (1939), which provides a useful survey of early work in this field], a term that refers to a population (portfolio in the insurance context) of individuals subject to death but also able to regenerate themselves so that a stable population can be achieved.

6

1. Early History

As a typical illustration, consider the evolution of a human population for which it is assumed that each female of age x has a probability φ(x) dt of giving birth to a daughter in a time interval of length dt, independently of the behaviour of other females in the population and also of any previous children she may have had. Let S(x) denote the survivor function for the (female) life distribution and n(t) the expected female birth rate at time t. Then n(t) satisfies the integral equation

t

n(t − x)S(x)φ(x) dx,

n(t) = 0

which represents a breakdown of the total female birth rate by age of parent. If the population is started at time zero with an initial age distribution having density r(x), the equation can be rewritten in the form

t

n(t − x)S(x)φ(x) dx,

n(t) = n0 (t) + 0

where n0 (t) =

∞

S(t + x) φ(t + x) dx S(x)

r(x) 0

is the contribution to the birth rate at time t from the initial population. In this form, the analogy with the integral equation of renewal theory is clear. Indeed, the latter equation corresponds to the special case where at death each individual is replaced by another of age zero and no other ‘births’ are possible. The population size then remains constant, and it is enough to consider a population with just one member. In place of n(t), we then have the renewal density m(t), with m(t) dt representing the probability that a replacement will be required in the small time interval (t, t + dt); also, φ(x) becomes the hazard function h(x) for the life distribution, and the combination S(x)h(x) can be replaced by the probability density function f (x) as in (1.1.3). Thus, we obtain the renewal equation in the form

t

m(t − u)f (u) du.

m(t) = n0 (t) + 0

If, finally, the process is started with a new component in place at time 0, then n0 (t) = f (t) and we have the standard form

t

ms (t − u)f (u) du.

ms (t) = f (t) + 0

The third field to mention is reliability theory. A few problems in this field, including Weibull’s discussion of brittle fracture, appeared before World War II, but its systematic development relates to the post-war period and the rapid growth of the electronics industry. Typical problems are the calculation

1.1.


7

of lifetime distributions of systems of elements connected in series (‘weakest link’ model) or in parallel. Weibull’s analysis is an example of the first type of model, which typically leads to an extreme-value distribution with a long right tail. An early example of a parallel model is Daniels’ (1945) treatment of the failure of fibre bundles; the distributions in this case have an asymptotically normal character. In between and extending these two primary cases lie an infinite variety of further failure models, in all of which the concepts and terminology invented to cover the life table problem play a central role. In retrospect, it is easy to see that the three fields referred to are closely interconnected. Together, they provide one of the main areas of application and development of point process theory. Of course, they do not represent the only fields where life table methods have been applied with success. An early paper by Watanabe (1933) gives a life table analysis of the times between major earthquake disasters, a technique that has been resurrected by several more recent writers under the name of theory of durability. An important recent field of application has been the study of trains of nerve impulses in neurophysiology. In fact, the tools are available and relevant for any phenomenon in which the events occur along a time axis and the intervals between the time points are important and meaningful quantities.

Exercises and Complements to Section 1.1 1.1.1 A nonnegative random variable (r.v.) X with distribution function (d.f.) F has an increasing failure rate (abbreviated to IFR) if the conditional d.f.s Fx (u) = Pr{X ≤ x + u | X > x} =

F (x + u) − F (x) 1 − F (x)

(u, x ≥ 0)

are increasing functions of x for every fixed u in 0 x) decreases with increasing x, and it is new better than used in expectation (NBUE) if E(X − x | X > x) ≤ EX (all x > 0). Show that IFR implies DMRL, DMRL implies NBUE, and NBUE implies that var X ≤ (EX)2 [see Stoyan (1983, Section 1.6)]. 1.1.2 Let X1 , X2 , . . . be a sequence of independent identically distributed r.v.s with d.f. F (·). Then, for any fixed nonnegative integer n,

Pr

max Xj ≤ u

1≤j≤n

n

= (F (u)) .

Replacing n by a Poisson-distributed r.v. N with mean µ yields G(u) ≡ Pr

max Xj ≤ u

1≤j≤N

≡ e−µ

∞

µk (k!)−1 (F (u)) = e−µ(1−F (u)) . k

k=0

When F (u) = 1 − e−λu , G is the Gumbel d.f., while when F (u) = 1 − λu−α , G is the Weibull d.f. [In the forms indicated, these extreme-value distributions include location and/or scale parameters; see e.g. Johnson and Kotz (1970, p. 272).]

8

1. Early History

= 1 − e−λu . Show 1.1.3 Let X1 , X2 , . . . be as in the previous exercise with F (u) n that Y ≡ max(X1 , . . . , Xn ) has the same distribution as Xj /j. j=1 [Hint: Regard X1 , . . . , Xn as lifetimes in a linear death process with death rate λ, so that y is the time to extinction of the process. Exercise 2.1.2 gives more general properties.] 1.1.4 Suppose that the lifetimes of rulers are independent r.v.s with common d.f. F and that conditional on reaching age 21 years, a ruler has a son (with lifetime d.f.s F ) every 2 years for up to six sons, with the eldest surviving son succeeding him. Conditional on there being a succession, what is the d.f. of the age at succession and the expected time that successor reigns (assuming a reign terminated by death from natural causes)? What types of error would be involved in matching chronologies from a knowledge of the orders of two sets of rulers (see the reference to Newton’s work in the text)? How would such chronologies be matched in the light of developments in statistical techniques subsequent to Newton? 1.1.5 Investigate the integral equation for the stationary age distribution in a supercritical age-dependent branching process. Using a suitable metric, evaluate the difference between this stationary age distribution and the backward recurrence time distribution of a stationary renewal process with the same lifetime distribution as a function of the mean of the offspring distribution. Note that Euler worked on the age distribution in exponentially growing populations.

1.2. Counting Problems The other basic approach to point process phenomena, and the only systematic approach yet available in spaces of higher dimension, is to count the numbers of events in intervals or regions of various types. In this approach, the machinery of discrete distributions plays a central role. Since in probability theory discrete problems are usually easier to handle than continuous problems, it might be thought that the development of general models for a discrete distribution would precede those for a continuous distribution, but in fact the reverse seems to be the case. Although particular examples, such as the Bernoulli distribution and the negative binomial distribution, occurred at a very early stage in the discussion of games of chance, there seems to be no discussion of discrete distributions as such until well into the nineteenth century. We may take as a starting point Poisson’s (1837) text, which included a derivation of the Poisson distribution by passage to the limit from the binomial (the claim that he was anticipated in this by de Moivre is a little exaggerated in our view: it is true that de Moivre appends a limit result to the discussion of a certain card problem, but it can hardly be said that the resulting formula was considered by de Moivre as a distribution, which may be the key point). Even Poisson’s result does not seem to have been widely noted at the time, and it is not derived in a counting process context. The first discussions of counting problems known to us are by Seidel (1876) and Abbé (1879),

1.2.

Counting Problems

9

who treated the occurrence of thunderstorms and the number of blood cells in haemocytometer squares, respectively, and both apparently independently of Poisson’s work. Indeed, Poisson’s discovery of the distribution seems to have been lost sight of until attention was drawn to it in Von Bortkiewicz’s (1898) monograph Vas Gesetz der kleinen Zahlen, which includes a systematic account of phenomena that fit the Poisson distribution, including, of course, the famous example of the number of deaths from horse kicks in the Prussian army. Lyon and Thoma (1881), on Abbé’s data, and Student (1907) gave further discussions of the blood cell problem, the latter paper being famous as one of the earliest applications of the chi-square goodness-of-fit test. Shortly afterward, the Poisson process arose simultaneously in two very important contexts. Erlang (1909) derived the Poisson distribution for the number of incoming calls to a telephone trunking system by supposing the numbers in disjoint intervals to be independent and considering the limit behaviour when the interval of observation is divided into an increasing number of equally sized subintervals. This effectively reproduces the Poisson distribution as the limit of the binomial, but Erlang was not aware of Poisson’s work at the time, although he corrected the omission in later papers. Then, in 1910, Bateman, brought in as mathematical consultant by Rutherford and Geiger in connection with their classical experiment on the counting of α particles, obtained the Poisson probabilities as solutions to the family of differential equations pn (t) = −λpn (t) + pn−1 (t) p0 (t)

(n ≥ 1),

= −λp0 (t).

[Concerning the relation p0 (t) = e−λt , Bateman (1910) commented that it “has been known for some time (Whitworth’s Choice and Chance, 4th Ed., Proposition LI),” while Haight (1967) mentioned the result as a theorem of Boltzmann (1868) and quoted the reference to Whitworth, who does not indicate the sources of his results; in a Gresham lecture reproduced in Whitworth (1897, p. xxxiii), he wrote of Proposition LI as “a general theorem which I published in 1886, which met with rather rough treatment at the hands of a reviewer in The Academy.” Whitworth’s (1867) book evolved through five editions. It is easy to envisage repeated independent discovery of his Proposition LI.] These equations represent a formulation in terms of a pure birth process and the first step in the rapid development of the theory of birth and death processes during the next two decades, with notable early papers by McKendrick (1914, 1926) and Yule (1924). This work preceded the general formulation of birth and death processes as Markov processes (themselves first studied by Markov more than a decade earlier) in the 1930s and is not of immediate concern, despite the close connection with point process problems. A similar remark can be made about branching processes, studied first by Bienaymé (see Heyde and Seneta, 1977) and of course by Galton and Watson

10

1. Early History

(1874). There are close links with point processes, particularly in the general case, but the early studies used special techniques that again lie a little outside the scope of our present discussion, and it was only from the 1940s onward that the links became important. Closer in line with our immediate interests is the work on alternatives to the Poisson distribution. In many problems in ecology and elsewhere, it is found that the observed distribution of counts frequently shows a higher dispersion (i.e. a higher variance for a given value of the mean) than can be accounted for satisfactorily by the Poisson distribution, for which the variance/mean ratio is identically unity. The earliest and perhaps still the most widely used alternative is the negative binomial distribution, which figures in early papers by Student (1907), McKendrick (1914), and others. A particularly important paper for the sequel was the study by Greenwood and Yule (1920) of accident statistics, which provided an important model for the negative binomial, and in so doing sparked a controversy, still not entirely resolved, concerning the identifiability of the model describing accident occurrence. Since the accident process is a kind of point process in time, and since shades of the same controversy will appear in our own models, we briefly paraphrase their derivation. Before doing so, however, it is convenient to summarize some of the machinery for handling discrete distributions. The principal tool is the probability generating function (p.g.f.) defined for nonnegative integer-valued random variables X by the equation P (z) =

∞

pn z n ,

0

where pn = Pr{X = n}. It is worth mentioning that although generating functions have been used in connection with difference equations at least since the time of Laplace, their application to this kind of problem in the 1920s and 1930s was hailed as something of a technological breakthrough. In Chapter 5, relations between the p.g.f., factorial moments, and cumulants are discussed. For the present, we content ourselves with the observation that the negative binomial distribution can be characterized by the form of its p.g.f., α µ P (z) = (α > 0, µ > 0), (1.2.1) 1+µ−z corresponding to values of the probabilities themselves, α n µ 1 (α − 1 + n)! . pn = (α − 1)! n! 1 + µ 1+µ 1 Note

that there is a lack of agreement on terminology. Other authors, for example Johnson and Kotz (1969), would label this as a compound Poisson and would call the distribution we treat below under that name a generalized Poisson. The terminology we use is perhaps more common in texts on probability and stochastic processes; the alternative terminology is more common in the statistical literature.

1.2.

Counting Problems

11

Greenwood and Yule derived this distribution as an example of what we call a mixed Poisson1 distribution; that is, it can be obtained from a Poisson distribution pn = e−λ λn /n! by treating the parameter λ as a random variable. If, in particular, λ is assumed to have the gamma distribution −1 −µλ e dλ, dF (λ) = µα λα−1 Γ(α) then the resultant discrete distribution has p.g.f. α ∞ µ P (z) = , eλ(z−1) dF (λ) = 1+µ−z 0 eλ(z−1) being the p.g.f. of the Poisson distribution with parameter λ. It is not difficult to verify that the mean and variance of this negative binomial distribution equal α/µ and (α/µ)(1 + µ−1 ), so that the variance/mean ratio of the distribution equals 1 + µ−1 , exceeding by µ−1 the corresponding ratio for a Poisson distribution. Greenwood and Yule interpreted the variable parameter λ of the underlying Poisson distribution as a measure of individual ‘accident proneness,’ which was then averaged over all individuals in the population. The difficulty for the sequel is that, as was soon recognized, many other models also give rise to the negative binomial, and these may have quite contradictory interpretations in regard to accidents. L¨ uders (1934) showed that the same distribution could be derived as an example of a compound Poisson distribution, meaning a random sum of independent random variables in which the number of terms in the sum has a Poisson distribution. If each term is itself discrete and has a logarithmic distribution with p.g.f. P (z) =

log(1 + µ − z) , log µ

(1.2.2)

and if the number of terms has a Poisson distribution with parameter α, then the resultant distribution has the identical p.g.f. (1.2.1) for the negative binomial (see Exercise 1.2.1). The interpretation here would be that all individuals are identical but subject to accidents in batches. Even before this, Eggenberger and P´ olya (1923) and P´ olya (1931) had introduced a whole family of distributions, for which they coined the term ‘contagious distributions’ to describe situations where the occurrence of a number of events enhances the probability of the occurrence of a further event, and had shown that the negative binomial distribution could be obtained in this way. If the mixed and compound models can be distinguished in principle by examining the joint distributions of the number of accidents in nonoverlapping intervals of a person’s life, Cane (1974, 1977) has shown that there is no way in which the mixed Poisson and P´ olya models can be distinguished from observations on individual case histories, for they lead to identical conditional distributions (see Exercise 1.2.2).

12

1. Early History

Another important contribution in this field is the work of Neyman (1939), who introduced a further family of discrete distributions, derived from consideration of a cluster model. Specifically, Neyman was concerned with distributions of beetle larvae in space, supposing these to have crawled some small distance from their initial locations in clusters of eggs. Further analysis of this problem resulted in a series of papers, written by Neyman in collaboration with E.L. Scott and other writers, which treated many different statistical questions relating to clustering processes in ecology, astronomy, and other subjects (see e.g. Neyman and Scott, 1958). Many of these questions can be treated most conveniently by the use of generating functionals and moment densities, a theory that had been developing simultaneously as a tool for describing the evolution of particle showers and related problems in theoretical physics. The beginnings of such a general theory appear in the work of the French physicist Yvon (1935), but the main developments relate to the post-war period, and we therefore defer a further discussion to the following section.

Exercises and Complements to Section 1.2 1.2.1 Poisson mixture of logarithmic distributions is negative binomial. Verify that if X1 , X2 , . . . are independent r.v.s with the logarithmic distribution whose p.g.f. is in (1.2.2), and if N , independent of X1 , X2 , . . . , is a Poisson r.v. with mean α, then X1 + · · · + XN has the negative binomial distribution in (1.2.1). 1.2.2 Nonidentifiability in a model for accident proneness. Suppose that an individual has n accidents in the time interval (0, T ) at t1 < t2 < · · · < tn . Evaluate the likelihood function for these n times for the two models: (i) accidents occur at the epochs of a Poisson process at rate λ, where λ is fixed for each individual but may vary between individuals; (ii) conditional on having experienced j accidents in (0, t), an individual has probability (k + j)µ dt/(1 + µt) of an accident in (t, t + dt), independent of the occurrence times of the j accidents in (0, t); each individual has probability kµ dt of an accident in (0, dt). Show that the probabilities of n events in (0, T ) are Poisson and negative binomial, respectively, and deduce that the conditional likelihood, given n, is the same for (i) and (ii). See Cane (1974) for discussion. 1.2.3 The negative binomial distribution can also arise as the limit of the Pólya– Eggenberger distribution defined for integers n and α, β > 0 by

pk =

n Γ(α + k)Γ(β + n − k)Γ(α + β) = Γ(α + β + n)Γ(α)Γ(β) k

−α Γ(α + β)n!Γ(β + n − k) . k Γ(β)(n − k)!Γ(β + n + α)

When β and n → ∞ with n/β → µ, a constant, and α fixed, show that {pk } has the p.g.f. in (1.2.1). [For further properties, see Johnson and Kotz (1969) and the papers cited in the text.]

1.3.

Some More Recent Developments

13

1.2.4 Neyman’s Type A distribution (e.g. Johnson and Kotz, 1969) has a p.g.f. of the form exp µ

αi exp[−λi (1 − z)] − 1

,

i

α = 1, λi > 0, and µ > 0, and arises as a cluster model. where αi ≥ 0, i i Give such a cluster model interpretation for the simplest case αi = 1 for i = 1, αi = 0 otherwise, and general λ ≡ λ1 and µ. 1.2.5 Suppose that a (large) population evolves according to a one-type Galton– Watson branching process in which the distribution of the number of children has p.g.f. P (z). Choose an individual at random in a particular generation. Show that the distribution of the number of sibs (sisters, say) of this randomly chosen individual has p.g.f. P (z)/P (1) and that this is the same as for the number of aunts, or great-aunts, of this individual. [Hint: Attempting to estimate the offspring distribution by using the observed family size distribution, when based on sampling via the children, leads to the distribution with p.g.f. zP (z)/P (1) and is an example of length-biased sampling that underlies the waiting-time paradox referred to in Sections 3.2 and 3.4. The p.g.f. for the number of great-aunts is used in Chapter 11.]

1.3. Some More Recent Developments The period during and following World War II saw an explosive growth in theory and applications of stochastic processes. On the one hand, many new applications were introduced and existing fields of application were extended and deepened; on the other hand, there was also an attempt to unify the subject by defining more clearly the basic theoretical concepts. The monographs by Feller (1950) and Bartlett (1955) (preceded by mimeographed lecture notes from 1947) played an important role in stressing common techniques and exploring the mathematical similarities in different applications; both remain remarkably succinct and wide-ranging surveys. From such a busy scene it is difficult to pick out clearly marked lines of development, and any selection of topics is bound to be influenced by personal preferences. Bearing such reservations in mind, we can attempt to follow through some of the more important themes into the post-war period. On the queueing theory side, a paper of fundamental importance is Connie Palm’s (1943) study of intensity fluctuations in traffic theory, a title that embraces topics ranging from the foundation of a general theory of the input stream to the detailed analysis of particular telephone trunking systems. Three of his themes, in particular, were important for the future of point processes. The first is the systematic description of properties of a renewal process, as a first generalization of the Poisson process as input to a service system. The notion of a regeneration point, a time instant at which the system reverts to a specified state with the property that the future evolution is independent of how the state was reached, has proved exceptionally fruitful in many different applications. In Palm’s terminology, the Poisson process

14

1. Early History

is characterized by the property that every instant is a regeneration point, whereas for a general renewal process only those instants at which a new interval is started form regeneration points. Hence, he called a Poisson process a process without aftereffects and a renewal process a process with limited aftereffects. Another important idea was his realization that two types of distribution function are important in describing a stationary point process—the distribution of the time to the next event from a fixed but arbitrary origin and the distribution of the time to the next event from an arbitrary event of the process. The relations between the two sets of distributions are given by a set of equations now commonly called the Palm–Khinchin equations, Palm himself having exhibited only the simplest special case. A third important contribution was his (incomplete) proof of the first limit theorem for point processes: namely, that superposition of a large number of independent sparse renewal processes leads to a Poisson process in the limit. Finally, it may be worth mentioning that it was in Palm’s paper that the term ‘point processes’ (Punktprozesse) was first used as such—at least to the best of our knowledge. All these ideas have led to important further development. H. Wold (1948, 1949), also a Swedish mathematician, was one of the first to take up Palm’s work, studying processes with Markov-dependent intervals that, he suggested, would form the next most complex alternative to the renewal model. Bartlett (1954) reviewed some of this early work. Of the reworkings of Palm’s theory, however, the most influential was the monograph by Khinchin (1955), which provided a more complete and rigorous account of Palm’s work, notably extended it in several directions, and had the very important effect of bringing the subject to the attention of pure mathematicians. Thus, Khinchin’s book became the inspiration of much theoretical work, particularly in the Soviet Union and Eastern Europe. Ryll-Nardzewski’s (1961) paper set out fundamental properties of point processes and provided a new and more general approach to Palm probabilities. Starting in the early 1960s, Matthes and co-workers developed many aspects concerned with infinitely divisible point processes and related questions. The book by Kerstan, Matthes and Mecke (1974) represented the culmination of the first decade of such work; extensive revisions and new material were incorporated into the later editions in English (1978) (referred to as MKM in this book) and in Russian (1982). In applications, these ideas have been useful not only in queueing theory [for continuing development in this field, see the monographs of Franken et al. (1981) and Brémaud (1981)] but also in the study of level-crossing problems. Here the pioneering work was due to Rice (1944) and McFadden (1956, 1958). More rigorous treatments, using some of the Palm–Khinchin theory, were given by Leadbetter and other writers [see e.g. Leadbetter (1972) and the monographs by Cramér and Leadbetter (1967) and Leadbetter, Lindgren and Rootzen (1983)]. On a personal note in respect of much of this work, it is appropriate to remark that Belyaev, Franken, Grigelionis, K¨ onig, Matthes, and one of us,

1.3.


15

among others, were affected by the lectures and personal influence of Gnedenko (see Vere-Jones, 1997), who was a student of Khinchin. Meanwhile, there was also rapid development on the theoretical physics front. The principal ideas here were the characteristic and generating functionals and product densities. As early as 1935, Kolmogorov suggested the use of the characteristic functional Φ(ξ) = E(eiX,ξ ) as a tool in the study of random elements X from a linear space L; ξ is then an element from the space of linear functionals on L. The study of probability measures on abstract spaces remained a favourite theme of the Russian school of probability theory and led to the development of the weak convergence theory for measures on metric spaces by Prohorov (1956) and others, which in turn preceded the general study of random measures [e.g. Ji˘rina (1966) and later writers including the Swedish mathematicians Jagers (1974) and Kallenberg (1975)]. After the war, the characteristic functional was discussed by LeCam (1947) for stochastic processes and Bochner (1947) for random interval functions. Bochner’s (1955) monograph, in particular, contains many original ideas that have only partially been followed up, for example, by Brillinger (1972). Kendall (1949) and Bartlett and Kendall (1951) appear to be the first to have used the characteristic functional in the study of specific population models. Of more immediate relevance to point processes is the related concept of a probability generating functional (p.g.fl.) defined by G[h] = E h(xi ) = E exp log h(x) N (dx) , i

where h(x) is a suitable test function and the xi are the points at which population members are located, that is, the atoms of the counting measures N (·). The p.g.fl. is the natural extension of the p.g.f., and, like the p.g.f., it has an expansion, when the total population is finite, in terms of the probabilities of the number of particles in the population and the probability densities of their locations. There is also an expansion, analogous to the expansion of the p.g.f. in terms of factorial moments, in terms of certain factorial moment density functions, or product densities as they are commonly called in the physical literature. Following the early work of Yvon noted at the end of Section 1.2, the p.g.fl. and product densities were used by Bogoliubov (1946), while properties of product densities were further explored in important papers by Bhabha (1950) and Ramakrishnan (1950). Ramakrishnan, in particular, gave formulae expressing the moments of the number of particles in a given set in terms of the product densities and Stirling numbers. Later, these ideas were considerably extended by Ramakrishnan, Janossy, Srinivasan, and others; an extensive literature exists on their application to cosmic ray showers summarized in the monographs by Janossy (1948) and Srinivasan (1969, 1974).

16

1. Early History

This brings us to another key point in the mathematical theory of point processes, namely the fundamental paper by Moyal (1962a). Drawing principally on the physical and ecological contexts, Moyal for the first time set out clearly the mathematical constructs needed for a theory of point processes on a general state space, clarifying the relations between such quantities as the product densities, finite-dimensional distributions, and probability generating functionals and pointing out a number of important applications. Independently, Harris (1963) set out similar ideas in his monograph on branching processes, subsequently (Harris, 1968, 1971) contributing important ideas to the general theory of point processes and the more complex subject of interacting particle systems. In principle, the same techniques are applicable to other contexts where population models are important, but in practice the discussions in such contexts have tended to use more elementary, ad hoc tools. In forestry, for example, a key problem is the assessment of the number of diseased or other special kinds of trees in a given region. Since a complete count may be physically very difficult to carry out and expensive, emphasis has been on statistical sampling techniques, particularly of transects (line segments drawn through the region) and nearest-neighbour distances. Matérn’s (1960) monograph brought together many ideas, models, and statistical techniques of importance in such fields and includes an account of point process aspects. Ripley’s (1981) monograph covers some more recent developments. On the statistical side, Cox’s (1955) paper contained seeds leading to the treatment of many statistical questions concerning data generated by point processes and discussing various models, including the important class of doubly stochastic Poisson processes. A further range of techniques was introduced by Bartlett (1963), who showed how to adapt methods of time series analysis to a point process context and brought together a variety of different models. This work was extended to processes in higher dimensions in a second paper (Bartlett, 1964). Lewis (1964a) used similar techniques to discuss the instants of failure of a computer. The subsequent monograph by Cox and Lewis (1966) was a further important development that, perhaps for the first time, showed clearly the wide range of applications of point processes as well as extending many of the probabilistic and statistical aspects of such processes. In the 1970s, perhaps the most important development was the rapid growth of interest in point processes in communications engineering (see e.g. Snyder, 1975). It is a remarkable fact that in nature, for example in nerve systems, the transfer of information is more often effected by pulse signals than by continuous signals. This fact seems to be associated with the high signal/noise ratios that it is possible to achieve by these means; for the same reason, pulse techniques are becoming increasingly important in communication applications. For such processes, just as for continuous processes, it is meaningful to pose questions concerning the prediction, interpolation, and estimation of signals, and the detection of signals against background noise (in this context, of random pulses). Since the signals are intrinsically nonnega-

1.3.


17

tive, the distributions cannot be Gaussian, so linear models are not in general appropriate. Thus, the development of a suitable theory for point processes is closely linked to the development of nonlinear techniques in other branches of stochastic process theory. As in the applications to processes of diffusion type, martingale methods provide a powerful tool in the discussion of these problems, yielding, for example, structural information about the process and its likelihood function as well as more technical convergence results. Amongst other books, developments in this area were surveyed in Liptser and Shiryayev (1974; English translation 1977, 1978; 2nd ed. 2000), Brémaud (1981), and Jacobsen (1982). The last quarter-century has seen both the emergence of new fields of applications and the consolidation of older ones. Here we shall attempt no more than a brief indication of major directions, with references to texts that can be consulted for more substantive treatments. Spatial point processes, or spatial point patterns as they are often called, have become a burgeoning subject in their own right. The many fields of application include environmental studies, ecology, geography, astrophysics, fisheries and forestry, as well as substantially new topics such as image processing and spatial epidemic theory. Ripley (1981) and Diggle (1983) discuss both models and statistical procedures, while Cressie (1991) gives a broad overview with the emphasis on applications in biology and ecology. Image processing is discussed in the now classical work of Serra (1982). Theoretical aspects of spatial point patterns link closely with the fields of stereology and stochastic geometry, stemming from the seminal work of Roger Miles and, particularly, Rollo Davidson (see Harding and Kendall, 1974) and surveyed in Stoyan, Kendall and Mecke (1987, 2nd ed. 1995) and Stoyan and Stoyan (1994). There are also close links with the newly developing subject of random set theory; see Mathéron (1975) and Molchanov (1997). The broad-ranging set of papers in Barndorff-Nielsen et al. (1998) covers many of these applications and associated theory. Time, space–time, and marked space–time point processes have continued to receive considerable attention. As well as in the earlier applications to queueing theory, reliability, and electrical engineering, they have found important uses in geophysics, neurophysiology, cardiology, finance, and economics. Applications in queueing theory and reliability were developed in the 1980s by Brémaud (1981) and Franken et al. (1981). Baccelli and Brémaud (1994) contains a more recent account. Second-order methods for the statistical analysis of such data, including spectral theory, are outlined in the now classic text of Cox and Lewis (1966) and in Brillinger (1975b). Snyder and Miller (1991) describe some of the more recent applications in medical fields. Extreme-value ideas in finance are discussed, from a rather different point of view than in Leadbetter et al. (1983) and Resnick (1987), in Embrechts et al. (1997). Prediction methods for point processes have assumed growing importance in seismological applications, in which context they are reviewed in Vere-Jones (1995).

18

1. Early History

Survival analysis has emerged as another closely related major topic, with applications in epidemiology, medicine, mortality, quality control, reliability, and other fields. Here the study of a single point process is usually replaced by the study of many individual processes, sometimes with only a small number of events in each, evolving simultaneously. Starting points include the early papers of Cox (1972b) and Aalen (1975). Andersen et al. (1993) give a major survey of modelling and inference problems in this field; their treatment includes an excellent introduction to point process concepts in general, emphasizing martingale concepts for inference, and the use of product-integral formulae. The growing range of applications has led to an upsurge of interest in inference problems for point process models. Many of the texts referred to above devote a substantial part of their discussion to the practical implementation of inference procedures. General principles of inference for point processes are treated in the text by Liptser and Shiryayev already mentioned and in Kutoyants (1980, 1984), Karr (1986, 2nd ed. 1991), and Kutoyants (1998). Theoretical aspects have also continued to flourish, particularly in the connections with statistical mechanics and stochastic geometry. Recent texts on basic theory include Kingman’s (1993) beautiful discussion of the Poisson process and Last and Brandt’s (1995) exposition of marked point processes. There are close connections between point processes and infinite particle systems (Liggett, 1999), while Georgii (1988) outlines ideas related to spatial processes and phase changes. Branching processes in higher-dimensional spaces exhibit many remarkable characteristics, some of which are outlined in Dawson et al. (2000). Very recently, Coram and Diaconis (2002), exploiting Diaconis and Evans (2000, 2001), have studied similarities between finite point processes of n points on the unit circle constructed from the eigenvalues of random unitary matrices from the unitary group Un , and blocks of n successive zeros of the Riemann zeta function, where n depends on the distance from the real axis of the block of zeros.

CHAPTER 2

Basic Properties of the Poisson Process

The archetypal point processes are the Poisson and renewal processes. Their importance is so great, not only historically but also in illustrating and motivating more general results, that we prefer to give an account of some of their more elementary properties in this and the next two chapters before proceeding to more complex examples and the general theory of point processes. For our present purposes, we shall understand by a point process some method of randomly allocating points to intervals of the real line or (occasionally) to rectangles or hyper-rectangles in a d-dimensional Euclidean space Rd . It is intuitively clear and will be made rigorous in Chapters 5 and 9 that a point process is completely defined if the joint probability distributions are known for the number of events in all finite families of disjoint intervals (or rectangles, etc.). We call these joint or finite-dimensional distributions fidi distributions for short.

2.1. The Stationary Poisson Process With the understanding just enunciated, the stationary Poisson process on the line is completely defined by the following equation, in which we use N (ai , bi ] to denote the number of events of the process falling in the half-open interval (ai , bi ] with ai < bi ≤ ai+1 : Pr{N (ai , bi ] = ni , i = 1, . . . , k} =

k [λ(bi − ai )]ni i=1

ni !

e−λ(bi −ai ) .

(2.1.1)

This definition embodies three important features: (i) the number of points in each finite interval (ai , bi ] has a Poisson distribution; 19

20

2. Basic Properties of the Poisson Process

(ii) the numbers of points in disjoint intervals are independent random variables; and (iii) the distributions are stationary: they depend only on the lengths bi − ai of the intervals. Thus, the joint distributions are multivariate Poisson of the special type in which the variates are independent. Let us first summarize a number of properties that follow directly from (2.1.1). The mean M (a, b] and variance V (a, b] of the number of points falling in the interval (a, b] are given by M (a, b] = λ(b − a) = V (a, b].

(2.1.2)

The constant λ here can be interpreted as the mean rate or mean density of points of the process. It also coincides with the intensity of the process as defined following Proposition 3.3.I. The facts that the mean and variance are equal and that both are proportional to the length of the interval provide a useful diagnostic test for the stationary Poisson process: estimate the mean M (a, b] and the variance V (a, b] for half-open intervals (a, b] over a range of different lengths, and plot the ratios V (a, b]/(b − a). The estimates should be approximately constant for a stationary Poisson process and equal to the mean rate. Any systematic departure from this constant value indicates some departure either from the Poisson assumption or from stationarity [see Exercise 2.1.1 and Cox and Lewis (1966, Section 6.3) for more discussion]. Now consider the relation, following directly from (2.1.1), that Pr{N (0, τ ] = 0} = e−λτ

(2.1.3)

is the probability of finding no points in an interval of length τ . This may also be interpreted as the probability that the random interval extending from the origin to the point first appearing to the right of the origin has length exceeding τ . In other words, it gives nothing other than the survivor function for the length of this interval. Equation (2.1.3) therefore shows that the interval under consideration has an exponential distribution. From stationarity, the same result applies to the length of the interval to the first point of the process to the right of any arbitrarily chosen origin and then equally to the interval to the first point to the left of any arbitrarily chosen origin. In this book, we follow queueing terminology in calling these two intervals the forward and backward recurrence times; thus, for a Poisson process both forward and backward recurrence times are exponentially distributed with mean 1/λ. Using the independence property, we can extend this result to the distribution of the time interval between any two consecutive points of the process, for the conditional distribution of the time to the next point to the right of the origin, given a point in (−∆, 0], has the same exponential form, which, being independent of ∆, is therefore the limiting form of this conditional distribution as ∆ → 0. When such a unique limiting form exists, it can be

2.1.

The Stationary Poisson Process

21

identified with the distribution of the time interval between two arbitrary points of the process (see also Section 3.4 in the next chapter). Similarly, by considering the limiting forms of more complicated joint distributions, we can show that successive intervals are independently distributed as well as having exponential distributions (see Exercises 2.1.2–4 and, for extensions to R2 and R3 , Exercises 2.1.7–8). On the other hand, the particular interval containing the origin is not exponentially distributed. Indeed, since it is equal to the sum of the forward and backward recurrence times, and each of these has an exponential distribution and is independent of the other, its distribution must have an Erlang (or gamma) distribution with density λ2 xe−λx . This result has been referred to as the ‘waiting-time paradox’ because it describes the predicament of a passenger arriving at a bus stop when the bus service follows a Poisson pattern. The intuitive explanation is that since the position of the origin (the passenger’s arrival) is unrelated to the process governing the buses, it may be treated as effectively uniform over any given time interval; hence, it is more likely to fall in a large rather than a small interval. See Sections 3.2 and 3.4 for more detail and references. Now let tk , k = 1, 2, . . . , denote the time from the origin t0 = 0 to the kth point of the process to the right of the origin. Then we have {tk > x} = {N (0, x] < k}

(2.1.4)

in the sense that the expressions in braces describe identical events. Hence, in particular, their probabilities are equal. But the probability of the event on the right is given directly by (2.1.1), so we have Pr{tk > x} = Pr{N (0, x] < k} =

k−1

j=0

(λx)j −λx e . j!

(2.1.5)

Differentiating this expression, which gives the survivor function for the time to the kth point, we obtain the corresponding density function fk (x) =

λk xk−1 −λx , e (k − 1)!

(2.1.6)

which is again an Erlang distribution. Since the time to the kth event can be considered as the sum of the lengths of the k random intervals (t0 , t1 ], (t1 , t2 ], . . . , (tk−1 , tk ], which as above are independently and exponentially distributed, this provides an indirect proof of the result that the sum of k independent exponential random variables has the Erlang distribution. In much the same vein, we can obtain the likelihood of a finite realization of a Poisson process. This may be defined as the probability of obtaining the given number of observations in the observation period, times the joint conditional density for the positions of those observations, given their number. Suppose that there are N observations on (0, T ] at time points t1 , . . . , tN . From (2.1.1), we can write down immediately the probability of obtaining

22


single events in (ti − ∆, ti ] and no points on the remaining part of (0, T ]: it is just N λ∆. e−λT j=1

Dividing by ∆ and letting ∆ → 0, to obtain the density, we find as the required likelihood function N

L(0,T ] (N ; t1 , . . . , tN ) = λN e−λT .

(2.1.7)

Since the probability of obtaining precisely N events in (0, T ] is equal to [(λT )N /N ! ]e−λT , this implies inter alia that the conditional density of obtaining points at (t1 , . . . , tN ), given N points in the interval, is just N !/T N , corresponding to a uniform distribution over the hyperoctant 0 ≤ t1 ≤ · · · ≤ tN ≤ T. One point about this result is worth stressing. It corresponds to treating the points as indistinguishable apart from their locations. In physical contexts, however, we may be concerned with the positions of N physically distinguishable particles. The factor N ! , which arises in the first instance as the volume of the unit hyperoctant, can then be interpreted also as the combinatorial factor representing the number of ways the N distinct particles can be allocated to the N distinct time points. The individual particles are then to be thought of as uniformly and independently distributed over (0, T ]. It is in this sense that the conditional distributions for the Poisson process are said to correspond to the distributions of N particles laid down uniformly at random on the interval (0, T ] (see Exercise 2.1.5). Furthermore, either from this result or directly from (2.1.1), we obtain Pr{N (0, x] = k, N (x, T ] = N − k} Pr{N (0, T ] = N } k N −k N 1 − px,T = , (2.1.8) px,T k

Pr{N (0, x] = k | N (0, T ] = N } =

where px,T = x/T , representing a binomial distribution for the number in the subinterval (0, x], given the number in the larger interval (0, T ]. Most of the results in this section extend both to higher dimensions and to nonstationary processes (see Exercises 2.1.6–8). We conclude the present section by mentioning the simple but important extension to a Poisson process with time-varying rate λ(t), commonly called the nonhomogeneous or inhomogeneous Poisson process. The process can be defined exactly as in b (2.1.1), with the quantities λ(bi − ai ) = aii λ dx replaced wherever they occur by quantities bi λ(x) dx. Λ(ai , bi ] = ai

Thus, the joint distributions are still Poisson, and the independence property still holds. Furthermore, conditional distributions now correspond to particles

2.1.


23

independently distributed on (0, T ] with a common distribution having density function λ(x)/Λ(0, T ] (0 ≤ x ≤ T ). The construction of sample realizations is described in Exercise 2.1.6, while the likelihood function takes the more general form L(0,T ] (N ; t1 , . . . , tN ) = e−Λ(0,T ]

λ(ti )

i=1 T

= exp

N

−

T

λ(t) dt + 0

(2.1.9)

log λ(t) N (dt) . 0

From this expression, we can see that results for the nonstationary Poisson process can be derived from those for the stationary case by a deterministic time change t → u(t) ≡ Λ(0, t]. In other words, if we write N (t) = N (0, t] (all t ≥ 0) and define a new point process by (t) = N u−1 (t) , N (t) has the rate quantity Λ(0, ˜ t) = u(u−1 (t)) = t and is therefore a then N stationary Poisson process at unit rate. In Chapters 7 and 14, we shall meet a remarkable extension of this last result, due to Papangelou (1972a, b): any point process satisfying a simple continuity condition can be transformed into a Poisson process if we allow a random time change in which Λ[0, t] depends on the past of the process up to time t. Papangelou’s result also implies that (2.1.9) represents the typical form of the likelihood for a point process: in the general case, all that is needed is to replace the absolute rate λ(t) in (2.1.9) by a conditional rate that is allowed to depend on the past of the process. Other extensions lead to the class of mixed Poisson processes (see Exercise 2.1.9) and Cox processes treated in Chapter 6.

Exercises and Complements to Section 2.1 2.1.1 Let N1 , . . . , Nn be i.i.d. like the Poisson r.v. N with mean µ = EN , and write N = (N1 + · · · + Nn )/n for the sample mean. When µ is sufficiently large, indicate why the sample index of dispersion Z=

n

(Nj − N )2 j=1

N

has a distribution approximating that of a χ2n−1 r.v. Darwin (1957) found approximations to the distribution of Z for a general distribution for N based on its cumulants, illustrating his work via the Neyman, negative binomial, and Thomas distributions (see also Kathirgamatamby, 1953). 2.1.2 Exponential distribution order properties. Let X1 , . . . , Xn be i.i.d. exponential r.v.s on (0, ∞) with Pr{X1 > x} = e−λx (x ≥ 0) for some positive finite λ. (a) Let X(1) < · · · < X(n) be the order statistics of X1 , . . . , Xn . Then (X(1) , . . . , X(n) ) has the same distribution as the vector whose kth component is Xn Xn−k+1 Xn−1 + ··· + . + n−1 n−k+1 n

24

2. Basic Properties of the Poisson Process (b) Write Y = X1 + · · · + Xn and set Y(k) = (X1 + · · · + Xk )/Y . Then Y(1) , . . . , Y(n−1) are the order statistics of n − 1 i.i.d. r.v.s uniformly distributed on (0, 1).

2.1.3 Exponential r.v.s have no memory. Let X be exponentially distributed as in Exercise 2.1.2, and for any nonnegative r.v. Y that is independent of X, define an r.v. XY as any r.v. whose d.f. has as its tail R(z) ≡ Pr{XY > z} = Pr{X > Y + z | X > Y }. Then XY and X have the same d.f. [There exist innumerable characterizations of exponential r.v.s via their lack of memory properties; many are surveyed in Galambos and Kotz (1978).] 2.1.4 A process satisfying (2.1.1) has Pr{N (t − x − ∆, t − ∆] = 0, N (t − ∆, t] = 1, N (t, t + y] = 0 | N (t − ∆, t] > 0} → e−λx e−λy

(∆ → 0),

showing the stochastic independence of successive intervals between points of the process. 2.1.5 Order statistics property of Poisson process. Denote the points of a stationary Poisson process on R+ by t1 < t2 < · · · < tN (T ) < · · · , where for any positive T , tN (T ) ≤ T < tN (T )+1 . Let u(1) < · · · < u(n) be the order statistics of n i.i.d. points uniformly distributed on [0, T ]. Show that, conditional on N (T ) = n, the distributions of {u(i) : i = 1, . . . , n} and {ti : i = 1, . . . , n} coincide. 2.1.6 Conditional properties of inhomogeneous Poisson processes. Given a finite measure Λ(·) on a c.s.m.s. X , let {t1 , . . . , tN (X ) } be a realization of an inhomogeneous Poisson process on X with parameter measure Λ(·). (a) I.i.d. property. Let r.v.s U1 , . . . , Un be i.i.d. on X with probability distribution Λ(·)/Λ(X ). Show that the joint distributions of {Ui } coincide with those of {ti } conditional on N (X ) = n. (b) Binomial distribution. When X = (0, T ], show that (2.1.8) still holds for the process N (·) with px,T = Λ(x)/Λ(T ). (c) Thinning construction. To construct a realization on (0, T ] of an inhomogeneous Poisson process Π1 for which the local intensity λ(·) satisfies 0 ≤ λ(u) ≤ λmax (0 < u ≤ T ) for some finite positive constant λmax , first construct a realization of a stationary Poisson process with rate λmax (using the fact that successive intervals are i.i.d. exponential r.v.s with mean 1/λmax ), yielding the points 0 < tl < t2 < · · ·, say. Then, independently for each k = 1, 2, . . . , retain tk as a point of Π1 with probability λ(tk )/λmax and otherwise delete it. Verify that the residual set of points satisfies the independence axiom and that

E(#{j: 0 < tj < u, tj ∈ Π1 }) =

u

λ(v) dv. 0

[See also Lewis and Shedler (1976) and Algorithm 7.5.II.]

2.1.


25

2.1.7 Avoidance functions of Poisson process in Rd . The distance X of the point closest to the origin of a Poisson process in Rd with rate λ satisfies Pr{X > y} = exp ( − λvd (y)), where vd (y) = y vd (1) is the volume of a sphere of radius y in Rd . particular, (i) in R1 , Pr{X > y} = e−2λy ; d

In

2

(ii) in R2 , Pr{X > y} = e−πλy ;

3

(iii) in R3 , Pr{X > y} = e−(4π/3)λy . These same expressions also hold for the nearest-neighbour distance of an arbitrarily chosen point of the process. 2.1.8 Simulating a Poisson process in Rd . Using the notation of Exercise 2.1.6, we can construct a realization of a Poisson process Πd in a neighbourhood of the origin in Rd by adapting Exercises 2.1.6 and 2.1.7 to give an inhomogeneous Poisson process on (0, T ) with intensity λ(d/dy)vd (y) and then, denoting these points by r1 , r2 , . . . , taking the points of Πd as having polar coordinates (rj , θj ), where θj are points independently and uniformly distributed over the surface of the unit sphere in Rd . [An alternative construction for rj is to use the fact that λ(vd (rj ) − vd (rj−1 )), with r0 = 0, are i.i.d. exponential r.v.s with unit mean. See also Quine and Watson (1984). The efficient simulation of a Poisson process in a ddimensional hypersphere, at least for small d, is to choose a point at random in a d-dimensional hypercube containing the hypersphere and use a rejection method of which Exercise 2.1.6(c) is an example.] 2.1.9 (a) Mixed Poisson process. A point process whose joint distributions are given by integrating λ in the right-hand side of (2.1.1) with respect to some d.f. defines a mixed Poisson process since the distributions come from regarding λ as a random variable. Verify that N (0, t]/t →a.s. λ

(t → ∞),

EN (0, t] = (Eλ)t, var N (0, t] = (Eλ)t + (var λ)t2 ≥ EN (0, t], with strict inequality unless var λ = 0. (b) Compound Poisson process. Let Y, Y1 , Y2 , . . . be i.i.d. nonnegative integervalued r.v.s with probability generating function g(z) = Ez Y (|z| ≤ 1), and let them be independent of a Poisson process Nc at rate λ; write Nc (t) = Nc (0, t]. Then

Nc (t)

N (0, t] ≡

Yi

i=1

defines the counting function of a compound Poisson process for which Ez N (0,t] = exp [ − λt(1 − g(z))], EN (0, t] = λ(EY )t, var N (0, t] = λ(var Y )t + λ(EY )2 t = λ[E(Y 2 )]t = [ENc (t)](var Y ) + [var Nc (t)](EY )2 ≥ EN (0, t], with strict inequality unless E[Y (Y − 1)] = 0, i.e. Y = 0 or 1 a.s.

26

2. Basic Properties of the Poisson Process [Both the mixed and compound Poisson processes are in general overdispersed compared with a Poisson process in the sense that (var N (0, t])/EN (0, t] ≥ 1, with equality holding only in the exceptional cases as noted.]

2.1.10 For a Poisson process with the cyclic intensity function (κ ≥ 0, ω0 > 0, 0 ≤ θ < 2π, λ > 0),

λ(t) = λ exp[κ sin(ω0 t + θ)]/I0 (κ)

2π

where I0 (κ) = 0 exp(κ sin u) du is the modified Bessel function of the first kind of zero order, the likelihood [see (2.1.9) above] of the realization t1 , . . . , tN on the interval (0, T ) where, for convenience of simplifying the integral below, T is a multiple of the period 2π/ω0 , equals

exp

− 0

T

λ exp[κ sin(ω0 t + θ)] dt I0 (κ)

= e−λT /2π

λ I0 (κ)

N

N

exp κ

λ I0 (κ)

N

N

exp κ

sin(ω0 ti + θ)

i=1

sin(ω0 ti + θ) .

i=1

Consequently, N is a sufficient statistic for λ, and, when the frequency ω0 is known,

N,

N

i=1

sin ω0 ti ,

N

cos ω0 ti

≡ (N, S, C)

say,

i=1

are jointly sufficient statistics for the parameters (λ, κ, θ), the maximum likeˆ κ ˆ being determined by λ ˆ = 2πN/T , tan θˆ = C/S, and lihood estimates (λ, ˆ , θ) ˆ = (S 2 + C 2 )1/2 /N (the constraints that (d/dκ) log I0 (κ)|κ=ˆκ = S/(N cos θ) κ ˆ ≥ 0 and that S and cos θ are of the same sign determine which root θˆ is taken). [See Lewis (1970) and Kutoyants (1984, Chapter 4) for more details.]

2.2. Characterizations of the Stationary Poisson Process: I. Complete Randomness In applications, the Poisson process is sometimes referred to simply as a random distribution of points on a line (as if there were no alternative random processes!) or slightly more specifically as a purely random or completely random process. In all these terminologies, what is in view is the fundamental independence property referred to in (ii) under (2.1.1). We start our discussion of characterizations by examining how far this property alone is capable of characterizing the Poisson process. More precisely, let us assume that we are given a point process satisfying the assumptions below and examine how far the distributions are determined by them. Assumptions 2.2.I. (i) The number of points in any finite interval is finite and not identically zero. (ii) The numbers in disjoint intervals are independent random variables. (iii) The distribution of N (a + t, b + t] is independent of t.

2.2.

Characterizations: I. Complete Randomness

27

For brevity, we speak of a process satisfying (i) as boundedly finite and nonnull, while property (ii) may be referred to as complete independence and (iii) as (crude) stationarity. Theorem 2.2.II. Under Assumptions 2.2.I, the probability generating function (p.g.f.) P (z, τ ) = E(z N (0,τ ] ) can be written uniquely in the form (2.2.1) P (z, τ ) = e−λτ [1−Π(z)] , ∞ where λ is a positive constant and Π(z) = n=1 πn z n is the p.g.f. of a discrete distribution having no zero term. Remark. From the stationarity and independence assumptions, all the joint distributions can be written down once the form of (2.2.1) is given, so that (2.2.1) is in fact sufficient to specify the process completely. Hence, the assumption of crude stationarity suffices in the case of the Poisson process to ensure its (complete) stationarity (see Definition 3.2.I below). Proof. Since N (a, b] is a monotonically increasing function of b, it is clear that P (z, τ ) is a monotonically decreasing function of τ for any fixed z with 0 ≤ z ≤ 1, while Q(z, τ ) = − log P (z, τ ), finite because of Assumption 2.2.I(i), is a monotonically increasing nonnegative function of τ . Also, since N (0, τ1 + τ2 ] = N (0, τ1 ] + N (τ1 , τ1 + τ2 ], it follows from the stationarity and independence assumptions that P (z, τ1 + τ2 ) = P (z, τ1 )P (z, τ2 ), Q(z, τ1 + τ2 ) = Q(z, τ1 ) + Q(z, τ2 ).

(2.2.2)

Now it is well known (see e.g. Lemma 3.6.III) that the only monotonic solutions of the functional equation (2.2.2) are of the form Q(z, τ ) = constant × τ, where in this case the constant is a function of z, C(z) say. Thus, for all τ > 0 we can write P (z, τ ) = e−τ C(z) (2.2.3) for some uniquely determined function C(z). Consider first the case z = 0. From Assumption 2.2.I(i), N (0, τ ] ≡ 0, so P (0, τ ) ≡ 1, and hence C(0) = 0. Now n k−1 k {N (0, 1] ≥ n} ⊇ N ≥1 , , k=1

n

n

so using the independence assumption and (2.2.3), we have n n Pr{N (0, 1] ≥ n} ≥ Pr{N (0, 1/n] ≥ 1} = 1 − e−C(0)/n .

28


If now C(0) = ∞, then Pr{N (0, 1] ≥ n} = 1 (all n = 1, 2, . . .), contradicting Assumption 2.2.I(i) that N (0, 1] is a.s. finite. Thus, we conclude that 0 < C(0) < ∞.

(2.2.4)

Define quantities λ and Π(z) by λ = C(0)

and

Π(z) =

log P (z, τ ) − log P (0, τ ) C(0) − C(z) = , C(0) − log P (0, τ )

the finiteness and nonnegativity of Π(z) on 0 ≤ z ≤ 1 being ensured by the monotonicity in z of P (z, ·). From (2.2.3) and (2.2.4), it follows that P (z, τ ) → 1 (τ → 0) for every fixed z in 0 ≤ z ≤ 1, so from (2.2.3) we have τ C(z) = 1 − P (z, τ ) + o(τ ) from which also Π(z) = lim τ ↓0

(τ ↓ 0),

P (z, τ ) − P (0, τ ) . 1 − P (0, τ )

This representation expresses Π(·) as the limit of p.g.f.s, namely the p.g.f.s of the conditional probabilities πk|τ ≡ Pr{N (0, τ ] = k | N (0, τ ] > 0}. The definition of Π(z) shows that it inherits from P (z, τ ) the property of continuity as z ↑ 1, and therefore the continuity theorem for p.g.f.s (see e.g. Feller, 1968, Section XI.6) ensures that Π(z) must also be a p.g.f., Π(z) = πk z k say, where πk = lim πk|τ = lim Pr{N (0, τ ] = k | N (0, τ ] > 0} τ ↓0

τ ↓0

(k = 0, 1, . . .). (2.2.5)

In particular, π0 = Π(0) = 0. We have thus established the required form of the representation in (2.2.1). Uniqueness follows from the uniqueness of P (z, τ ), which defines C(z) by (2.2.3), and C(z) in turn defines λ and Π(z). The process defined by Assumptions 2.2.I is clearly more general than the Poisson process, to which it reduces only in the case π1 = 1, πk = 0 (k = 1). The clue to its interpretation comes from the limit relation (2.2.5), which suggests that {πk } should be interpreted as a ‘batch-size’ distribution, where ‘batch’ refers to a collection of points of the process located at the same time point. None of our initial assumptions precludes the possibility of such batches. The distribution of the number of such batches in (0, 1) is found by replacing Π(z) by z in (2.2.1), and therefore it is Poisson with rate λ. Thus, the general process defined by Assumptions 2.2.I can be described as consisting of a succession of batches, the successive batch sizes or multiplicities being independent random variables [as follows readily from Assumption 2.2.I(ii)] having the common distribution {πk }, and the number of batches following

2.2.

Characterizations: I. Complete Randomness

29

a Poisson process with constant rate λ. Recognizing that (2.2.1) specifies the p.g.f. of a compound Poisson distribution, we refer to the process as the compound Poisson process [see the footnote on p.10 regarding terminology]. Processes with batches represent an extension of the intuitive notion of a point process as a random placing of points over a region. They are variously referred to as nonorderly processes, processes with multiple points, compound processes, processes with positive integer marks, and so on. For a general proof of the existence of a batch-size distribution for stationary point processes, see Proposition 3.3.VII. It should be noted that the uniqueness of the representation (2.2.1) breaks down once we drop the convention π0 = 0. Indeed, given any p.g.f. Π(·) as in (2.2.1), let π0∗ be any number in 0 ≤ π0∗ < 1, and define λ∗ = λ/(1 − π0∗ ), πn∗ = (1 − π0∗ )πn . Then ∞ Π∗ (z) ≡ n=0 πn∗ z n = π0∗ + (1 − π0∗ )Π(z), and λ∗ 1 − Π∗ (z) = λ(1 − π0∗ )−1 {(1 − π0∗ )[1 − Π(z)]} = λ 1 − Π(z) . The interpretation of this nonuniqueness is that if we increase the rate of occurrence of batches, we may compensate for this increase by observing only those batches with nonzero batch size. We obtain an alternative interpretation of the process by writing (2.2.1) in the form ∞ P (z, τ ) = exp[−λπk τ (1 − z k )], k=1

corresponding to a representation of the total as the sum of independent contributions from a countable family of simpler processes, the kth of which may be regarded as a modified Poisson process in which the rate of occurrence of points is equal to λπk and each such point is treated as a batch of fixed size k. In this representation, the process is regarded as a superposition of independent component processes, each of Poisson type but with fixed batch size. Since both interpretations lead to the same joint distributions and hence to the same probability structures, they must be regarded as equivalent. Theorem 2.2.II may also be regarded as a special case of the more general theorem of Lévy on the structure of processes with stationary independent increments (see e.g. Loève, 1963, Section 37). In our case, there can be no Gaussian component (since the realizations are monotonic), no drift component (since the realizations are integer-valued), and the Poisson components must have positive integral jumps. Because a process has independent increments if and only if the distributions of the increment over any finite interval are infinitely divisible, (2.2.1) also gives the general form of an infinitely divisible distribution taking values only on the nonnegative integers [see Exercise 2.2.2 and Feller (1968, Section XII.2)]. Analytically, the condition corresponding to the requirement of no batches, or points occurring one at a time, is clearly π1 = 1, or equivalently Pr{N (0, τ ] > 1} = o(Pr{N (0, τ ] > 0}) = o(1 − e−λτ ) = o(τ )

for τ ↓ 0.

(2.2.6)

30


More generally, a stationary process satisfying this condition was called by Khinchin (1955) an orderly process (Russian ordinarnii), and we follow this terminology for the time being, as contrasted with the sample path terminology of a simple point process. The relations between analytical and sample path properties are discussed later in Section 3.3 and Chapter 9. For the present, suffice it to be noted that the analytical condition (2.2.6) is equivalent to the absence of batches with probability 1 (see Exercise 2.2.4). Using the notion of an orderly process, we obtain the following characterization of the Poisson process as a corollary to Theorem 2.2.II. Theorem 2.2.III. A stationary point process satisfying Assumption 2.2.I(i) is a Poisson process if and only if (a) it has the complete independence property 2.2.I(ii) and (b) it is orderly.

Exercises and Complements to Section 2.2 2.2.1 In equation (2.2.3), P (z, τ ) → 1 (z → 1) for every finite τ (why?), and equation (2.2.2) and λτ > 0 suffice to check that Π(1) = 1. (A general proof, using only stationarity and not the Poisson assumption, is given in Proposition 3.3.VIII below.) 2.2.2 Call the p.g.f. P (z) infinitely divisible when for 0 ≤ z ≤ 1 its uniquely defined nonnegative kth root P1/k (z) ≡ (P (z))1/k is a p.g.f. for every positive integer. Then show that unless P (z) = 1 for all 0 ≤ z ≤ 1: (a) p0 = P (0) > 0; (b) (P (z)/p0 )1/k → 1 (k → ∞); P1/k (z) − P1/k (0) log P (z) − log P (0) = lim ; (c) k↑∞ − log P (0) 1 − P1/k (0) (d) the left-hand side of (c) represents a p.g.f. on {1, 2, . . .}. Hence, deduce that every nontrivial infinitely divisible p.g.f. is of the form ∞ exp[−λ(1−Π(z))] for finite λ (in fact, p0 = e−λ ), and p.g.f. Π(z) = n=1 πn z n [for details see e.g. Feller (1968, Section XII.2)]. 2.2.3 (Continuation). Show that an r-variate p.g.f. P (z1 , . . . , zr ), which is nontrivial r in the sense that P (z1 , . . . , zr ) ≡ 1 in |1 − zj | > 0, is infinitely divisible j=1 if and only if it is expressible in the form exp[−λ(1 − Π(z1 , . . . , zr ))] for some p.g.f. Π(z1 , . . . , zr ) =

∞

n1 =0

···

∞

πn1 ,...,nr z n1 · · · zrnr

nr =0

for which π0...0 = 0. 2.2.4 If a point process N has N ((k − 1)/n, k/n] ≤ 1 for k = 1, . . . , n, then there can be no batches on (0, 1]. Use the complete independence property in Assumption 2.2.I(ii) and the fact that (1 − o(1/n))n → 1 (n → ∞) to show that a Poisson process satisfying the analytic orderliness property in (2.2.6) has a.s. no batches on the unit interval, and hence on R.

2.3.

Characterizations: II. The Form of the Distribution

31

2.3. Characterizations of the Stationary Poisson Process: II. The Form of the Distribution The discussion to this point has stressed the independence property, and it has been shown that the Poisson character of the finite-dimensional distributions is really a consequence of this property. To what extent is it possible to work in the opposite direction and derive the independence property from the Poisson form of the distributions? Observe that for any partition A1 , . . . , Ar of a Borel set A, the avoidance probability P0 (A) of a Poisson process satisfies P0 (A) = Pr{N (A) = 0} = exp(−λ(A)) =

r

exp(−λ(Ai )) =

i=1

r

P0 (Ai ),

i=1

(2.3.1) so the events {N (Ai ) = 0} are independent [in (2.3.1), (·) denotes Lebesgue measure]. Rényi (1967) weakened this assumption by requiring (2.3.1) to hold merely on all sets A that are finite unions of finite intervals, and then, adding the requirement that N be orderly, he deduced that N must be Poisson. In the converse direction, it is not enough to take A to be the class of unions of any fixed number of intervals: in particular, it is not enough to know that N (A) has a Poisson distribution for all single intervals A = [a, b], as shown in a series of counterexamples provided by Shepp in Goldman (1967), Moran (1967, 1976a, b), Lee (1968), Szasz (1970), and Oakes (1974); two such counterexamples are described in Exercises 2.3.1 and 4.5.12. Theorem 2.3.I. Let N be an orderly point process on R. Then, for N to be a stationary Poisson process it is necessary and sufficient that for all sets A that can be represented as the union of a finite number of finite intervals, P0 (A) = e−λ(A) .

(2.3.2)

It is as easy to prove a more general result for a Poisson process that is not necessarily stationary. To this end, define a simple Poisson process in d-dimensional space Rd as a point process N for which the joint distributions of the counts N (Ai ) on bounded disjoint Borel sets Ai satisfy [see equation (2.1.1)] Pr{N (Ai ) = ki (i = 1, . . . , r)} =

r [µ(Ai )]ki i=1

ki !

e−µ(Ai )

(r = 1, 2, . . .)

for some nonatomic measure µ(·) that is bounded on bounded sets. Thus, the N (Ai ) are Poisson-distributed and independent, E[N (A)] = µ(A), and µ being nonatomic, µ(An ) → 0 for any monotonic sequence of bounded sets An ↓ ∅ or {x } for any singleton set {x } (see Lemma A1.6.II). It is an elementary property

of the Poisson distribution that this then implies that Pr{N (An ) ≥ 2} Pr{N (An ) ≥ 1} → 0 for the same sequence {An }; thus, N has the property of orderliness noted below (2.2.6).

32


Theorem 2.3.II. Let µ be a nonatomic measure on Rd , finite on bounded sets, and suppose that the simple point process N is such that for any set A that is a finite union of rectangles, Pr{N (A) = 0} = e−µ(A) .

(2.3.3)

Then N is a Poisson process with mean µ(A). Proof. We use the idea of a dissecting system (see Appendix A1.6). For any set A as in (2.3.3), let the set Tn of rectangles {Ani : i = 1, . . . , rn } be an element of a dissecting system {Tn } of partitions for A [so, for given n, the union of the Ani is A, Ani and Anj are disjoint for i = j, and each Anj is the union of some subset An+1,is (s = 1, . . . , rn,i ) of Tn+1 , and for any x ∈ A, there is a sequence {An (x)}, An (x) ∈ Tn with n An (x) = {x}]. Since µ is nonatomic, µ(An (x)) → 0 as n → ∞. Given a partition Tn , define the indicator random variables Ini =

1 if N (Ani ) > 0, 0 otherwise,

rn Ini . Because the sets Ani are disjoint, the random and set Nn (A) = i=1 variables of the set {Inij : j = 1, . . . , s} are mutually independent because they are {0, 1}-valued and Pr{Inij = 0 (j = 1, . . . , s)} = Pr{N (Anij ) = 0 (j = 1, . . . , s)} s = Pr{N j=1 Anij = 0} s = exp − µ j=1 Anij =

s

exp[−µ(Anij )] .

j=1

Also, E(z Ini ) = 1 − (1 − z)(1 − e−µ(Ani ) ), so Nn (A) has p.g.f. E(z Nn (A) ) =

i

E(z Ini ) =

1 − (1 − z)(1 − e−µ(Ani ) ) . i

Because µ is nonatomic, supi µ(Ani ) ≡ n → 0 as n → ∞ (see Lemma A1.6.II), and thus, using 1 − δ < e−δ < 1 − δ + δ 2 for all δ sufficiently small, the p.g.f. of Nn (A) converges to exp[−(1 − z)µ(A)] as n → ∞. Since N is simple, for each realization there exists n0 such that, for all n ≥ n0 , each of the N (A) points xj is in a distinct set Anj , say. Then, for n ≥ n0 , Nn (A) = N (A). Also, the random variables Nn (A) are monotonically increasing in n and thus have the a.s. limit N (A). It follows that E(z N (A) ) = exp[−(1 − z)µ(A)]; i.e. N (A) is Poisson-distributed with mean µ(A) for sets A as in the theorem.

2.3.

Characterizations: II. The Form of the Distribution

33

Next, let {Aj } be a finite family of disjoint sets that are unions of rectangles. Repeating the argument above shows that the random variables {N (Aj )} are mutually independent Poisson random variables with means µ(Aj ). Now let A be an open set. Then there is a sequence of families Tn of rectangles Ani that are disjoint, as for Tn , with union a subset of A and the unions converging monotonically to A. Analysis similar to that just given shows that N (A) is Poisson distributed with mean µ(A). Similarly, for a finite family of disjoint open sets Aj , the random variables N (Aj ) are independent. Finally, we extend these properties to arbitrary disjoint bounded Borel sets Aj by using generating functionals (see Definition 5.5.I) with functions that equal 1 on open sets contained by Aj , vanish on a closed set containing Aj , and are continuous (and between 0 and 1). Such approximating functions yield generating functions that are of Poisson variables and that decompose into products of the separate functions (for each distinct Aj ), so the N (Aj ) are Poisson-distributed and independent. Theorem 2.3.II is due to Rényi (1967); the proof above is adapted from Kingman (1993). This result includes Theorem 2.3.I as a special case, while in the other direction, it is a corollary of a more general result, proved in Chapter 9 and due to Kurtz, that for a simple point process N , it is enough to know the avoidance probabilities P0 (A) on a sufficiently rich class of sets A in order to determine its distribution. In turn, this leads to a characterization of those set functions P0 (A) that can be avoidance functions.

Exercises and Complements to Section 2.3 2.3.1 (see Theorem 2.3.II). Let N (·) be a point process on R having as its fidi distributions those of a stationary Poisson process of unit rate except for the following eight probabilities relating to the interval (0, 4]: p0010 = p0101 = p1011 = p1100 = e−4 + , p0100 = p1010 = p1101 = p0011 = e−4 − , where pijkl = Pr{N (0, 1] = i, N (1, 2] = j, N (2, 3] = k, N (3, 4] = l}, 0
0} > 0 is a fixed atom of the process. Thus, we conclude that every atom of Λ(·) is a fixed atom of N (·). Conversely, if x0 is a fixed atom of N (·), then N {x0 } must have a Poisson distribution with nonzero parameter λ0 , say. From this, it follows that x0 is an atom of Λ(·) with mass λ0 . Hence, the following is true. Lemma 2.4.I. The point x0 is an atom of the parameter measure Λ if and only if it is a fixed atom of the process. Note that whether a given point x0 represents a fixed atom of the process is not discernible from a single realization: any point of the process is an atom of its particular realization. For x0 to constitute a fixed atom, there must be positive probability of it recurring over a whole family of realizations. Thus, the fixed atoms relate to the probability structure of the process, not to the structure of individual realizations. In the Poisson case, the fixed atoms are also the key to the question of orderliness. The definition given earlier in (2.2.6) is most naturally extended to the present context by requiring Pr{N (S (x)) > 1} = o(Pr{N (S (x)) > 0})

( → 0),

(2.4.2)

for each x ∈ X , where S (x) denotes the open sphere with radius and centre x. In the case of a Poisson process, N (S (x)) has a Poisson distribution, with parameter Λ(S (x)) = Λ , say, so that Pr{N (S (x)) > 0} = 1 − e−Λ , Pr{N (S (x)) > 1} = 1 − e−Λ − Λ e−Λ . Now if x is a fixed atom of Λ, Λ → Λ0 = Λ{x} > 0 as ↓ 0, whereas if x is not a fixed atom, Λ → 0. In the first case, the ratio Pr{N (S (x)) > 1}/ Pr{N (S (x)) > 0} tends to the positive constant 1 − Λ0 /(eΛ0 − 1), whereas in the second case it tends to zero. Thus, the process is orderly, in the sense of (2.4.2), if and only if Λ(·) has no atoms. Theorem 2.4.II. The Poisson process defined by (2.4.1) is orderly if and only if it has no fixed atoms; equivalently, if and only if the parameter measure has no discrete component. When X is the real line, the distribution function FΛ (x) ≡ Λ(0, x] is continuous if and only if Λ has no discrete component, so in this case Λ itself could

36


be called continuous. One should beware of claiming any such conclusions for more general X , however, for even though Λ(·) may have no atoms, it may well have concentrations on lines, surfaces, or other lower-dimensional subsets that may cause an associated distribution function to be discontinuous. In such situations, in contrast to the case of a homogeneous Poisson process, there will be some positive probability of points of the process appearing on such lines, surfaces, and so on. We turn next to the slightly more difficult problem of extending the characterizations based on the complete independence property stated below. Assumption 2.4.III. For each finite family of bounded, disjoint Borel sets {Ai , i = 1, . . . , k}, the random variables N (A1 ), . . . , N (Ak ) are mutually independent. The most important result is contained in the following lemma. Lemma 2.4.IV. Suppose (i) N is boundedly finite a.s. and has no fixed atoms, and (ii) N has the complete independence property of Assumption 2.4.III. Then, there exists a boundedly finite nonatomic Borel measure Λ(·) such that P0 (A) = Pr{N (A) = 0} = e−Λ(A)

(all bounded Borel sets A).

Proof. Set Q(A) = − log P0 (A), observing immediately that Q(A) ≥ 0 and that by (ii) it is finitely additive. Countable additivity is equivalent to having Q(An ) → 0 for any decreasing sequence {An } of bounded Borel sets for which Q(An ) < ∞ and An ↓ ∅. For An ↓ ∅, we must have N (An ) → 0 a.s., and thus e−Q(An ) = P0 (An ) = Pr{N (An ) = 0} → 1, establishing Q(An ) → 0 as required. To show that Q(·) is nonatomic, observe that, by (i), 0 = Pr{N {x} > 0} = 1 − e−Q({x}) , so that Q({x}) = 0 for every x. It remains to show that Q(·) is boundedly finite, which is equivalent to P0 (A) > 0 for any bounded Borel set A. Suppose the contrary for some set A, which without loss of generality we may assume to be closed, for if not, ¯ ≤ P0 (A) = 0, whence P0 (A) ¯ = 0. Since X is separable, A can be 0 ≤ P0 (A) covered by a countable ∞number of disjoint Borel sets An , each with diameter less than 1, so A = n=1 An . Let pn = Pr{N (An ) > 0}, ∞so that N (A) = 0 (1 − pn ). This only if N (An ) = 0 for all n, and thus 0 = P0 (A) = n=1 ∞ infinite product vanishes only if pn = 1 for some n, or else n=1 pn diverges. In the latter event, the Borel–Cantelli lemma implies that a.s. infinitely many N (An ) are nonzero, and hence N (A) = ∞ a.s., contradicting the assumption that N (·) is boundedly finite. Consequently, we must have pn = 1 for some set An , A(1) say, and A(1) has diameter less than 1 and as with A may be assumed to be closed. By repeating the argument, we can find a closed set A(2) with diameter less than 2−1 such that P0 (A(2) ) = 0. Proceeding by induction, a

2.4.

The General Poisson Process

37

sequence {A(n) } of nested closed sets is constructed with diameters → 0, and P0 (A(n) ) = 0 (all n). Choose xn ∈ A(n) , so that {xn } is a Cauchy sequence, xn → x0 say, and, each A(n) being closed, x0 ∈ A(n) , and therefore An ↓ {x0 }. Then N (A(n) ) ↓ N ({x0 }), and by monotone convergence, P0 ({x0 }) = limn→∞ P0 (A(n) ) = 0. Equivalently, Pr{N {x0 } > 0} = 1, so that x0 is a fixed atom of the process, contradicting (i). Now suppose that the process is orderly in addition to satisfying the conditions of Lemma 2.4.IV. Then, it follows from Theorem 2.3.II that we have a Poisson process without fixed atoms. Thus, the following theorem, due to Prekopa (1957a, b), is true. Theorem 2.4.V. Let N (·) be a.s. boundedly finite and without fixed atoms. Then N (·) is a Poisson process if and only if (i) it is orderly, and (ii) it has the complete independence property of Assumption 2.4.III. To extend this result to the nonorderly case, consider for fixed real z in 0 ≤ z ≤ 1 the set function Qz (A) ≡ − log E(z N (A) ) ≡ − log Pz (A) defined over the Borel sets A. It follows immediately that 0 ≤ Qz (A) < Q(A), and using also the argument of Lemma 2.4.VI, it follows that Qz (·) is a measure, absolutely continuous with respect to Q(·). Consequently, there exists a density, qz (x) say, such that Qz (A) = qz (x) Q(dx) (2.4.3) A

and for Q-almost-all x qz (x) = lim ↓0

Qz (S (x)) , Q(S (x))

where S (x) is as in (2.4.2); see also e.g. Lemma A1.6.III for this property of Radon–Nikodym derivatives. If we continue to assume that the process has no fixed atoms, Q(S (x)) and hence also Qz (S (x)) both → 0 as → 0, for then S (x) → {x}. We can then imitate the argument leading to Theorem 2.2.II and write for Q-almost-all x Πz (x) = 1 − qz (x) = lim ↓0

Pz (S (x)) − P0 (S (x)) . 1 − P0 (S (x))

(2.4.4)

Now, for fixed A, Qz (A) is monotonically decreasing in z for 0 ≤ z ≤ 1, so by taking a countably dense set of z values in [0, 1], (2.4.4) holds for such z except possibly on a Q-null set formed by the union of the Q-null sets where it may fail for the separate values of z.

38


For each , (2.4.4) is the p.g.f. of the conditional distribution Pr{N (S (x)) = k | N (S (x)) > 0}. Now a sequence of p.g.f.s converging on a countably dense set of z values in [0, 1) converges for all 0 ≤ z < 1, with the limit being a p.g.f. of a possibly dishonest distribution. In the present case, the limit is in fact Q-a.e. honest because by monotone convergence and (2.4.3), 0 = log P1 (A) = lim Qz (A) = lim qz (x) Q(dx), z↑1

A

z→1

implying that limz→1 qz (x) = 0 Q-a.e. Consequently, except for a Q-null set, (2.4.4) holds for all 0 ≤ z ≤ 1, and for the limit qz (x), 1 − qz (x) is the p.g.f. of a proper distribution, {πk (x)} say, for which ∞

π0 (x) = 0, Πz (x) = πk (x)z k , k=1

and Pz (A) = exp

−

[1 − Πz (x)] Q(dx) .

(2.4.5)

A

There is the alternative form for (2.4.5), Pz (A) = exp − Q(A)[1 − Πz (A)] , in which there appears the p.g.f. Πz (A) of the ‘averaged’ probabilities 1 πk (A) = πk (x) Q(dx). Q(A) A Thus, the distributions in this process still have the compound Poisson form. Finally, suppose we reinstate the fixed atoms of the process. Note that these are also atoms of Q(·) and can therefore be at most countable in number, and also that the number of points of the process at each fixed atom must be a discrete random variable independent of the rest of the process. We thus arrive at the following structure theorem for the general point process satisfying the complete independence property. Theorem 2.4.VI. Let N (·) be a point process that has the complete independence property of Assumption 2.4.III. Then N (·) can be written in the form of a superposition N = N1 + N2 , where N1 and N2 are independent and (i) N1 consists of a finite or countable family of fixed atoms, {x1 , x2 , . . .}, where for each i, N1 {xi } has a proper, discrete distribution and is independent of the rest of the process; and (ii) N2 is a process without fixed atoms, which can be represented in the compound Poisson form (2.4.5), where Q(·) is a fixed, boundedly finite, nonatomic measure, and for Q-almost-all x, Πz (x) is the p.g.f. of a proper discrete distribution, satisfying Π0 (x) = 0.

2.4.

The General Poisson Process

39

We remark that, analogously to the situation described by Theorem 2.2.II, the realizations of N2 consist a.s. of random batches of points, where the number of batches is governed by a Poisson process with parameter measure Q(·) and, conditional on a batch occurring at x, its probability distribution is given by {πk (x)}. These sample-path results can be established directly for this special case, but we prefer to treat them as special cases of the theorems established in Chapter 3.

Exercises and Complements to Section 2.4 2.4.1 Let N1 , N2 be independent Poisson processes with parameter measures Λ1 , Λ2 . Show that N1 + N2 is a Poisson process with parameter measure Λ1 + Λ2 . 2.4.2 Poisson process on the surface of a sphere. There is an area-preserving map of the surface of a sphere of radius r onto the curved surface of a cylinder of radius r and height 2r. Conclude that a homogeneous Poisson process on the surface of such a sphere can be represented as a Poisson process on a rectangle with side-lengths 2r and 2πr. How may a homogeneous Poisson process on the surface of an oblate or prolate elliptical spheroid be constructed? [Hint: An oblate spheroid is the solid of revolution obtained by rotating an ellipse with major and minor axes of lengths 2a and 2b, respectively, about its minor axis, so it has the same surface area as the curved surface of a cylinder of π/2 radius a and height 2 0 cos θ a2 sin2 θ + b2 cos2 θ dθ. For a prolate spheroid, use a cylinder of radius b and height 2

π/2 0

sin θ

a2 sin2 θ + b2 cos2 θ dθ.]

2.4.3 Poisson process on a lattice. A homogeneous Poisson process with density λ on a given (countably infinite) lattice of points, {zi } say, is a sequence of i.i.d. Poisson r.v.s, {Ni } say, with common mean λ. A homogeneous binary process on such a lattice is a sequence, {Yi } say, of i.i.d. {0, 1}-valued r.v.s {Yi } for which Pr{Yi = 1} = p for some p ∈ (0, 1). It is only approximately Poisson, and then only for small p. 2.4.4 Define a homogeneous Poisson process on a cylinder of unit radius as a Poisson process of points {(xi , θi )} on the doubly infinite strip R × (0, 2π] at rate λ dx dθ. Such a point process can also be interpreted as a Poisson process of directed lines in the plane since any such line is specified by its orientation relative to a given direction and its distance from the origin (negative if the origin is to the left of the line rather than the right). (a) In this line-process interpretation, check that the largest circle that can be drawn around a randomly chosen point in the plane without intersecting a line has radius R with distribution Pr{R > y} = Pr{strip of width 2y has no point (xi , θi )} = exp(−λ 2πy). (b) Show that the expected number of intersections lying within the circle SR (0) between the line (x, 0) and lines of the process, where 0 < x < R R, equals 4 x arsin(y/R) 2λ dy. Deduce that the expected number of intersections between any two lines of the process and lying in a circle of radius R equals

2π

R

2λ dx 0

x

R

8λ arsin(y/R) dy = (2λ πR)2 .

40

2. Basic Properties of the Poisson Process Observe that such a point process (from line intersections) cannot be Poisson because with probability 1, given any two points, there are infinitely many other points collinear with the two given points.

2.4.5 Poisson process in Hilbert space. (i) Find an example of a Hilbert-space-valued random variable that does not have its distribution concentrated ona finite-dimensional subspace. [Hint: Consider a series of the form Y = ak Uk ek , where the ak form a scalar series, the Uk are i.i.d., and ek is the unit vector in the kth dimension. Other examples follow from the Hilbert-space Gaussian measures discussed in Chapter 9.] By combining copies of this probability measure suitably, build up examples of σ-finite measures. (ii) Using the measures above, construct examples of well-defined Poisson processes on a Hilbert space. Discuss the nature of the realizations in increasing sequences of spheres or cubes. (iii) Show that if a σ-finite measure is invariant under Hilbert-space translations, then it cannot be boundedly finite. Hence, show that no Poisson process can exist that is invariant under the full set of Hilbert-space translations.

CHAPTER 3

Simple Results for Stationary Point Processes on the Line

The object of this chapter is to give an account of some of the distinctive aspects of stationary point processes on the line without falling back on the measure-theoretic foundations that are given in Chapter 9. Some aspects that are intuitively reasonable and that can in fact be given a rigorous basis are taken at face value in order that the basic ideas may be exposed without the burden of too much mathematical detail. Thus, the results presented in this chapter may be regarded as being made logically complete when combined with the results of Chapter 9. Ideas introduced here concerning second-order properties are treated at greater length in Chapters 8 and 12, and Palm theory in Chapter 13.

3.1. Specification of a Point Process on the Line A point process on the line may be taken as modelling the occurrences of some phenomenon at the time epochs {ti } with i in some suitable index set. For such a process, there are four equivalent descriptions of the sample paths: (i) (ii) (iii) (iv)

counting measures; nondecreasing integer-valued step functions; sequences of points; and sequences of intervals.

In describing a point process as a counting measure, it does not matter that the process is on the real line. However, for the three other methods of describing the process, the order properties of the reals are used in an essential way. While the methods of description may be capable of extension into higher dimensions, they become less natural and, in the case of (iv), decidedly artificial. 41

42

3. Simple Results for Stationary Point Processes on the Line

In Chapters 1 and 2, we mostly used the intuitive notion of a point process as a counting measure. To make this notion precise, take any subset A of the real line and let N (A) denote the number of occurrences of the process in the set A; i.e. N (A) = number of indices i for which ti lies in A = #{i: ti ∈ A}.

(3.1.1)

When A is expressed as the union of the disjoint sets A1 , . . . , Ar , say, that is, A=

r

Ai

where Ai ∩ Aj = ∅ for i = j,

i=1

it is a consequence of (3.1.1) that N

r i=1

Ai

=

r

N (Ai )

for mutually disjoint A1 , . . . , Ar .

(3.1.2)

i=1

It also follows from (3.1.1) that N (A) is nonnegative integer-(possibly ∞-)valued.

(3.1.3)

In order that we may operate conveniently on N (A) for different sets A—in particular, in order that the probability of events specified in terms of N (A) may be well defined—we must impose a restriction on the sets A that we are prepared to consider. Since we want to include intervals and unions thereof, the usual constraint is that N (A) is defined for all Borel subsets A of the real line.

(3.1.4)

Finally, in order to exclude the possibility of ‘too many’ points occurring ‘too close’ together, we insist that, for the point processes we consider, N (A) is finite for bounded sets A.

(3.1.5)

The assumptions in (3.1.2–5) with (3.1.2) extended to allow r = ∞ are precisely those that make N (·) a counting measure on the σ-field BR of all Borel subsets of the real line R. The constraint in (3.1.3) that N (·) be integervalued distinguishes it from other more general nonnegative measures as a counting measure. To be consistent with N (·) being a set function, we ought to write, for example, N ((a, b]) when A is the half-open interval (a, b]; our preference for the less cumbersome abbreviation N (a, b] should lead to no confusion. We have already used in Chapters 1 and 2 the further contraction N (t) = N (0, t] = N ((0, t])

(0 < t ≤ ∞);

(3.1.6)

3.1.

Specification of a Point Process on the Line

43

the difference in argument should suffice to distinguish the real function N (t) (t > 0) from the set function N (A). This function N (t) is nondecreasing, right-continuous, and integer-valued, and hence a step function. For point processes on the positive half-line, knowledge of N (t) for all t ≥ 0 suffices to determine N (A) for Borel sets A ⊂ (0, ∞) in precisely the same manner as a distribution function determines a probability measure on Borel sets. When the point process is defined on the whole line, we extend the definition (3.1.6) to ⎧ (t > 0), ⎪ ⎨ N ((0, t]) (t = 0), (3.1.7) N (t) = 0 ⎪ ⎩ −N ((t, 0]) (t < 0). In this way, N (t) retains the properties of being a right-continuous integervalued function on the whole line. Moreover, N (t) determines N (A) for all Borel sets A and hence describes the point process via a step function. Thus, instead of starting with N (A) (all A ∈ B), we could just as well have specified the sample path as a right-continuous function N (t) (−∞ < t < ∞) that is nonnegative and integer-valued for t > 0, nonpositive and integer-valued for t < 0, and has N (0) = 0. The simplest case of the third method listed above occurs where the process is defined on the half-line t > 0. Setting ti = inf{t > 0: N (t) ≥ i}

(i = 1, 2, . . .),

(3.1.8)

it follows that for i = 1, 2, . . ., we have the seemingly obvious but most important relation ti ≤ t if and only if N (t) ≥ i. (3.1.9) This relation makes it clear that specifying the sequence of points {ti } is equivalent to specifying the function N (t) in the case where N (−∞, 0] = 0. It should be noted that the set of points {ti } in (3.1.8) is in increasing order; such a restriction is not necessarily implied in talking of a set of time epochs {ti } as at the beginning of the present section. If the point process has points on the whole line and not just the positive axis, the simplest extension consistent with (3.1.8) is obtained by defining ti = inf{t: N (t) ≥ i} inf{t > 0: N (0, t] ≥ i} (i = 1, 2, . . .), = − inf{t > 0: N (−t, 0] ≥ −i + 1} (i = 0, −1, . . .).

(3.1.10)

Such a doubly infinite sequence of points has the properties that ti ≤ ti+1 (all i)

t0 ≤ 0 < t1 .

(3.1.11)

with {ti } as in (3.1.10)

(3.1.12)

and

Finally, by setting τi = ti − ti−1

44


[or else, in the case of only a half-line as in (3.1.8), with the added conventions that t0 = 0 and τi is defined only for i = 1, 2, . . . ], the process is fully described by the sequence of intervals {τi } and one of the points {ti }, usually t0 . Observe n that τi ≥ 0 and that if N (t) → ∞ as t → ∞, then i=1 τi → ∞ as n → ∞, while if N (t) → ∞ as t → ∞, then τi is not defined for i > limt→∞ N (t). We now make the intuitively plausible assumption that there exists a probability space on which the functions N (A), N (t), ti , τi are well-defined random variables and furthermore that we can impose various constraints on these random variables in a manner consistent with that assumption. The question of the existence of such a probability space is discussed in Chapter 9.

Exercises and Complements to Section 3.1 3.1.1 Suppose that the r.v.s {ti } in (3.1.8) are such that Pr{ti+1 > ti } = 1, and define Gi (x) = Pr{ti ≤ x}. (a) Show that limx→0 Gi (x) = 0 for all integers i > 0. (b) Show that the assumption in (3.1.5) of N (·) being boundedly finite implies that, for all real x > 0, lim Gi (x) = 0. i→∞

3.1.2 (Continuation). Show that for x > 0, M (x) ≡ EN (x) = more generally, that E([N (x)]r ) =

∞

(ir − (i − 1)r )Gi (x) =

i=1

∞

∞ i=1

Gi (x) and,

ir (Gi (x) − Gi+1 (x))

i=1

in the sense that either both sides are infinite or, if one is finite, so is the other and the two sides are equal. 3.1.3 (Continuation). Show that for |z| ≤ 1 and x > 0, P (x; z) ≡ Ez N (x) = 1 + (z − 1)

∞

Gi+1 (x)z i .

i=0

3.2. Stationarity: Definitions The notion of stationarity of a point process at first sight appears to be a simple matter: at the very least, it means that the distribution of the number of points lying in an interval depends on its length but not its location; that is, pk (x) ≡ Pr{N (t, t + x] = k} (x > 0, k = 0, 1, . . .) depends on the length x but not the location t. Lawrance (1970) called this property simple stationarity, while we follow Chung (1972) in calling it crude stationarity. It is in fact weaker than the full force of the definition below (see Exercise 3.2.1).

3.2.

Stationarity: Definitions

45

Definition 3.2.I. A point process is stationary when for every r = 1, 2, . . . and all bounded Borel subsets A1 , . . . , Ar of the real line, the joint distribution of {N (A1 + t), . . . , N (Ar + t)} does not depend on t (−∞ < t < ∞). In the case where the point process is defined only on the positive half-line, the sets Ai must be Borel subsets of (0, ∞) and we require t > 0. There is also the intuitive feeling that the intervals {τi } should be stationary, and accordingly we introduce the following definition. Definition 3.2.II. A point process is interval stationary when for every r = 1, 2, . . . and all integers ii , . . . , ir , the joint distribution of {τi1 +k , . . . , τir +k } does not depend on k (k = 0, ±1, . . .). Note that this definition makes no reference to the point t0 required to complete the specification of a sample path as below (3.1.12). It is most natural to take t0 = 0 [see (3.1.11)]. Such processes may then be regarded as a generalization of renewal processes in that the intervals between occurrences, instead of being mutually independent and identically distributed, constitute merely a stationary sequence. The relation that exists between the probability distributions for interval stationarity on the one hand and stationarity on the other is taken up in Section 3.4 and elsewhere, notably Chapter 13, under its usual heading of Palm–Khinchin theory. Some authors speak of arbitrary times and arbitrary points in connection with point processes. A probability distribution with respect to an arbitrary time epoch of a stationary point process is one that is stationary as under Definition 3.2.I; a probability distribution with respect to an arbitrary point of a point process is one determined by the interval stationary distributions as under Definition 3.2.II. The importance of maintaining a distinction between interval stationarity and ordinary stationarity is underlined by the waiting-time paradox. If in some town buses run exactly on schedule every ∆ minutes and a stranger arrives at a random time to wait for the next bus, then his expected waiting time EW is 12 ∆ minutes. If, on the other hand, buses run haphazardly according to a Poisson process with an average time ∆ between buses, then the expected waiting time of the same stranger is ∆. The core of the so-called paradox lies in the use of ∆ as an average interval length from the arrival of one bus to the next, and the waiting time EW being half the mean interval between bus arrivals when the probabilities of different intervals being chosen are proportional to their lengths. In renewal theory, the resolution of the paradox is known as length-biased sampling [see Feller (1966, Section I.4), Exercise 1.2.5 above, and (3.4.17) below].

46


Exercises and Complements to Section 3.2 3.2.1 (a) Construct an example of a crudely stationary point process that is not stationary (for one example, see Exercise 2.3.1). (b) Let N (·) be crudely stationary. Is it necessarily true that Pr{N ({t}) ≥ 2 for some t in (−1, 0]} = Pr{N ({t}) ≥ 2 for some t in (0, 1]} ? [See the proof of Proposition 3.3.VI, where equality is shown to hold when the probabilities equal zero.]

3.3. Mean Density, Intensity, and Batch-Size Distribution A natural way of measuring the average density of points of a point process is via its mean, or in the case of a stationary point process, its mean density, which we define as m = E(N (0, 1]). (3.3.1) Defining the function M (x) = E(N (0, x]),

(3.3.2)

it is a consequence of the additivity properties of N (·) as in (3.1.2) and of expectations of sums, and of the crude stationarity property in (3.2.1), that for x, y ≥ 0, M (x + y) = E N (0, x + y] = E N (0, x] + N (x, x + y] = E N (0, x] + E N (x, x + y] = E N (0, x] + E N (0, y] = M (x) + M (y). In other words, M (·) is a nonnegative function satisfying Cauchy’s functional equation M (x + y) = M (x) + M (y) (0 ≤ x, y < ∞). Consequently, by Lemma 3.6.III, M (x) = M (1)x = mx

(0 ≤ x < ∞),

(3.3.3)

irrespective of whether M (x) is finite or infinite for finite x > 0. There is another natural way of measuring the rate of occurrence of points of a stationary point process, due originally to Khinchin (1955). Proposition 3.3.I (Khinchin’s Existence Theorem). For a stationary (or even crudely stationary) point process, the limit λ = lim h↓0

exists, though it may be infinite.

Pr{N (0, h] > 0} h

(3.3.4)

3.3.

Mean Density, Intensity, and Batch-Size Distribution

47

Proof. Introduce the function φ(x) = Pr{N (0, x] > 0}.

(3.3.5)

Then φ(x) ↓ 0 as x ↓ 0, and φ(·) is subadditive on (0, ∞) because for x, y > 0, φ(x + y) = Pr{N (0, x + y] > 0} = Pr{N (0, x] > 0} + Pr{N (0, x] = 0, N (x, x + y] > 0} ≤ Pr{N (0, x] > 0} + Pr{N (x, x + y] > 0} = φ(x) + φ(y). The assertion of the proposition now follows from the subadditive function Lemma 3.6.I. The parameter λ is called the intensity of the point process, for when it is finite, it makes sense to rewrite (3.3.4) as Pr{N (x, x + h] > 0} = Pr{there is at least one point in (x, x + h]} = λh + o(h)

(h ↓ 0).

(3.3.6)

Examples of a point process with λ = ∞ are given in Exercises 3.3.2–3. These two measures of the ‘rate’ of a stationary point process coincide when the point process has the following property. Definition 3.3.II. A point process is simple when Pr{N ({t}) = 0 or 1 for all t} = 1.

(3.3.7)

Daley (1974) called this sample-path property almost sure orderliness to contrast it with the following analytic property due to Khinchin (1955). Definition 3.3.III. A crudely stationary point process is orderly when Pr{N (0, h] ≥ 2} = o(h)

(h ↓ 0).

(3.3.8)

Notice that stationarity plays no role in the definition of a simple point process, nor does it matter whether the point process is defined on the real line or even a Euclidean space. While orderliness can be defined for point processes that either are nonstationary or are on some space different from the real line, the defining equation (3.3.8) must then be suitably amended [see Exercise 3.3.1, Chapter 9, and Daley (1974) for further discussion and references]. It is a consequence of Korolyuk’s theorem and Dobrushin’s lemma, given below, that for stationary point processes with finite intensity, Definitions 3.3.II and 3.3.III coincide. Proposition 3.3.IV (Korolyuk’s Theorem). For a crudely stationary simple point process, λ = m, finite or infinite.

48


Remark. In Khinchin’s (1955, Section 11) original statement of this proposition, the point process was assumed to be orderly rather than simple. In view of the possible generalizations of the result to nonstationary point processes and to processes on spaces other than the real line where any definition of orderliness may be more cumbersome, it seems sensible to follow Leadbetter (1972) in connecting the present result with Korolyuk’s name. Proof. We use a sequence of nested intervals that in fact constitute a dissecting system (see Section A1.6 and the proof of Theorem 2.3.II). For any positive integer n and i = 1, . . . , n, define indicator random variables > 0, 1 i−1 i Ini = (3.3.10) according as N , n n = 0. 0 Then, as n → ∞ through the integers 2p , p = 1, 2, . . . , n

Ini ↑ N (0, 1]

(3.3.11)

i=1

for those realizations N (·) for which N (0, 1] < ∞ and N ({t}) = 0 or 1 for all 0 < t ≤ 1; that is, in view of (3.1.5) and (3.3.7), (3.3.11) holds a.s. Then ! " n

m = E N (0, 1] = E lim Ini n→∞

! = lim E n→∞

n

i=1 −1

= lim nφ(n n→∞

=λ

" Ini )

i=1

by Lebesgue’s monotone convergence theorem, by (3.3.5), (3.3.10), and crude stationarity,

by Khinchin’s existence theorem.

Proposition 3.3.V (Dobrushin’s Lemma). A crudely stationary simple point process of finite intensity is orderly. Proof. For any positive integer n, E(N (0, 1]) = n E(N (0, n−1 ]) by crude stationarity, so m = E(N (0, 1]) = n

∞

Pr{N (0, n−1 ] ≥ j}

j=1

≥ nφ(n−1 ) + n Pr{N (0, n−1 ] ≥ 2}.

(3.3.12)

Being crudely stationary, Khinchin’s existence theorem applies, so nφ(n−1 ) → λ as n → ∞, and being simple also, Korolyuk’s theorem applies, so λ = m. Combining these facts with (3.3.12), n Pr{N (0, n−1 ] ≥ 2} → 0 as n → ∞, which by (3.3.8) is the same as orderliness. Dobrushin’s lemma is a partial converse of the following result in which there is no finiteness restriction on the intensity. Proposition 3.3.VI. A crudely stationary orderly point process is simple.

3.3.


49

Proof. Simpleness is equivalent to 0=

∞

# $ Pr N ({t}) ≥ 2 for some t in (r, r + 1] ,

r=−∞

which in turn is equivalent to # $ 0 = Pr N ({t}) ≥ 2 for some t in (r, r + 1]

(r = 0, ±1, . . .). (3.3.13)

For every positive integer n, i−1 i Pr{N ({t}) ≥ 2 for some t in (0, 1]} ≤ ≥2 , Pr N n n i=1 n

= n Pr{N (0, n−1 ] ≥ 2} →0

(n → ∞)

by crude stationarity, when N (·) is orderly,

so (3.3.13) holds for r = 0 and, by trite changes, for all r. In the results just given, a prominent role is played by orderliness, which stems from the notion that the points {ti } can indeed be ordered; that is, in the notation of (3.1.10), we have ti < ti+1 for all i. Without orderliness, we are led to the idea of batches of points: we proceed as follows. Proposition 3.3.VII. For a crudely stationary point process, the limits λk = lim h↓0

Pr{0 < N (0, h] ≤ k} h

(3.3.14)

exist for k = 1, 2, . . . , and λk ↑ λ

(k → ∞), finite or infinite ;

(3.3.15)

when λ is finite, πk ≡

λk − λk−1 = lim Pr{N (0, h] = k | N (0, h] > 0} h↓0 λ

(3.3.16)

is a probability distribution on k = 1, 2, . . . . Proof. Define, by analogy with (3.3.5), φk (x) = Pr{0 < N (0, x] ≤ k}

(x > 0, k = 1, 2, . . .).

(3.3.17)

Then, like φ(·), φk (x) → 0 for x ↓ 0 and it is subadditive on (0, ∞) because, for x, y > 0, φk (x + y) = Pr{0 < N (0, x] ≤ k, N (x, x + y] = 0} + Pr{N (0, x] ≤ k − N (x, x + y], 0 < N (x, x + y] ≤ k} ≤ Pr{0 < N (0, x] ≤ k} + Pr{0 < N (x, x + y] ≤ k} = φk (x) + φk (y),

50


invoking crude stationarity at the last step. Thus, (3.3.14) follows from the subadditive function lemma, which is also invoked in writing λ = sup sup h>0 k>0

φk (h) φk (h) = sup sup = sup λk . h h k>0 h>0 k>0

The monotonicity of λk in k is obvious from (3.3.14), so (3.3.15) is now proved. Equation (3.3.16) follows from (3.3.14), (3.3.15), and (3.3.17). The limit of the conditional probability in (3.3.16) can be rewritten in the form (h ↓ 0, k = 1, 2, . . .).

Pr{N (0, h] = k} = λπk h + o(h)

(3.3.18)

This equation and (3.3.16) suggest that the points {ti } of sample paths occur in batches of size k = 1, 2, . . . with respective intensities λπk . To make this idea precise, recall that for bounded Borel sets A we have assumed N (A) to be integer-valued and finite so that we can define Nk (A) = #{distinct t ∈ A: N ({t}) = k}

(k = 1, 2, . . .)

and thereby express N (A) as N (A) =

∞

kNk (A).

(3.3.19)

k=1

By definition, these point processes Nk (·) are simple and stationary, and (k) for them we can define indicator random variables Ini , analogous to Ini in (3.3.10), by % = k, 1 i−1 i (k) , Ini = (3.3.20) according as N n n = k. 0 By letting n → ∞ through n = 2p for p = 1, 2, . . . , it follows from (3.3.20) and the construction of Nk (·) that Nk (0, 1] = lim

n→∞

n

(k)

Ini

a.s.

(3.3.21)

i=1

(k)

Now Ini ≤ Ini , so when λ < ∞, it follows from (3.3.21) by using dominated convergence that E(Nk (0, 1]) < ∞, being given by n (k) Ini E(Nk (0, 1]) = lim E n→∞

i=1

= lim n[φk (n−1 ) − φk−1 (n−1 )] n→∞

= λπk .

(3.3.22)

3.3.


51

The sample-path definition of Nk (·) having intensity λπk as in (3.3.22) warrants the use of the term batch-size distribution for the probability distribution {πk }. Note that a stationary orderly point process has the degenerate batch-size distribution for which π1 = 1, πk = 0 (all k = 1). Otherwise, the sample paths are appropriately described as having multiple points; this terminology is reflected in the frequently used description of a simple point process as one without multiple points. The moments of the distribution {πk } can be related to those of N (·) as in the next two propositions, in which we call equation (3.3.23) a generalized Korolyuk equation. Proposition 3.3.VIII. For a crudely stationary point process of finite intensity, ∞

m = E(N (0, 1]) = λ kπk , finite or infinite. (3.3.23) k=1

Proof. Take expectations in (3.3.19) with A = (0, 1] and use Fubini’s theorem and (3.3.22) to deduce (3.3.23). Proposition 3.3.IX. For a crudely stationary point process of finite intensity λ and finite γth moment, γ ≥ 1, ∞

E [N γ (0, h]]γ lim exists and equals λ k γ πk . h↓0 h

(3.3.24)

k=1

Proof. Introduce

Mγ (x) = E(N γ (0, x]),

and observe that for x, y > 0, using γ ≥ 1, Mγ (x + y) = E (N (0, x] + N (x, x + y])γ ≥ E(N γ (0, x]) + E(N γ (x, x + y]) = Mγ (x) + Mγ (y); that is, the function Mγ (x) is superadditive for x > 0. When Mγ (x) is finite for 0 < x < ∞, Mγ (x) → 0 (x ↓ 0), so the subadditive function Lemma 3.6.IV applied to −Mγ (x) proves the existence part of (3.3.24). Since N γ (0, 1] ≥

n ∞

i=1

(k)

k γ Ini

→

k−1

∞

k γ Nk (0, 1] a.s.

(n → ∞),

k=1

we can use dominated convergence and crude stationarity to conclude that ∞ ∞

lim nMγ (n−1 ) = E k γ Nk (0, 1] = λ k γ πk .

n→∞

k=1

k=1

52


Exercises and Complements to Section 3.3 3.3.1 Verify that a simple point process (Definition 3.3.II) can be defined equivalently as one for which the distances between points of a realization are a.s. positive. [Hint: When the realization consists of the points {tn }, (3.3.7) is equivalent (Vasil’ev, 1965) to the relation Pr{|ti − tj | > 0 (all i = j)} = 1. ] 3.3.2 Show that a mixed Poisson process for which

∞

Pr{N (0, t] = j} = 1

e−λt (λt)j j!

1 −3/2 λ 2

dλ

is simple but not orderly. A mixed Poisson process with

∞

Pr{N (0, t] = j} = 1

e−λt (λt)j −2 λ dλ j!

also has infinite intensity, but it does satisfy the orderliness property (3.3.8). 3.3.3 (a) Let the r.v. X be distributed on (0, ∞) with distribution function F (·) and, conditional on X, let the r.v. Y be uniformly distributed on (0, X). Now define a point process to consist of the set of points {nX + Y : n = 0, ±1, . . .}. Verify that such a process is stationary and that

∞

Pr{N (0, h] = 0} =

1−

h

h

Pr{N (0, h] ≥ 2} = h

h dF (x) = 1 − h x

∞

x−2 F (x) dx,

h

x−2 F (x) dx.

(1/2)h

When F (x) = x for 0 < x < 1, show that (i) the intensity λ = ∞; (ii) the process is not orderly; and (iii) it has the Khinchin orderliness property [Khinchin (1956); see also Leadbetter (1972) and Daley (1974)] Pr{N (0, h] ≥ 2 | N (0, h] ≥ 1} → 0

(h → 0).

(3.3.25)

(b) Let the realizations of a stationary point process come, with probability 12 each, either from a process of doublets consisting of two points at each of {n + Y : n = 0, ±1, . . .}, where Y is uniformly distributed on (0, 1), or from a simple point process as in part (a). Then Pr{N ({t}) ≤ 1 for all t} = 12 , so the process is not simple, but it does have the Khinchin orderliness property in (3.3.25). 3.3.4 Suppose that N (·) is a simple point process on (0, ∞) with finite first moment M (x) = EN (x), x and suppose that M (·) is absolutely continuous in the sense that M (x) = 0 m(y) dy (x > 0) for some density function m(·). Show that the distribution functions Gi (·) of Exercise 3.1.1 are also absolutely continuous with density functions gi (·), where

Gi (x) =

x

gi (y) dy, 0

and

m(x) =

∞

i=1

gi (x) a.e.

3.4.

Palm–Khinchin Equations

53

3.3.5 (Continuation). Now define Gi (x; t) as the d.f. of the ith forward recurrence time after t, i.e. Gj (x; t) is the d.f. of inf{u > t: N (t, u] ≥ i}. Supposing that N (·) has finite first moment and is absolutely continuous in the sense of Exercise 3.3.4, show that when N (·) is simple, g1 (0; t) = m(t),

gi (0; t) = 0

(i ≥ 2).

Use these results to give an alternative proof of Korolyuk’s Theorem 3.3.IV. Show also that when the rth moment of N (·) is finite, E[(N (t, t + h])r ] = m(t). h↓0 h 3.3.6 Given any point process with sample realizations N , define another point process with sample realization N ∗ by means of lim

N ∗ (A) = #{distinct x ∈ A: N ({x}) ≥ 1}

(all Borel sets A)

(in the setting of marked point processes in Section 6.4 below, N ∗ here is an example of a ground process, denoted Ng there). Show that if, for any real finite s > 0, ∗ E(e−sN (A) ) ≥ E(e−sN (A) ) (all Borel sets A), then N is simple. Irrespective of whether or not it is simple, N (A) = 0 iff N ∗ (A) = 0. Show that if N is a compound Poisson process as in Theorem 2.2.II, then N ∗ is a stationary Poisson process with rate λ. 3.3.7 Consider a compound Poisson process as in Theorem 2.2.II, and suppose that the mean batch size Π (1) = kπk is infinite. Let the points of the process be subject to independent shifts with a common distribution that has no atoms. The resulting process is no longer Poisson, is simple, and has infinite intensity. When the shifts are i.i.d. and uniform on (0, 1), show that, for 0 < h < 1,

Pr{N (0, h] = 0} = exp

− λ(1 + h) + λ(1 − h)Π(1 − h) + 2λ

h

Π(1 − u) du . 0

3.4. Palm–Khinchin Equations Throughout this section, we use P to denote the probability measure of a stationary point process (Definition 3.2.I). Our aim is to describe an elementary approach to the problem raised by the intuitively reasonable idea that the stationarity of a point process as in Definition 3.2.I should imply some equivalent interval stationarity property as in Definition 3.2.II. For example, for positive x and y and small positive h, stationarity of the point process N (·) implies that P{N (t, t + h] = N (t + x, t + x + h] = N (t + x + y, t + x + y + h] = 1, N (t, t + x + y + h] = 3} = P{N (−h, 0] = N (x − h, x] = N (x + y − h, x + y] = 1, N (−h, x + y] = 3} ≡ P{Ax,y,h }, say. (3.4.1) Now the event Ax,y,h describes a sample path with a point near the origin

54


and intervals of about x and y, respectively, to the next two points. Our intuition suggests that, as far as the dependence on the variables x and y is concerned, P{Ax,y,h } should be related to the probability measure P0 (·) for an interval stationary point process; that is, there should be a simple relation between P{Ax,y,h } and P0 {τ1 x, τ2 y}. We proceed to describe the partial solution that has its roots in Khinchin’s monograph (1955) and that connects P{N (0, x] ≤ j} to what we shall show is a distribution function Rj (x) = lim P{N (0, x] ≥ j | N (−h, 0] > 0} h↓0

(j = 1, 2, . . .).

(3.4.2)

What emerges from the deeper considerations of Chapter 13 is that, granted orderliness, there exists an interval stationary point process {τj } with probability measure P0 , so P0 {t0 = 0} = 1, for which we can indeed set P0 (·) = lim P(· | N (−h, 0] > 0). h↓0

It then follows, for example, that P0 {τ1 + · · · + τj ≤ x} = Rj (x)

(3.4.3)

[see (3.4.2) and (3.1.9)], thereby identifying a random variable having Rj (·) as its distribution function. Instead of the expression in (3.4.1), we consider first the probability ψj (x, h) ≡ P{N (0, x] ≤ j, N (−h, 0] > 0}

(3.4.4)

and prove the following proposition. Proposition 3.4.I. For a stationary point process of finite intensity, the limit Qj (x) = lim P{N (0, x] ≤ j | N (−h, 0] > 0} (3.4.5) h↓0

exists for x > 0 and j = 0, 1, . . . , being right-continuous and nonincreasing in x with Qj (0) = 1. Proof. Observe that for u, v > 0, ψj (x, u + v) = P{N (0, x] ≤ j, N (−u, 0] > 0} + P{N (0, x] ≤ j, N (−u, 0] = 0, N (−u − v, −u] > 0}. In the last term, {N (0, x] ≤ j, N (−u, 0] = 0} = {N (−u, x] ≤ j, N (−u, 0] = 0} ⊆ {N (−u, x] ≤ j} ⊆ {N (−u, x − u] ≤ j},

3.4.


55

and then using stationarity of P(·), we have ψj (x, u + v) ≤ ψj (x, u) + ψj (x, v). Consequently, the subadditivity lemma implies that the limit as h → 0 of ψj (x, h)/h exists, being bounded by λ [because ψj (x, h) ≤ φj (h)], so by writing ψj (x, h) ψj (x, h)/h P{N (0, x] ≤ j | N (−h, 0] > 0} = = , φ(h) φ(h)/h we can let h → 0 to prove the assertion in (3.4.5) concerning existence. By subadditivity, and right-continuity and monotonicity in x of ψj (x, h), ψj (x, h) ψj (y, h) = sup sup = sup Qj (y), λh λh y>x h>0 h>0 y>x

Qj (x) = sup

so Qj (x) is right-continuous and nonincreasing in x, with Qj (0) = 1 since ψj (0, h) = φ(h). It follows from this result that every Rj (x) ≡ 1 − Qj−1 (x)

(j = 1, 2, . . .)

(3.4.6)

is a d.f. on (0, ∞) except for the possibility, to be excluded later under the conditions of Theorem 3.4.II, that limx→∞ Rj (x) may be less than 1. The plausible interpretation of (3.4.5), or equivalently, of (3.4.6), is that Rj (x) represents the conditional probability (in which the conditioning event has zero probability) # $ P N (0, x] ≥ j | N ({0}) > 0 = P{τ1 + · · · + τj ≤ x | t0 = 0, t1 > 0}. (3.4.7) Example 3.4(a) Renewal process. Consistent with (3.4.7), for a renewal process starting at 0 with lifetime d.f. F for which F (0+) = 0, Rj (x) = F j∗ (x), where F n∗ (·) is the n-fold convolution of F . In this case then, Rj (·) is the d.f. of the sum of j random variables that are not merely stationary but also independent. On the other hand, if we have a renewal process with a point at 0 and having lifetime d.f. F for which 0 < F (0+) < 1, then the constraint in (3.4.7) that τ1 = t1 − t0 > 0 means that τ1 has d.f. F+ (x) = (F (x) − F (0+))/(1 − F (0+)), while τ2 , τ3 , . . . have d.f. F and

x

F (j−1)∗ (x − u) dF+ (u)

Rj (x) =

(j = 1, 2, . . .).

0

Thus, Rj (x) is here the d.f. of the sum of nonstationary r.v.s, and so for a renewal process we have the stationarity property at (3.4.3) only when F (0+) = 0; that is, when the process is orderly (or equivalently, simple).

56


This last assumption is also what enables us to proceed simply in general [but, note the remarks around (3.4.12) below]. Theorem 3.4.II. For an orderly stationary point process of finite intensity λ and such that P{N (−∞, 0] = N (0, ∞) = ∞} = 1, ∞ qj (u) du (j = 0, 1, . . .), P{N (0, x] ≤ j} = λ

(3.4.8) (3.4.9)

x

where qj (x) = lim P{N (0, x] = j | N (−h, 0] > 0}, h↓0

(3.4.10)

j−1 and Rj (x) = 1 − k=0 qk (x) is a distribution function on (0, ∞) with mean jλ−1 for each j = 1, 2, . . . . Proof. Set

Pj (x) = P{N (0, x] ≤ j}

and observe by Proposition 3.4.I and the assumption of orderliness that Pj (x + h) =

j

P{N (0, x] ≤ j − i, N (−h, 0] = i}

i=0

= P{N (0, x] ≤ j} − P{N (0, x] ≤ j, N (−h, 0] > 0} + P{N (0, x] ≤ j − 1, N (−h, 0] = 1} + o(h). Thus, Pj (x + h) − Pj (x) = P{N (0, x] ≤ j − 1, N (−h, 0] ≥ 1} − P{N (0, x] ≤ j, N (−h, 0] > 0} + o(h) = −λhqj (x) + o(h), where the existence of qj (x) in (3.4.10) is assured by (3.4.5) directly for j = 0 and then by induction for j = 1, 2, . . . . Using D+ to denote the right-hand derivative operator, it follows that D+ Pj (x) = −λqj (x). Setting Q−1 (x) ≡ 0, the nonnegative function qj (x) = Qj (x) − Qj−1 (x) is the difference of two bounded nonincreasing functions and hence is integrable on bounded intervals with y qj (u) du. (3.4.11) Pj (x) − Pj (y) = λ x

The assumption in (3.4.8) implies that Pj (y) → 0 for y → ∞, so (3.4.9) now follows from (3.4.11).

3.4.


57

Letting x ↓ 0 in (3.4.9), it follows that λ

−1

∞

=

qj (u) du

(j = 0, 1, . . .),

0

and hence, using (3.4.6) as well, that for j = 1, 2, . . . ,

∞

1 − Rj (u) du =

0

∞

Qj−1 (u) du = jλ−1 .

0

There is a most instructive heuristic derivation of (3.4.9) as follows. By virtue of (3.4.8), if we look backward from a point x, there will always be some point u < x for which N (u, x] ≤ j and N [u, x] > j. In fact, because of orderliness, we can write (with probability 1) {N (0, x] ≤ j} =

{N (u, x] = j, N ({u}) = 1},

u≤0

in which we observe that the right-hand side is the union of the mutually exclusive events that the (j + 1)th point of N (·) looking backward from x occurs at some u ≤ 0. Consequently, we can add their ‘probabilities’, which by (3.4.7), (3.3.4), and orderliness equal qj (x − u)λ du, yielding the Palm– Khinchin equation (3.4.9) in the form Pj (x) = λ

0

−∞

qj (x − u) du.

Without the orderliness assumption, made from (3.4.8) onward above, we can proceed as follows. First (see Proposition 3.4.I), we show that the function ψj|i (x, h) ≡ P{N (0, x] ≤ j, 0 < N (−h, 0] ≤ i}

(3.4.12)

is subadditive in h and so deduce that, for those i for which πi > 0 [see (3.3.16)], there exists the limit Qj|i (x) = lim P{N (0, x] ≤ j | N (−h, 0] = i}, h↓0

(3.4.13)

with P{N (0, x] ≤ j, N (−h, 0] = i} = λπi Qj|i (x)h + o(h)

(h ↓ 0)

irrespective of πi > or = 0 by setting Qj|i (x) ≡ 0 when πi = 0. Then, the argument of the proof of Theorem 3.4.II can be mimicked in establishing that Pj (x) = λ

∞ ∞

x

i=1

πi [Qj|i (u) − Qj−i|i (u)] du,

(3.4.14)

58


setting Qk|i (u) ≡ 0 for k < 0, and it can also be shown that, when πi > 0, Rj|i (x) ≡ 1 − Qj−1|i (x) ≡ 1 −

j−1

qk|i (x)

k=0

is a proper distribution function on (0, ∞). For any point process N , the random variable Tu ≡ inf{t > 0: N (u, y + t] > 0}

(3.4.15)

is the forward recurrence time r.v. For a stationary point process, Tu =d T0 for all u, and we can study its distribution via the Palm–Khinchin equations since {T0 > x} = {N (0, x] = 0}. Assuming that (3.4.8) holds, ∞ q0 (u) du (3.4.16) P{T0 > x} = λ x

when N (·) is orderly as in Theorem 3.4.II. Recall that q0 (·) is the tail of the d.f. R1 (·), which can be interpreted as the d.f. of the length τ1 of an arbitrarily chosen interval. Then, still assuming that (3.4.8) holds, ∞ ∞ P{T0 > x} dx = λ uq0 (u) du ET0 = 0 0 ∞ =λ u 1 − R1 (u) du = 12 λ(Eτ12 ). (3.4.17) 0

When all intervals are of the same length, ∆ say, λ = ∆−1 and ET0 = 12 ∆, whereas for a Poisson process, τ1 has mean ∆ and second moment Eτ12 = 2∆2 , so then ET0 = ∆. These remarks amplify the comments on the waiting-time paradox at the end of Section 3.2. In both Theorem 3.4.II and the discussion of the forward recurrence time r.v. Tu , the caveat that P{N (0, ∞) = ∞} = 1 has been added. This is because stationary point processes on the real line R have the property that P{N (0, ∞) = ∞ = N (−∞, 0)} = 1 − P{N (R) = 0},

(3.4.18)

which is equivalent to P{0 < N (R) < ∞} = 0.

(3.4.19)

A similar property in a more general setting is proved in Chapter 12. Inspection of the statements onward from (3.4.8) shows that they are either conditional probability statements (including limits of such statements), which in view of (3.4.18) reduce to being conditional also on {N (R) = ∞}, or unconditional statements, which without (3.4.8) need further elaboration. This is quickly given: (3.4.8) is equivalent by (3.4.18) to P{T0 < ∞} = 1, and without (3.4.8), equations (3.4.16) and (3.4.17) must be replaced by assertions

3.4.


59

of the form

P{T0 > x} = λ

∞

q0 (u) du + 1 − ,

(3.4.20)

x

E(T0 | T0 < ∞) = 12 λE(τ12 ),

(3.4.21)

where = P{N (R) = ∞} = P{T0 < ∞}.

Exercises and Complements to Section 3.4 3.4.1 Analogously to (3.4.15), define a backward recurrence time r.v. Bu ≡ inf{t > 0: N (u − t, u] > 0} (assuming this to be finite a.s.). Show that when N (·) is a stationary point process, Bu =d B0 =d T0 . The r.v. Lu = Bu + Tu denotes the current lifetime r.v.; when N is orderly and stationary, show that EL0 = (Eτ12 )/(Eτ1 ) [see (3.4.16)] and that

x

[q0 (u) − q0 (x)] du = λ

P{L0 < x} = λ 0

x

u dR1 (u). 0

3.4.2 Use Palm–Khinchin equations to show that when the hazard functions q and r of the interval and forward recurrence r.v.s τ0 and T0 , respectively, are such x that r(x) = r(0) + 0 r (u) du for some density function r , then q and r are related by r(x) = q(x) + r (x)/r(x) (x > 0). 3.4.3 Show that for an orderly point process,

1

P{N (dx) ≥ 1},

EN (0, 1] = 0

where the right-hand side is to be interpreted as a Burkill integral [see Fieger (1971) for further details]. 3.4.4 For a point process N on R, define the event Bk ≡ Bk ((xi , ji ): i = 1, . . . , k ) = {N (0, xi ] ≤ ji (i = 1, . . . , k)} for positive xi , nonnegative integers ji (i = 1, . . . , k), and any fixed finite positive integer k. (a) When N is stationary with finite intensity λ, ψ(Bk , h) = P(Bk ∩ {N (−h, 0] > 0}) is subadditive in h > 0, the limit Q(Bk ) = limh↓0 P (Bk | {N (−h, 0] > 0}) exists finite, is right-continuous and nonincreasing in each xi and nondecreasing in ji , is invariant under permutations of (x1 , j1 ), . . . , (xk , jk ), satisfies the consistency conditions Q(Bk ) = Q(Bk+1 ((0, jk+1 ), (xi , ji ) (i = 1, . . . , k))) = Q(Bk+1 ((xk+1 , ∞), (xi , ji ) (i = 1, . . . , k))), and Q(Bk ) = lim ψ(Bk , h)/λh = sup ψ(Bk , h}/λh. h↓0

h>0

60

3. Simple Results for Stationary Point Processes on the Line (b) Define a shift operator Sh (h > 0) and a difference operator ∆ on Bk by Sh Bk = Bk ((xi + h, ji ) (i = 1, . . . , k)), ∆Bk = Bk ((xi , ji − 1) (i = 1, . . . , k)), and put q(Bk ) = Q(Bk ) − Q(∆Bk ), with the convention that if any ji = 0, then ∆Bk is a null set with Q(∆Bk ) = 0. Under the condition (3.4.8) of Theorem 3.4.II, the right-hand derivative D+ P(Bk ) exists in the sense that D+ P(Sh Bk ) |h=0 = −λq(Bk ), and

P(Bk ) − P(Sx Bk ) = λ

x

q(Su Bk ) du. 0

[See Daley and Vere-Jones (1972, Section 7) and Slivnyak (1962, 1966). Note that Slivnyak used a slightly different operator Sh0 defined by Sh0 Bk = Bk+1 ((h, 0), (xi + h, ji ) (i = 1, . . . , k)), so that ψ(Bk , h) = P(Bk )−P(Sh0 Bk ), and deduced the existence of a derivative in h of P(Sh0 Bk ) from the convexity in h of this function, assuming stationarity of N but not necessarily that it has finite intensity.]

3.5. Ergodicity and an Elementary Renewal Theorem Analogue Let N (·) be a stationary point process with finite mean density m = EN (0, 1]. Then, the sequence {Xn } of random variables defined by Xn = N (n − 1, n]

(n = 0, ±1, . . .)

is stationary with finite first moment m = EXn (all n), and by the strong law for stationary random sequences, N (0, n] X1 + · · · + Xn = →ξ n n

a.s.

for some random variable ξ for which Eξ = m. Using x to denote the largest integer ≤ x, it then follows on letting x → ∞ in the inequalities N 0, x x N 0, x N 0, x + 1 x + 1 · ≤ ≤ · (x ≥ 1) x x x x + 1 x that we have proved the following proposition. Proposition 3.5.I. For a stationary point process with finite mean density m = EN (0, 1], ζ ≡ limx→∞ N (0, x]/x exists a.s. and is a random variable with Eζ = m.

3.5.

Ergodicity and an Elementary Renewal Theorem Analogue

61

In our discussion of limit properties of stationary point processes we shall have cause to use various concepts of ergodicity; for the present we simply use the following definition. Definition 3.5.II. A stationary point process with finite mean density m is ergodic when P{N (0, x]/x → m (x → ∞)} = 1. Suppose that in addition to being ergodic, the second moment E[(N (0, 1])2 ] is finite, so by stationarity and the Cauchy–Schwarz inequality, E[(N (0, x])2 ] < ∞ for all finite positive x. Then, we can use an argument similar to that leading to Proposition 3.5.I to deduce from the convergence in mean square of (X1 + · · · + Xn )/n = N (0, n]/n to the same limit [see e.g. (2.15) of Doob (1953, p. 471) or Chapter 12 below] that var(N (0, x]/x) = E(N (0, x]/x − m)2 → 0

(x → ∞)

(3.5.1)

when N (·) is ergodic with finite second moment. This is one of the key probabilistic steps in the proof of the next theorem, in which the asymptotic result in (3.5.3), combined with the remarks that follow, is an analogue of the elementary renewal theorem [see Exercise 4.1.1(b) and Section 4.4 below]. The function U (·), called the expectation function in Daley (1971), is the analogue of the renewal function. Theorem 3.5.III. For a stationary ergodic point process with finite second moment and mean density m, the second-moment function

x

M2 (x) ≡ E[(N (0, x])2 ] =

2U (u) − 1 m du

(3.5.2)

0

for some nondecreasing function U (·) for which U (x)/x → m when the process is orderly, U (x) =

(x → ∞); ∞

Rj (x).

(3.5.3)

(3.5.4)

j=0

Remarks. (1) It is consistent with the interpretation of Rj (·) in (3.4.3) as the d.f. of the sum Sj = τ1 + · · · + τj that U (x) = lim E(N (0, x] + 1 | N (−h, 0] > 0) h↓0

in the case where N (·) is orderly. In the nonorderly case, it emerges that, given an ergodic stationary sequence {τj } of nonnegative random variables

62


with Eτj = 1/m and partial sums {Sn } given by S0 = 0 and Sn = τ1 + · · · + τn , S−n = −(τ0 + · · · + τ−(n−1) )

(n = 1, 2, . . .),

we can interpret U (·) as ∞

2U (x) − 1 = E#{n = 0, ±1, . . . : |Sn | ≤ x} =

Pr{|Sn | ≤ x}.

(3.5.5)

n=−∞

In the case where the random variables {τj } are independent and identically distributed, ∞

U (x) = F n∗ (x) (3.5.6) n=0

and hence U (·) is then the renewal function. (2) It follows from (3.5.2) that var N (0, x] =

x

2[U (u) − mu] − 1 m du.

(3.5.7)

0

(3) It is a simple corollary of (3.5.3) that for every fixed finite y, U (x + y) →1 U (x)

(x → ∞).

(3.5.8)

Proof of Theorem 3.5.III. From the definition in (3.5.2) with N (x) = N (0, x], 2 M2 (x) = E [N (x)]2 = var N (x) + EN (x) = x2 [var(N (x)/x) + m2 ] ∼ m2 x2

(x → ∞)

when N (·) is ergodic, by (3.5.1). If we can assume that M2 (·) is absolutely continuous and that the function U (·), which can then be defined as in (3.5.2), is monotonically nondecreasing, we can appeal to a Tauberian theorem (e.g. Feller, 1966, p. 421) and conclude that (3.5.3) holds. It remains then to establish (3.5.2), for which purpose we assume first that N (·) is orderly so that the representation (3.4.9) is at our disposal. It is a matter of elementary algebra that ∞ M2 (x) + mx = E N (x)(N (x) + 1) = j(j + 1)P{N (x) = j} j=1

=2

∞

k=1

kP{N (x) ≥ k}

3.5.

Ergodicity and an Elementary Renewal Theorem Analogue

=2

∞

k=1 x

x

(k + 1) 1+

=2

qk (u)λ du 0

0

63

∞

1 − Qj (u)

λ du = 2

∞ x

0

j=0

Rj (u)λ du,

j=0

where R0 (u) ≡ 1. Thus, we have (3.5.2) in the case of orderly N (·) with the additional identification that ∞

U (x) = Rj (x), (3.5.9) j=0

of which (3.5.6) is a special case. Note in (3.5.9) that the nondecreasing nature of each Rj (·) ensures the same property for U (·). When N (·) is no longer orderly, we must appeal to (3.4.14) in writing M2 (x) + mx = 2 =2

∞

k=0 ∞

(k + 1) 1 − Pk (x) (k + 1)

∞ x

0

k=0

πi Qk|i (u) − Qk−i|i (u) λ du. (3.5.10)

i=1

Without loss of generality, we may set Qk|i (x) ≡ 1 when πi = 0. Fubini’s theorem is then applicable as before in the manipulations below: 2

∞

k=0

(k + 1)

∞

i=1

πi

k

qj|i (u) = 2

∞

i=1

j=(k−i+1)+

=

=

∞

i=1 ∞

πi

∞

πi

∞

(k + 1)

k=0

k

qj|i (u)

j=(k−i+1)+

i(2j + i + 1)qj|i (u)

j=0

∞

1 − Qj|1 (u) . (3.5.11) iπi i + 1 + 2

i=0

j=0

Substitute (3.5.11) in (3.5.10) and recall that Qj|i (u) is nonincreasing; this establishes the existence of nondecreasing U (·) in (3.5.2) as required.

Exercises and Complements to Section 3.5 3.5.1 (see Theorem 3.5.III). Use the Cauchy–Schwarz inequality to show that, when M2 (x) ≡ EN 2 (0, x] < ∞ for finite x, (M2 (x))1/2 is subadditive in x > 0 and hence that there is then a finite constant λ2 ≥ m2 such that M2 (x) ∼ λ2 x2 (x → ∞). 3.5.2 Let N (·) be a stationary mixed Poisson process with P{N (0, t] = j} = 1 −t j e t /j! + 12 e−2t (2t)j /j! . Show that λ = 32 = m 0) (cf. 2 Theorem 3.5.III; this process is not ergodic) and that N (0, t]/t → ξ (t → ∞), where ξ = 1 or 2 with probability 12 each.

64


3.6. Subadditive and Superadditive Functions We have referred earlier in this chapter to properties of subadditive and superadditive functions, and for convenience we now establish these properties in a suitable form. For a more extensive discussion of such functions, see Hille and Phillips (1957). A function g(x) defined for 0 ≤ x < a ≤ ∞ is subadditive when g(x + y) ≤ g(x) + g(y)

(3.6.1)

holds throughout its domain of definition; similarly, a function h(x) for which h(x + y) ≥ h(x) + h(y)

(3.6.2)

holds is superadditive. A function f (x) for which f (x + y) = f (x) + f (y)

(3.6.3)

holds is additive, and (3.6.3) is known as Cauchy’s functional equation or (see e.g. Feller, 1966, Section IV.4) the Hamel equation. Lemma 3.6.I. For a subadditive function g(·) that is bounded on finite intervals, µ ≡ inf x>0 g(x)/x is finite or −∞, and g(x) →µ x

(x → ∞).

(3.6.4)

Proof. There exists y for which g(y)/y < µ for any µ > µ. Given any x, there is a unique integer n for which x = ny + η, where 0 ≤ η < y, and n → ∞ as x → ∞. Then g(x) g(ny) + g(η) ng(y) g(η) ≤ ≤ + x x ny + η x g(η) g(y) g(y) + → (x → ∞). = y + η/n x y Thus, lim supx→∞ g(x)/x ≤ µ , and µ being an arbitrary quantity > µ, this proves the lemma. The function −h(x) is subadditive when h(·) is superadditive, and an additive function is both subadditive and superadditive, so Lemma 3.6.I implies both of the following results. Lemma 3.6.II. For a superadditive function h(·) that is bounded on finite intervals, µ ≡ supx>0 h(x)/x is finite or +∞ and h(x) →µ x

(x → ∞).

(3.6.5)

3.6.

Subadditive and Superadditive Functions

65

Lemma 3.6.III. An additive function f (·) that is bounded on finite intervals satisfies f (x) = f (1)x (0 ≤ x < ∞). (3.6.6) In passing, note that there do exist additive functions that do not have the linearity property (3.6.6): they are unbounded on every finite interval and moreover are not measurable (see e.g. Hewitt and Zuckerman, 1969). Observe also that nonnegative additive functions satisfy (3.6.6) with the understanding that f (1) = ∞ is allowed. The behaviour near 0 of subadditive and superadditive functions requires the stronger condition of continuity at 0 in order to derive a useful result [a counterexample when f (·) is not continuous at 0 is indicated in Hille and Phillips (1957, Section 7.11)]. Lemma 3.6.IV. Let g(x) be subadditive on [0, a] for some a > 0, and let g(x) → 0 as x → 0. Then λ ≡ supx>0 g(x)/x is finite or +∞, and g(x) →λ x

(x → 0).

(3.6.7)

Proof. The finiteness of g(x) for some x > 0 precludes the possibility that λ = −∞. Consider first the case where 0 < λ < ∞, and suppose that g(an )/an < λ − 2 for some > 0 for all members of a sequence {an } with an → 0 as n → ∞. For any given x > 0, we can find an sufficiently small that sup0≤δ0 g(x)/x ≤ λ − , contradicting the definition of λ. The case −∞ < λ ≤ 0 is established by considering g1 (x) ≡ g(x) + λ x for some finite λ > −λ. Finally, the case λ = ∞ is proved by contradiction starting from the supposition that g(an )/an → λ < ∞ for some {an } with an → 0. Lemma 3.6.V. Let h(x) be superadditive on [0, a] for some a > 0, and let h(x) → 0 as x → 0. Then λ ≡ inf x>0 h(x)/x is finite or −∞, and h(x) →λ x

(x → 0).

(3.6.8)

CHAPTER 4

Renewal Processes

The renewal process and variants of it have been the subject of much study, both as a model in many fields of application (see e.g. Cox, 1962; Cox and Lewis, 1966; Cox and Isham, 1980) and as a source of important theoretical problems. It is not the aim of this chapter to repeat much of the material that is available, for example, in Volume II of Feller (1966); rather, we have selected some features that are either complementary to Feller’s treatment or relevant to more general point processes. The first two sections are concerned with basic properties, setting these where possible into a point process context. The third section is concerned with some characterization theorems and the fourth section with aspects of the renewal theorem, a topic so important and with such far-reaching applications that it can hardly be omitted. Two versions of the theorem are discussed, corresponding to different forms of convergence of the renewal measure to Lebesgue measure. Some small indication of the range of applications is given in Section 4.5, which is concerned with ‘neighbours’ of the renewal process, notably the Wold process of correlated intervals. A final section is concerned with the concept of a hazard measure for the lifetime distribution, a topic that is of interest in its own right and of central importance to the discussion of compensators and conditional intensity functions in Chapters 7 and 14.

4.1. Basic Properties Let X, X1 , X2 , . . . be independent identically distributed nonnegative random variables, and define the partial sums S0 = 0,

Sn = Sn−1 + Xn = X1 + · · · + Xn 66

(n = 1, 2, . . .).

(4.1.1)

4.1.

Basic Properties

67

For Borel subsets A of (0, ∞), we attempt to define the counting measure of a point process by setting N (A) = #{n: Sn ∈ A}.

(4.1.2)

Even if we exclude the trivial case X = 0 a.s., as we do throughout this chapter, it may not be completely obvious that (4.1.2) is finite. To see that this is so, observe that for X = 0 a.s. there must exist positive ε, δ such that Pr{X > ε} > δ so that with probability 1 the event {Xn > ε} must occur infinitely often (by the Borel–Cantelli lemmas) and hence Sn → ∞ a.s. It follows that the right-hand side of (4.1.2) is a.s. finite whenever A is bounded, thus justifying the definition (4.1.2). (Here we ignore measurability aspects, for which see Chapter 9.) The process so defined is the (ordinary ) renewal process. In the notation and terminology of Chapter 3, provided X1 > 0, we have ti = Si and τi = Xi for i = 1, 2, . . . , while the assumption that the {Xn } are i.i.d. implies that N (·) is interval stationary. Orderliness of the process here means Sn+1 > Sn for n = 0, 1, . . . ; that is, Xn > 0 for all n ≥ 0, all with probability 1. But the probability that Xn > 0 for n = 0, 1, . . . , N − 1 is equal to (Pr{X > 0})N → 0 as N → ∞ unless Pr{X > 0} = 1. Thus, the process is orderly if and only if Pr{X > 0} = 1; that is, if and only if the lifetime distribution has zero mass at the origin. Taking expectations of (4.1.2) yields the renewal measure U (A) = E(#{n: Sn ∈ A, n = 0, 1, 2, . . .}) = E[N (A)],

(4.1.3)

an equation that remains valid even if A includes the origin. U (A) is just the first moment or expectation measure of N (·). Writing F (·) for the common lifetime distribution and F k∗ for its k-fold convolution (which is thus the distribution function for Sk ), and immediately abusing the notation by writing F (·) for the measure induced on the Borel sets of BR by F , we have ∞ ∞

I{Sk ∈A} = δ0 (A) + F k∗ (A). (4.1.4) U (A) = E k=0

k=1

We note in passing that the higher moments of N (A) can also be expressed in terms of U (·) (see Exercise 4.1.2). The quantity most commonly studied is the cumulative function, commonly called the renewal function, U (x) ≡ U ([0, x]) = 1 +

∞

F k∗ (x)

(x ≥ 0).

(4.1.5)

k=1

Again, U (x) is always finite. To see this, choose any δ > 0 for which F (δ) < 1 (possible since we exclude the case X = 0 a.s.). Then, since F (0−) = 0, we have for any positive integers i, j and x, y > 0, 1 − F (i+j)∗ (x + y) ≥ 1 − F i∗ (x) 1 − F j∗ (y) ,

68

4. Renewal Processes

and for 0 < y < x, F i∗ (x − y)F j∗ (y) ≤ F (i+j)∗ (x) ≤ F i∗ (x)F j∗ (x). Thus, F k∗ (δ) ≤ (F (δ))k < 1, and therefore the series in (4.1.5) certainly converges for x < δ. For general x in 0 < x < ∞, there exists finite positive k for which x/k < δ. For given x and such k, 1 − F k∗ (x) > [1 − F (x/k)]k > 0, so ∞ (k−1)∗ U (x) ≤ 1 + F (x) + · · · + F (x) F nk∗ (x) n=0

≤ 1 + F (x) + · · · + F

(k−1)∗

(x) / 1 − F k∗ (x) < ∞.

Thus, (4.1.5) converges for all x > 0. Taking Laplace–Stieltjes transforms in (4.1.5), we have for Re(θ) > 0 χ(θ) ≡

∞

e−θx dU (x) =

0

where ψ(θ) =

∞ 0

∞

k ψ(θ) =

k=0

1 , 1 − ψ(θ)

(4.1.6)

e−θx dF (x). Equivalently, for Re(θ) > 0, ψ(θ) = 1 − 1/χ(θ),

which shows (using the uniqueness theorem for Laplace–Stieltjes transforms) that U determines F uniquely and hence that there is a one-to-one correspondence between lifetime distributions F and renewal functions U . From (4.1.5), we have for x > 0 x U (x) = 1 + U (x − y) dF (y), (4.1.7) 0

this being the most important special case of the general renewal equation x Z(x) = z(x) + Z(x − y) dF (y) (x > 0), (4.1.8) 0

where the solution function Z is generated by the initial function z. If the function z(x) is measurable and bounded on finite intervals, one solution to (4.1.8) is given by Z0 (x) = z(x) +

∞

k=1

0

x

z(x − y) dF k∗ (y) =

x

z(x − y) dU (y),

(4.1.9)

0

the convergence of the series in the middle member being justified by comparison with (4.1.5). Using the monotonicity of the relation z → Z0 , we easily see that if z ≥ 0, (4.1.9) is the minimal nonnegative solution to (4.1.8). In fact, considerably more is true, for if z(x) is merely measurable and bounded on finite inter-

4.1.

Basic Properties

69

vals, the difference D(x) between any two solutions of (4.1.8) with the same property satisfies x D(x) = D(x − y) dF k∗ (y) for each k = 1, 2, . . . ; 0

hence, D(x) ≡ 0 from the fact that F k∗ (x) → 0 as k → ∞ and the assumed boundedness of D. We summarize as follows. Lemma 4.1.I (Renewal Equation Solution). When z(x) is measurable and bounded on finite intervals, the general renewal equation (4.1.8) has a unique measurable solution that is also bounded on finite intervals, and it is given by (4.1.8). In particular, U (x) is the unique monotonic and finite-valued solution of (4.1.7). Example 4.1(a) Exponential intervals. The lack of memory property of the exponential distribution bequeaths on the renewal process that it generates the additional independence properties of the Poisson process. Suppose specifically that (λ > 0, 0 ≤ x < ∞). F (x) = 1 − e−λx The renewal function for the corresponding Poisson process is U (x) = 1 + λx, as can be checked either by using the transform equation in (4.1.6), by summing the convolution powers as in (4.1.5), or by direct verification in the integral equation in (4.1.7). Example 4.1(b) Forward recurrence time. We gave below (3.4.15) an expression for the distribution of the forward recurrence time r.v. Tu of a stationary point process. The definition at (3.4.15) does not require stationarity, and in the present case of a renewal process, it can be written as Tu = inf{Sn : Sn > u} − u = inf{Sn − u: Sn − u > 0} X1 − u if X1 > u, = inf{Sn − X1 : Sn − X1 > u − X1 } − (u − X1 ) otherwise. Now when X1 ≤ u, Tu has the same distribution as the forward recurrence time r.v. Tu−X , defined on the renewal process with lifetime r.v.s {Xn } ≡ 1 {Xn+1 }, so u Pr{Tu > y} = Pr{X1 > y + u} + Pr{Tu−v > y} dF (v). (4.1.10) 0

But this equation is of the form (4.1.8), with z(x) = Pr{X1 > y + x} = 1 − F (y + x), so by (4.1.9) u Pr{Tu > y} = [1 − F (y + u − v)] dU (v). (4.1.11) 0−

In particular, putting y = 0, we recover the identity that is implicit in (4.1.5), x [1 − F (x − v)] dU (v) (all x ≥ 0). (4.1.12) 1= 0−

70


Example 4.1(c) Renewal equation with linear solution. As another important application of (4.1.8), consider the generator z(·) that corresponds to the solution Z(x) = λx (all x > 0), assuming such a solution function exists, and ∞ that λ−1 = EXn = 0 [1 − F (x)] dx is finite. Rearranging (4.1.8) yields x x z(x) = λx − λ (x − y) dF (y) = λ [1 − F (y)] dy. 0

0

We can recognize this expression as the distribution function of the forward recurrence time of a stationary point process. This argument identifies the only initial distribution for which the delayed renewal function is linear. We conclude this section with a few brief remarks concerning the more general case where the random variables Xn are not necessarily nonnegative or even one-dimensional; thus we admit the possibility that the Xn are ddimensional vectors for some integer d > 1. In such cases, the sequence {Sn } constitutes a random walk. Such a walk is said to be transient if (4.1.2) is finite for all bounded Borel sets A; otherwise, it is recurrent, in which case the walk revisits any nonempty open set infinitely often. Thus, it is only for transient random walks that (4.1.2) can be used to define a point process, which we shall call the random walk point process. In R1 , it is known that a random walk is transient if the mean E(X) is finite and nonzero; if E(X) exists but E(X) = 0, the random walk is recurrent. If the expectation is not defined (the integral diverges), examples of both kinds can occur. In R2 , the random walk can be transient even if E(X) = 0, but only if the variance is infinite. In higher dimensions, every random walk is transient unless perhaps it is concentrated on a one- or two-dimensional subspace. Proofs and further details are given, for example, in Feller (1966). Most of the renewal equation results also carry over to this context with only nominal changes of statement but often more difficult proofs. Thus, the expectation or renewal measure may still be defined as in (4.1.4), namely U (A) = δ0 (A) +

∞

F k∗ {A},

(4.1.4 )

k=1

and is finite for bounded Borel sets whenever the random walk is transient (but not otherwise, at least if A has nonempty interior). Furthermore, if z(x) is bounded, measurable, and vanishes outside a bounded set, we may consider the function ∞

Z0 (x) = z(x) + z(x − y) F k∗ (dy) = z(x − y) U (dy), (4.1.13) k=1

Rd

Rd

which is then a solution, bounded on finite intervals, of the generalized renewal equation Z(x) = z(x) + Rd

Z(x − y) F (dy).

(4.1.14)

4.1.

Basic Properties

71

Note that in (4.1.8) we were constrained not only to distributions F (·) concentrated on the half-line but also to functions z(x) and solutions Z(x) that could be taken as zero for x < 0. Without such constraints, the proof of uniqueness becomes considerably more subtle: one possible approach is outlined in Exercise 4.1.4. Note too that both (4.1.13) and (4.1.14) remain valid on replacing the argument x by a bounded Borel set A, provided Z(·) is then a set function uniformly bounded under translation for such A. Example 4.1(d) Random walks with symmetric stable distributions. Here we define the symmetric stable distributions to be those distributions in R with characteristic functions of the form 0 < α ≤ 2.

φα (s) = exp(−c|s|α )

Let us consider the associated random walks for the cases α ≤ 1 for which the first moment does not exist. The case α = 1 corresponds to the Cauchy distribution with density function for some finite positive c f (x) =

c π(c2 + x2 )

(−∞ < x < ∞).

The nth convolution is again a Cauchy distribution with parameter cn = nc. If the renewal measure were well defined, we would expect it to have a renewal density ∞ ∞

cn 1 u(x) = f n∗ (x) = . 2 2 π n=1 c n + x2 n=1 The individual terms are O(n−1 ) as n → ∞, so the series diverges. It follows readily that the first-moment measure is infinite, so the associated random walk is recurrent. For α < 1, it is difficult to obtain a convenient explicit form for the density, but standard results for stable distributions imply that f n∗ and f differ only by a scale factor, fαn∗ (x) = n−1/α fα (xn−1/α ), so that, assuming fα is continuous at zero, fαn∗ (x) ∼ xn−1/α fα (0). Thus, the series is convergent for 1/α > 1 (i.e. for α < 1), and divergent otherwise, so the associated random walk is transient only for α < 1. Example 4.1(e) A renewal process in two dimensions. We consider independent pairs (Xn , Yn ) where each pair has a bivariate exponential distribution with density vanishing except for x ≥ 0, y ≥ 0, where f (x, y) =

λ1 λ2 exp 1−ρ

λ1 x + λ2 y 1−ρ

I0

2(ρλ1 λ2 xy)1/2 , 1−ρ

72


λ1 , λ2 , and ρ are positive constants, 0 ≤ ρ < 1, and In (x) is the modified Bessel function of order n defined by the series In (x) =

∞

(x/2)2k+n . k! (k + n)!

(4.1.15)

k=0

The marginal distributions are exponential with parameters λ1 , λ2 ; ρ is the correlation between X1 and Y1 ; and the joint distribution has bivariate Laplace–Stieltjes transform ψ(θ, φ) = {(1 + θ/λ1 )(1 + φ/λ2 ) − ρθφ/λ1 λ2 }−1 . Much as in the one-dimensional case, the renewal function can be defined as U (x, y) = E(#{n: Sn ≤ x, Tn ≤ y}), n where Sn = k=1 Xk and Tn = k=1 Yk and has Laplace–Stieltjes transform χ(θ, φ) given by 1 . χ(θ, φ) = 1 − ψ(θ, φ) n

Substituting for ψ(θ, φ) and simplifying, we obtain χ(θ, φ) − 1 = [θ/λ1 + φ/λ2 + (1 − ρ)θφ/λ1 λ2 ]−1 , corresponding to the renewal density λ 1 x + λ2 y 2(λ1 λ2 xy)1/2 λ1 λ2 exp − I0 u(x, y) = 1−ρ 1−ρ 1−ρ

(x > 0, y > 0).

It should be noted that while the renewal density has uniform marginals, corresponding to the fact that each marginal process is Poisson, the bivariate renewal density is far from uniform, and in fact as x → ∞ and y → ∞, it becomes relatively more and more intensely peaked around the line λ1 x = λ2 y, as one might anticipate from the central limit theorem. The example is taken from Hunter (1974a, b), where more general results can be found together with a bibliography of earlier papers on bivariate renewal processes. See also Exercise 4.1.5.

Exercises and Complements to Section 4.1 4.1.1 (a) Using a sandwich argument and the strong law of large numbers for the i.i.d. sequence of lifetimes, prove that N (x)/x → λ a.s. as x → ∞. (b) Deduce from (a) the Elementary Renewal Theorem: The renewal function U (x) satisfies U (x)/x → λ as x → ∞, i.e. U (x) ∼ λx. [Hint: See Smith (1958) and Doob (1948). This is not the only possible proof.] (c) Similarly, if the lifetime distribution has finite second moment with variance σ 2 , deduce √ from the central limit theorem for the Xn that as x → ∞, (N (x) − λx)/λσ λx converges in distribution to a standard N (0, 1) random variable. [Hint: N (x) ≥ n if and only if Sn ≤ x, and if n, x → ∞ √ such that (x − n/λ)/(σ n ) → z for finite z, then λx/n → 1.]

4.1.

Basic Properties

73

4.1.2 Higher moments of the number of renewals. (a) Show that for 0 < x < y < ∞, E[N (dx) N (dy)] = U (dx) U (dy − x), where U is the renewal measure. Similarly, for any finite sequence 0 < x1 < x2 < · · · < xk < ∞, E[N (dx1 ) · · · N (dxk )] = U (dx1 ) U (dx2 − x1 ) · · · U (dxk − xk−1 ). [These are differential forms for the moment measures. When the densities exist, they reduce to the moment or product densities as discussed in Chapter 5; see, in particular, Example 5.4(b).] [k]

(b) Prove directly that E[(N (0, x]) ] ≤ k! [U0 (x)]k < ∞, where n[k] = n(n − 1) · · · (n − k + 1) and U0 (x) = U (x) − 1. (c) In terms of the renewal function U (x), use (a) to show that E[(N [0, x])2 ] = U (x) + 2

x

U0 (x − y) dU (y) 0−

and hence that when the renewal process is simple,

x

[U0 (x − y) − U0 (y)] dU0 (y).

var N [0, x] = var N (0, x] = U0 (x) + 2 0+

Check that in the case of a Poisson process at rate λ, E[(N [0, x])2 ] = 1 + 3λx + λ2 x2 and var N (0, x] = λx. 4.1.3 Let Q(z; x) =

∞

n=0

z n Pr{N [0, x] ≥ n}. Show that

x

Q(z; x − y) dF (y)

Q(z; x) = 1 + z 0

and hence that the Laplace–Stieltjes transform is given by

θ) = Q(z;

∞

e−θx dx Q(z; x) =

0−

1 , 1 − zψ(θ)

where ψ(θ) is the Laplace–Stieltjes of F . Obtain corresponding ∞ transform n results for the p.g.f. P (z; x) = z Pr{N [0, x] = n}. Deduce that the n=0 factorial moment E[(N [0, x])[k] ] is the k-fold convolution of U (x) − 1. 4.1.4 For the one-dimensional random walk with nonlattice step distribution F , prove that the only bounded measurable solutions of the equation

∞

D(x − y) F (dy)

D(x) = −∞

are constant. An outline of one method nis as follows. (1◦ ) Let Yn = D(−Sn ), where Sn = i=1 X1 . Use the equation to show that for any bounded measurable solution D, the random variables {Yn } constitute a bounded martingale (see Appendix 3) and hence converge a.s. to some limit random variable Y∞ . (2◦ ) Since Y∞ is defined on the tail σ-algebra of the i.i.d. sequence {Xn }, it must be degenerate; that is, Y∞ = c for some finite real number c.

74

4. Renewal Processes (3◦ ) Since for all X1 independent of Sn , D(−X1 − Sn ) =d D(−Sn+1 ) → c a.s., deduce that E(D(−X1 − Sn ) | X1 ) → c and hence, using the equation again, that D(−X1 ) = c a.s., whence also D(−Sn ) = c a.s. for n = 1, 2, . . . . Thus, finally, D(x) = c a.e. whenever X has a nonlattice distribution. [Hint: See Doob, Snell and Williamson (1960); for an alternative proof, see Feller (1966, Section XI.2), and for a review, see Rao and Shanbhag (1986).]

4.1.5 Two-dimensional renewal process. In the context n of Example 4.1(e), n let N (x, y) = #{n : Sn ≤ x, Tn ≤ y}, where Sn = X and T = Y, i n i=1 i=1 i and put Q(z; x, y) =

n

z n Pr{N (x, y) ≥ n},

n=0

P (z; x, y) =

∞

z n Pr{N (x, y) = n}.

n=0

Extend the result of Exercise 4.1.3 to show that the double Laplace–Stieltjes transform of P (z; x, y) is given by 1 − ψ(θ, φ) , P˜ (z; θ, φ) = 1 − zψ(θ, φ)

∞

ψ(θ, φ) = 0

∞

e−θx−φy dx,y F (x, y).

0

For the particular bivariate exponential distribution in Example 4.1(e), the ∞ n∗ renewal measure has the density f , where for x, y > 0, n=1

n−1 f

n∗

(x, y) = f (x, y)

ζ ρ

In−1 (2ζ/(1 − ρ)) , I0 (2ζ/(1 − ρ))

ζ=

ρλ1 λ2 xy .

4.2. Stationarity and Recurrence Times A modified or delayed renewal process, {Sn } say, is defined much as in (4.1.1) but with X1 replaced by X1 , which is independent of, but not necessarily identically distributed with, the remaining variables X2 , X3 , . . . . Let F1 (x) = Pr{X1 ≤ x}. Then, in terms of a forward recurrence time r.v. Tu for a renewal process as in Example 4.1(b), the forward recurrence time r.v. Tu for such a process {Sn } is defined by Tu = inf{Sn : Sn > u} − u and satisfies % X1 − u if X1 > u, Tu =d (4.2.1) Tu−X1 otherwise, hence (see (4.1.10)) Pr{Tu > y} = 1 − F1 (y + u) +

u

Pr{Tu−v > y} dF1 (v). 0

(4.2.2)

4.2.

Stationarity and Recurrence Times

75

The most important delayed renewal process arises when X1 has the probability density function f1 (x) = λ 1 − F (x) x ≥ 0, λ−1 = E(X) , (4.2.3) for then the resulting point process in (0, ∞), with counting measure N (A) = #{n: Sn ∈ A}, is stationary, as we might anticipate from (3.4.16) and Example 4.1(c). Note that here we are dealing with stationarity on the half-line, in the sense that Definition 3.2.I is required to hold only for Borel subsets of (0, ∞) and for shifts t ≥ 0. To establish this stationarity property more formally, define another delayed renewal process, {Sn } say, with initial lifetime r.v. X1 = Tu that is followed by a further sequence of i.i.d. random variables with common d.f. F . Stationarity of {Sn } is proved by showing that the distributions of the two sequences {Sn } and {Sn } coincide. From the assumed independence and distributional properties, it is enough to show that the distributions of the two initial intervals X1 and X1 coincide; i.e. Pr{X1 > y} = Pr{Tu > y} for all nonnegative u and y. Using (4.2.2) and (4.1.11), Pr{Tu > y} equals u u−v ∞ [1 − F (x)] dx + [1 − F (y + u − v − w)] dU (w) λ[1 − F (v)] dv, λ 0

y+u

0−

(4.2.4) and the last term here equals u u−w λ dU (w) 1 − F (v) 1 − F (y + u − v − w) dv 0−

0

1 − F (u − w − v) 1 − F (y + v) dv 0 0 u−v u 1 − F (y + v) dv 1 − F (u − v − w) dU (w) =λ 0 0 u =λ 1 − F (y + v) dv, using (4.1.12). =λ

u

dU (w)

u−w

0

Substituting back in (4.2.4) and simplifying leads by (4.2.3) to Pr{Tu > y} ∞ = λ y [1 − F (x)] dx = Pr{X1 > y}, as required. These remarks prove the first part of the following proposition (see Exercise 4.2.2 for an alternative proof of this part). Proposition 4.2.I. If the lifetime d.f. has finite first moment λ−1 , then the delayed renewal process with initial density (4.2.3) is stationary, and for all u > 0 the forward recurrence time Tu has this density. If the mean of the lifetime distribution is infinite, then no delayed renewal process with this lifetime distribution can be stationary. Proof. To prove the last statement, start by noting from the key renewal theorem, proved later in Proposition 4.4.II, that the forward recurrence time r.v. Tu for a renewal process {Sn } whose lifetime distribution has infinite

76


mean satisfies (see also Example 4.4(a)) for every finite y,

lim Pr{Tu ≤ y} = 0.

u→∞

Then, by dominated convergence, letting u → ∞ in (4.2.2) shows that, irrespective of the distribution F1 of X1 , Pr{Tu > y} → 1 for every y, so no stationary form for the distribution of Tu is possible. The intuitive interpretation of the last somewhat paradoxical limit statement is that if λ−1 = ∞, we shall spend an ever greater proportion of time traversing intervals of exceptional length and find ourselves in a situation where the current interval has a length greater than y still to run. Now recall from Exercise 3.4.1 the definition of a backward recurrence time r.v. Bu as a companion to the forward recurrence time r.v. Tu : Tu = inf{y: N (u, u + y] > 0},

Bu = inf{x: N (u − x, u] > 0}.

(4.2.5)

Note that there is an asymmetry in the definitions of Bu and Tu : because N (·) is a.s. finite on bounded intervals, Tu > 0 a.s. but it is quite possible to have Pr{Bu = 0} > 0. The current lifetime r.v. Lu can then be defined by Lu ≡ Bu + Tu . The joint distribution of any two of these r.v.s thus gives the distribution of all three: the simplest is that of Bu and Tu for which, when N (·) is stationary and orderly, Pr{Bu > x, Tu > y} = Pr{N (u − x, u + y] = 0} = Pr{N (u, u + x + y] = 0} ∞ 1 − F (v) dv. = Pr{Tu > x + y} = λ

(4.2.6)

x+y

Note that under stationarity and orderliness, Bu has the same marginal d.f. as Tu , while z Pr{Tu > z − x, Bu ∈ (x, x + dx)} + Pr{Bu > z} Pr{Lu > z} = 0 z ∞ = λ 1 − F (x + z − x) dx + λ 1 − F (v) dv 0 z ∞ =λ 1 − F (max(v, z)) dv. (4.2.7) 0

Thus, ELu = 2ETu = 2EBu = λEX 2 = EX 2 /EX ≥ EX,

(4.2.8)

with equality only in the case where X = EX a.s.; that is, all lifetimes are equal to the same constant, when the renewal process is variously called a deterministic renewal process or a process of equidistant points. By identifying 1 − F (·) with q0 (·) in (3.4.9), equations (4.2.6–8) continue to hold for any stationary orderly point process as discussed in Section 3.4.

4.2.

Stationarity and Recurrence Times

77

Without the assumption of stationarity, we may use the alternative definition for Bu , Bu = u − sup{Sn : Sn ≤ u} (u ≥ 0}. Arguing as in (4.1.10), it is not difficult to show (see Exercise 4.2.1) that for the basic renewal process {Sn }, (u−x)+ 1 − F (u + y − v) dU (v). (4.2.9) Pr{Bu > x, Tu > y} = 0

In the case of a Poisson process, we have F (x) = 1 − e−λx , and it is then not difficult to check from these relations that EX < ∞ and the distribution of Tu is independent of u; EX < ∞ and Bu and Tu are independent for each u > 0;

(4.2.10b)

ETu < ∞ (all u) and is independent of u.

(4.2.10c)

(4.2.10a)

Properties such as (4.2.10) have been used to characterize the Poisson process amongst renewal processes, as detailed in part in Galambos and Kotz (1978). For example, when ETu < ∞, integration of (4.1.10) shows that u ∞ ETu = 1 − F (y) dy + E(Tu−v ) dF (v), 0

u

so that when (4.2.10c) holds, 1 − F (u) ETu = 1 − F (u) ET0 =

∞

1 − F (y) dy

(all u > 0).

u

Thus, F (y) = 1−c e−λy for some constant c = 1−F (0+); since F (0+) = 0 for an orderly renewal process, c = 1. The proof of the rest of Proposition 4.2.II is indicated in Exercises 4.2.3–4. Proposition 4.2.II. Any one of the statements (4.2.10a), (4.2.10b), and (4.2.10c) characterizes the Poisson process amongst orderly renewal processes.

Exercises and Complements to Section 4.2 4.2.1 By following the argument leading to (4.2.3), show that for an orderly renewal process N (·) for which N ({0}) = 1 a.s., Pr{Bu > x, Tu > y} = Pr{N (u − x, u + y] = 0}

(u−x)+

[1 − F (y + u − v)] dU (v),

= 0− u

[1 − F ( max(z, u − v))] dU (v).

Pr{Lu > z} = 0−

4.2.2 Suppose that the delayed renewal process {Sn } with counting function N (·) and lifetime distribution F (·) with finite mean λ−1 is stationary. Show that X1 must have the density (4.2.3). [Hint: Stationarity implies that EN (0, x] = λx (all x > 0); now use Example 4.1(c).] 4.2.3 Use (4.1.10) to show that (4.2.10a) characterizes the Poisson process among orderly renewal processes.

78


4.2.4 Use (4.2.9) with x ↑ u to deduce that when (4.2.10b) holds, Pr{Tu > y} =

1 − F (y + u) 1 − F (u)

for each u and y ≥ 0. Consequently, for all v in the support of U (·),

[1 − F (0+)][1 − F (y + v)] = [1 − F (y)][1 − F (v)], so that F (·) is either geometric or exponential. If F (x) is constant for 0 < x < δ, then Bu and Tu cannot be independent—hence the characterization in Proposition 4.2.II via (4.2.10b). 4.2.5 For a renewal process with lifetime d.f. F (x) = 1 − (1 + µx)e−µx , evaluate the renewal function as U (x) = 1 + 12 µx − 14 (1 − e−2µx ) and hence derive the d.f.s of the forward and backward recurrence time r.v.s Tu and Bu . Verify their asymptotic properties for u → ∞.

4.3. Operations and Characterizations Because a single d.f. F suffices to describe a renewal or stationary renewal process, it is of interest to ask in various contexts involving the manipulation of point processes what conditions lead again to a renewal process as a result of the transformation or operation concerned. More often than not, the solution to such a question is a characterization of the Poisson process, a conclusion that can be disappointing when it might otherwise be hoped that more general renewal processes could be realized. Roughly speaking, when such a Poisson process characterization solution holds, it indicates that the interval independence property of a renewal process can be preserved only as a corollary of the stronger lack-of-memory property of the Poisson process. We have already given examples of characterizations of the Poisson process in Proposition 4.2.II. The three operations considered in this section concern thinning, superposition, and infinite divisibility. Example 4.3(a) Thinning of renewal processes. Given a renewal process {Sn }, let each point Sn for n = 1, 2, . . . be omitted from the sequence with probability 1 − α and retained with probability α for some constant α in 0 < α < 1, each such point Sn being treated independently. This independence property means that if {Sn(r) , r = 1, 2, . . .} is the sequence of retained points with 0 = n(0) < n(1) < n(2) < . . . , then Nr ≡ n(r) − n(r − 1) is a family of i.i.d. positive integer-valued r.v.s with Pr{Nr = j} = α(1 − α)j−1 for j = 1, 2, . . . , and hence {Yr } ≡ {Sn(r) − Sn(r−1) } (4.3.1)

4.3.

Operations and Characterizations

79

is a family of i.i.d. r.v.s with d.f. Pr{Yr ≤ x} =

∞

α(1 − α)j−1 F j∗ (x).

j=1

Consequently, {Sn(r) } is still a renewal process, and it is not hard to verify that its renewal function, Uα say, is related to that of {Sn } by rescaling as in (4.3.2) Uα (x) − 1 = α U (x) − 1 . It is readily seen that whenever {Nr } here is a family of i.i.d. positive integer-valued r.v.s, {Sn(r) } is a renewal process, but it is only for the geometric distribution for Nr that (4.3.2) holds. In connection with this equation, the converse question can be asked as to when it can be taken as defining a renewal function for α > 1. In general, for a given renewal function U , there is a finite largest α ≥ 1 for which 1 + α(U (x) − 1) is a renewal function, although there is a class of lifetime d.f.s, including the exponential and others besides, for which 1 + α(U (x) − 1) is a renewal function for all finite positive α [Daley (1965); see also van Harn (1978) and Exercise 4.3.1]. Any renewal function U satisfies U (x)/λx → 1 as x → ∞, and consequently the renewal function Uα of the thinned renewal process {Sn(r) }, when rescaled so as to have the same mean lifetime, becomes Uαs , say, defined by (α ↓ 0). Uαs (x) − 1 = α U (x/α) − 1 → λx Thus, if Uαs is independent of α, it must equal the renewal function of a Poisson process, which is therefore the only renewal process whose renewal function is preserved under thinning and rescaling, i.e. Uαs = U (all 0 < α < 1). Example 4.3(b) Superposition of renewal processes. Let N1 , . . . , Nr be independent nontrivial stationary renewal processes. When is the superposed process N = N1 + · · · + N r (4.3.3) again a renewal process? Certainly, N is a renewal process (indeed a Poisson process) when each of the components N1 , . . . , Nr is a Poisson process. Conversely, since by Raikov’s theorem (e.g. Lukacs, 1970) independent random variables can have their sum Poisson-distributed only if every component of the sum is Poisson-distributed also, it follows from writing N (A) = N1 (A) + · · · + Nr (A) (all Borel sets A) and appealing to Renyi’s characterization in Theorem 2.3.II that if N is a Poisson process, then so also is each Nj . Because a renewal process is characterized by its renewal function, and this is linear only if the process is Poisson, one way of proving each of the two assertions below is to show that the renewal function concerned is linear. Proposition 4.3.I. A stationary renewal process is the superposition of two independent nontrivial stationary renewal processes only if the processes are Poisson.

80


Proposition 4.3.II. A stationary renewal process is the superposition of r ≥ 2 independent identically distributed stationary renewal processes only if the processes are Poisson. Proof. We start by allowing the renewal processes Nj to have possibly different lifetime d.f.s Fj , denoting each mean by λ−1 j , so by Proposition 4.1.I, each λj is finite and positive. Write λ = λ1 +· · ·+λr , pj = λj /λ, πj = Fj (0+), and π = F (0+), where F is the lifetime d.f. of the superposed process N . For any such renewal process, we have, for small h > 0 and |z| ≤ 1, E z N (0,h) = 1 −

λh(1 − z) + o(h) (1 − π)(1 − zπ) r r = 1− E z Nj (0,h) = j=1

j=1

λj h(1 − z) + o(h) . (1 − πj )(1 − zπj )

It follows by equating powers of z that for i = 1, 2, . . . , lim Pr{N (0, h] = i | N (0, h] > 0} = π i−1 (1 − π) = (1 − π)λ−1 h↓0

r

λj πji−1 .

j=1

All these equations can hold for nonzero π and πj (and nonzero λ) only if π = πj for j = 1, . . . , r; that is, only if all renewal processes concerned have the same probability of zero lifetimes. Consequently, it is enough to establish the propositions in the orderly case, which we assume to hold from here on. In place of the renewal function U in (4.1.5), we use H(x) =

∞

F n∗ (x),

so H(x) = λx for a Poisson process.

(4.3.4)

n=1

Then, from (3.5.3), for a stationary renewal process N , x var N (0, x) = var N (0, x] = λ [2H(u) + 1] du − (λx)2 0 x 2[H(u) − λu] + 1 du ≡ V (x) =λ 0

and thus

cov N [−x, 0), N (0, y] = 12 V (x + y) − V (x) − V (y) y =λ G(x + u) − G(u) du, 0

where G(x) = H(x)−λx. It is convenient to write below, for r.v.s Y for which the limits exist, E0 (Y ) = lim E(Y | N (0, h] > 0). h↓0

Since pj = limh↓0 Pr{Nj (0, h] > 0 | N (0, h] > 0},

4.3.

Operations and Characterizations

81

H(x) = E0 (N (0, x] | N ({0}) > 0) r r &

Pr{Nj (−h, 0] > 0}[1 + o(1)] & = lim E0 Ni (0, x] & Nj ({0}) > 0 h→0 Pr{N (−h, 0] > 0} j=1 i=1 r

= pj Hj (x) + pj λi x , (4.3.5) j=1

so G(x) =

i =j

r

pj Gj (x). Similar, somewhat lengthier, algebra leads to G(x, y) ≡ lim E0 N (−x, 0) − λx N (0, y) − λy | N ({0}) > 0 h→0 y r r

2 G(x + u) − G(u) − pj Gj (x, y) + λ pj Gj (x + u) − Gj (u) du. = j=1

j=1

0

j=1

Thus, when N1 , . . . , Nr are identically distributed, pj = 1/r, Gj (x) = G1 (x) (all j), and G1 (x) = G(x). Also, for a renewal process, G(x, y) = G(x)G(y), so y G(x + u) − G(u) du. G(x)G(y) = G(x)G(y) + λ(1 − 1/r) 0

It follows that G(x + y) = G(y) = G(0) (all x, y > 0). Thus, H(x) = λx, and Proposition 4.3.II is proved. On the other hand, for r = 2 and possibly different F1 and F2 , replacing G(x, y) by G(x)G(y) with G(x) = p1 G1 (x) + p2 G2 (x), p1 + p2 = 1, leads to −p1 p2 G1 (x) − G2 (x) G1 (y) − G2 (y) y G1 (x + u) + G2 (x + u) − G1 (u) − G2 (u) du. = λp1 p2 0

The function K(y) ≡ G1 (y) − G2 (y) thus has a right-derivative k(·) given by −K(x)k(y) = λ G1 (x + y) + G2 (x + y) − G1 (y) − G2 (y) . Either K(x) = 0, in which case G1 = G2 and the earlier argument shows that G(x) = 0, or else by letting y ↓ 0 and using G1 (0) = G2 (0) = 0, it follows that G1 (x) is proportional to G2 (x), with G1 (x) having the derivative g1 (x), say. Consequently, g1 (x)g1 (y) = αg1 (x + y) for some nonzero α, so g1 (x) = αe−βx for some 0 < β < ∞ because G1 (x)/x → 0 as x → ∞. Transform calculus now shows that each 1 − Fj (u) = e−bj u . An earlier version of Proposition 4.3.I is in McFadden and Weissblum (1963), and a different proof is in Mecke (1969). Another argument is used in Mecke (1967) to prove the following result (the proof is omitted here). Proposition 4.3.III. Let the stationary renewal process N be the superposition of the independent stationary point processes N1 and N2 with N1 renewal. If the lifetime d.f.s F and F1 of N and N1 have density functions that are continuous on (0, ∞) and right-continuous at 0, then N1 is a Poisson process.

82


By taking N1 to be Poisson with rate parameter λ and N2 to be an alternating renewal process with exponential distributions for the alternating lifetime d.f.s, their parameters α and β being such that λ2 = αβ, Daley (1973a) furnished an example showing that Mecke’s result cannot characterize N2 as a Poisson process. If only the differentiability assumptions could be omitted, the restriction in Proposition 4.3.II that the components Nj of the sum N at (4.3.3) should be identically distributed could be dropped. Example 4.3(c) Infinite divisibility. A natural complement to Example 4.3(b) is to ask whether there are any stationary renewal processes other than the Poisson that are infinitely divisible. Here we ask whether for (any or all) integers r, the stationary renewal process N in (4.3.3) is expressible as the superposition of i.i.d. stationary point processes N1 , . . . , Nr . Assuming that the lifetime distribution concerned has a density function, [MKM] state that H¨ aberlund (1975) proved that the Poisson process is the only one, while under the additional assumption of the existence of density functions for all the joint distributions of the component process N1 , Ito (1980) has asserted the stronger result that if N is expressible as N = N1 + · · · + Nr for one integer r ≥ 2, then it is Poisson and hence infinitely divisible. There are innumerable characterizations of the exponential distribution and Poisson processes (see reviews in Galambos and Kotz (1978) and Johnson and Kotz (1994, Section 19.8)). Fosam and Shanbhag (1997) has a useful list of papers exploiting variants of the Choquet–Deny functional equation approach.

Exercises and Complements to Section 4.3 4.3.1 (a) When F (x) = 1 − (1 + x)e−x , show (e.g. by using Laplace–Stieltjes transforms) that 1 + α(U (x) − 1) is a renewal function if and only if 0 < α ≤ 1. (b) Let {X(t): t ≥ 0} be a stochastic process with X(0) = 0 and stationary nonnegative independent increments, with Lévy–Khinchin representation E(e−θX(t) ) = etψ(θ) , where

ψ(θ) = −θµ0 +

(e−θx − 1) µ(dx),

(0,∞)

with µ0 ≥ 0 and µ(·) a nonnegative measure on (0, ∞) satisfying

min(x, 1) µ(dx) < ∞, and µ(0, ∞) = ∞ if µ0 = 0. Let 0 = t0 < t1 < · · · be the successive epochs of a Poisson process in (0, ∞) with intensity so that the r.v.s X(tn ) − X(tn−1 ) are i.i.d. with d.f. F (x) = unit ∞ F (x, t)e−t dt, where F (x, t) = Pr{X(t) ≤ x}. Show that with U (·) the 0 renewal function corresponding to F and U0 (x) = U (x) − 1, 1 + αU0 (x) is a renewal function for all 0 < α < ∞, and that U0 (x) is subadditive (see Kingman, 1972, p. 100). (0,∞)

4.3.2 Let the stationary point process N1 arise as the jump epochs of a Markov process on countable state space, and let N2 be a stationary Poisson process independent of N1 . Daley (1975b) showed that for N ≡ N1 + N2 to be a stationary renewal process different from Poisson, not only must the Markov chain transition rates underlying N1 have a particular structure but also there is a unique rate λ for N2 for which N can have the renewal property.

4.4.

Renewal Theorems

83

4.4. Renewal Theorems Considerable effort has been expended in the mathematics of renewal theory on establishing Theorem 4.4.I below and its equivalents; they are stronger statements than the elementary renewal theorem [i.e. the property U (x) ∼ λx given in Exercise 4.1.1(b) of which there is a generalization in (3.5.3)]. Theorem 4.4.I is variously known as Blackwell’s renewal theorem or the key renewal theorem, depending basically on how it is formulated. Theorem 4.4.I (Blackwell’s Renewal Theorem). For fixed positive y, restricted to finite multiples of the span of the lattice when the lifetime d.f. is lattice, and otherwise arbitrary, U (x + y) − U (x) → λy

(x → ∞).

(4.4.1)

Equation (4.4.1) says roughly that the renewal measure ultimately behaves like a multiple of Lebesgue measure. To make this more precise, let St U denote the shifted version of the renewal measure U so that St U (A) = U (t + A). Then (4.4.1) implies that on any finite interval (0, M ), St U converges weakly to the multiple λ of Lebesgue measure (·) (or, equivalently, St U as a whole converges vaguely to λ; see Section A2.3 for definitions and discussion of weak and vague convergence). Blackwell’s theorem represents the ‘set’ form of the criterion for weak convergence, while the key renewal theorem (Theorem 4.4.II below) represents a strengthened version of the corresponding ‘function’ form, the strengthening taking advantage of the special character of the limit measure and its approximants. On the other hand, the theorem is not so strong as to assert anything concerning a density u(·) for U . Such results require further assumptions about the lifetime distributions and are explored, together with further strengthenings of Blackwell’s theorem, following Theorem 4.4.II. Proof of Theorem 4.4.I. The proof given here is probabilistic and uses a coupling method [see Lindvall (1977, 1992) and Thorisson (2000, Section 2.8)]. We compare each sample path {Sn } with the sample path {Sn } of a stationary renewal process as defined in Section 4.2, {Sn } and {Sn } being defined on a common probability space (Ω, F, P ) so as to be mutually independent. For each ω ∈ Ω, and every integer i ≥ 0, define for {Sn } the forward recurrence time r.v.s Zi ω = TS i (ω) so that Zi (ω) = min{Sj (ω) − Si (ω): Sj (ω) > Si (ω)}. Because the sequence {Si+n − Si } has a distribution independent of i and is independent of {Sn }, and because Tu is stationary, it follows that the sequence {Zi } is also stationary. Thus, the events Ai ≡ {Zj < δ for some j ≥ i},

84


which we define for any fixed δ > 0, have the same probability for each i = 0, 1, . . . , and in particular therefore P (A0 ) = P (A∞ ), where A0 ⊇ A1 ⊇ · · · ⊇ A∞ ≡

∞

Ai = {Zi < δ i.o.}.

i=1

Now A∞ is a tail event on the conditional σ-field (namely, conditional on X1 ) of the i.i.d. r.v.s {X1 , X1 , X2 , X2 , . . .} and therefore by the zero–one law for tail events (see e.g. Feller, 1966, Section IV.6), for -a.e. x, P (A∞ | X1 = x) = 0 or 1

(0 < x < ∞).

Because F is nonlattice, P {u − x < Sj − X1 0 (see Feller, 1966, Section V.4a, Lemma 2), and hence P (A0 | X1 = x) > 0 for every x. Thus, the equations ∞ 0 0, P {Zi < δ for some i} = 1. To establish (4.4.1), it is enough to show that, for any δ > 0, we can find x0 such that x ≥ x0 implies that |EN (x, x + y] − λy| ≤ δ. Observe that λy = EN (x, x + y], where N is the counting function for the stationary renewal process with intervals {Xn }. Let Iδ = inf{i: Zi < δ}, so that P {Iδ < ∞} = 1. Defining J ≡ inf{j: Sj (ω) > SIδ (ω)}, we then have 0 < ZIδ (ω) = SJ (ω) − SIδ (ω) < δ. Define a new point process by means of the sequence of intervals {X1 , . . . , XIδ , XJ+1 , XJ+2 , . . .},

and denote its counting function by N so that for any Borel set A, N (A) = N A ∩ (0, SIδ ) + N (A + ZIδ ) ∩ (Sj , ∞) = N A ∩ (0, SIδ ) + N (A + ZIδ ) − N (A + ZIδ ) ∩ (0, Sj ) . When A is the interval (x, x+y], the shifted interval A+ZIδ has EN (A+ZIδ ) lying between λ(y − δ) and λ(y + δ) because (x + δ, x + y] ⊆ (x + ZIδ , x + y + ZIδ ] ⊆ (x, x + y + δ]. For every x, the r.v.s N (x, x + y] are stochastically dominated by the r.v. 1 + N (0, y], and since this has finite expectation, {N (x, x + y]: x ≥ 0} is a

4.4.

Renewal Theorems

85

uniformly integrable family of r.v.s. This ensures that as x → ∞ E N (x, x + y]I{x<SIδ } → 0 since then P {x < SIδ } → 0. Similarly, N (x+ZIδ , x+y +ZIδ ] is stochastically dominated by 1 + N (0, y] and P {x < Sj } → 0 as x → ∞, so E(N (x + ZIδ , x + y + ZIδ ]I{x<Sj } ) → 0. Consequently, for x sufficiently large, U (x + y) − U (x) = EN (x, x + y] is arbitrarily close to EN (A + ZIδ ), and since δ is arbitrarily positive, (4.4.1) is established. We now turn to an equivalent but very important form of Theorem 4.4.I for nonlattice lifetimes. A function g(·) defined on [0, ∞) is directly Riemann integrable there when, for any h > 0, the normalized sums ∞ ∞

h h h g− (nh) and h g+ (nh) n=1

n=1

converge to a common finite limit as h → 0; here, h g− (x) = inf g(x − δ),

h g+ (x) = sup g(x − δ).

0≤δ≤h

0≤δ≤h

Exercise 4.4.1 states sufficient conditions for g to be directly Riemann integrable. For such a function, with U (x) ≡ 0 for x < 0 and monotonically increasing on x ≥ 0, x ∞ ≤ h g(x − y) dU (y) g± (nh) U x − (n − 1)h − U (x − nh) . ≥ 0 n=1 These sums can be truncated to finite sums with truncation error bounded by x−C |g(x − y)| dU (y) 0

[x−C]

≤

|g|1+ (C + n) U (x + 1 − C − n) − U (x − C − n)

n=1

≤ U (1)

∞

|g|1+ (C + n),

n=1

which can be made arbitrarily small, uniformly in x > 0, by taking C sufficiently large. Thus, the sums are approximated by x [C/h] ≤ h g(x − y) dU (y) g (nh)[U (x − nh + h) − U (x − nh)] ≥ n=1 ± x−C

[C/h]

→ λh

h g± (nh)

(x → ∞)

n=1 C

→ λ

g(u) du 0

(h → 0).

86


The following equivalent form of Theorem 4.4.I can now be given. Theorem 4.4.II (Key Renewal Theorem). For nonlattice lifetime distributions and directly Riemann integrable functions g(·), x ∞ g(x − y) dU (y) → λ g(y) dy (x → ∞). (4.4.2) 0

0

Some results for monotonically decreasing but not necessarily integrable functions g(·) are sketched in Exercise 4.4.5(c). The following examples may serve as prototypes for the application of the renewal theorem to problems of convergence to equilibrium. Example 4.4(a) Convergence of the forward recurrence time distribution. Our starting point is (4.1.11), which after subtracting from (4.1.12) can be written u Fu (y) ≡ Pr{Tu ≤ y} = [F (y + u − v) − F (u − v)] dU (v). (4.4.3) 0−

This is in the form (4.4.2) with g(x) = F (y + x) − F (x). This function is integrable and of bounded variation over the whole half-line; it then follows easily (see Exercise 4.4.1) that the function is directly Riemann integrable, so that the theorem can be applied. It asserts that, provided the lifetime distribution is nonlattice, ∞ y Fu (y) → λ [F (y + x) − F (x)] dx = λ [1 − F (v)] dv (u → ∞). 0

0

If λ−1 < ∞, this is the usual form of the length-biased distribution associated with F , the fact that the distribution is proper following from the identity ∞ 1 = λ 0 1 − F (v) dv. In this case, (4.4.2) asserts directly that the forward recurrence time distribution converges weakly to its limit form. The extension of this result to a delayed renewal process with arbitrary initial distribution follows then from (4.4.4). When λ−1 = ∞, Fu (y) → 0 for all y and no stationary form can exist. Example 4.4(b) Convergence of the renewal density. As a further corollary, we shall prove (see Feller, 1966, Section XI.4) that if the lifetime distribution F has finite mean and bounded density f (t), then U (t) has density u(t) such that u(t) − f (t) → λ. (4.4.4) This follows from the fact that u(t), when it exists, satisfies the renewal equation in its traditional form t u(t) = f (t) + u(t − x)f (x) dx. 0

[To check this, note that equation (4.1.9) implies that the solution has the form s t u(s) = 0 f (s − x) dU (x), which on integrating yields 0 u(s) ds = U (t) − 1.]

4.4.

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87

Moreover, the function u(t) − f (t) =

∞

f k∗ (t)

k=2

satisfies the renewal equation u(t) − f (t) = f (t) +

t

[u(t − x) − f (t − x)]f (x) dx.

2∗

(4.4.5)

0

Now if f (t) is bounded, f 2∗ (t) is directly Riemann integrable. Indeed, as the convolution of a bounded and an integrable function, it is uniformly continuous (Exercise 4.4.2), while the inequality

t/2

t

f (t − y)f (y) dy +

f 2∗ (t) = 0

=2 0

f (t − y)f (y) dy t/2

t/2

f (t − y)f (y) dy ≤ 2C[1 − F ( 12 t)],

where C = sup |f (t)|, shows that when µ = λ−1 < ∞, f 2∗ (t) is also bounded above by an integrable monotonic function and is therefore directly Riemann integrable by Exercise 4.4.1(c). Thus, Proposition 4.4.II applies, yielding (4.4.4). The argument can be extended to the case where, if not f itself, at least one of its convolution powers has bounded density (see Exercise 4.4.3). Even a partial assumption of absolute continuity allows the conclusions of the renewal theorems to be substantially strengthened—for example, from local weak convergence of the renewal measure to local convergence in variation norm, namely St U − λM → 0, (4.4.6) where µM is the variation norm of the (signed) measure µ over [0, M ]. Equation (4.4.6) would imply that, in Blackwell’s theorem, U (t + A) → λ(A) not only for A an interval, as in (4.4.1), but for any bounded Borel A, a strengthening considered by Breiman (1965) [see Feller (1966, Section XI.1) for counterexamples]. An appropriate condition is embodied in the following definition. Definition 4.4.III. A probability distribution F is spread out if there exists a positive integer n0 such that F n0 ∗ has a nonzero absolutely continuous component with respect to Lebesgue measure. The definition implies that F n0 ∗ can be written in the form F n0 ∗ = Σ + A,

(4.4.7)

88


where Σ is singular and A is absolutely continuous with respect to Lebesgue measure, and A has a nonzero density a(x), so that ∞ σ = Σ = 1 − a(x) dx < 1. 0

Since the convolution of A with any power of F or Σ is again absolutely continuous, it follows that the total masses of the absolutely continuous components F n∗ can only increase as n → ∞, and in fact must approach 1, since Σk∗ = σ k → 0. Thus, we might anticipate that the asymptotic behaviour of the renewal measure for a spread out distribution would approximate the behaviour to be expected when a density exists. This is the broad content of the following proposition (see Stone, 1966) from which our further results will follow as corollaries. Proposition 4.4.IV. Let F be spread out, U the renewal measure associated with F , and UG = G ∗ U the renewal measure associated with the corresponding delayed renewal process with initial distribution G. Then UG can be written in the form UG = U1G + U2G , (4.4.8) where U1G is absolutely continuous with density u1G (x) satisfying ∞ −1 λ = x dF (x), u1G (x) → λ,

(4.4.9)

0

and U2G is totally finite. Proof. Consider first the ordinary renewal measure U associated with F . Since the convolution of A with itself can always be taken to dominate a uniformly continuous function (Exercise 4.4.2), there is no loss of generality in supposing that the density a(x) of A in (4.4.6) is continuous, bounded, and vanishes outside some finite interval (0, M ). With this understanding, let U3 denote the renewal measure associated with the distribution F n0 ∗ so that we may write u3 = δ0 + F n0 ∗ + F 2n0 ∗ + · · · and

U = [δ0 + F + F 2∗ + · · · + F (n0 −1)∗ ] ∗ U3 = ρ ∗ U3 ,

where ρ has total mass n0 . Also, since U3 satisfies the renewal equation U3 = δ0 + F n0 ∗ ∗ U3 = δ0 + (Σ + A) ∗ U3 , we have U3 ∗ (δ0 − Σ) = δ0 + A ∗ U3 . Since δ0 − Σ has total mass less than unity, this factor may be inverted to yield U3 = Uσ + A ∗ Uσ ∗ U3 , 2∗

where Uσ = δ0 + Σ + Σ U , and then for UG ,

+ · · · has total mass (1 − σ)−1 . Thus, we obtain for

UG = G ∗ ρ ∗ Uσ + A ∗ G ∗ ρ ∗ Uσ ∗ U3 . This will serve as the required decomposition, with U2G = G ∗ ρ ∗ Uσ totally finite and U1G = A ∗ G ∗ ρ ∗ Uσ ∗ U3 absolutely continuous, since it is a

4.4.

Renewal Theorems

89

convolution in which one of the terms is absolutely continuous. To show that its density has the required properties, we note first that the key renewal theorem applies to U3 in the form ∞ λ (U3 ∗ g)(t) → g(x) dx n0 0 whenever g is directly Riemann integrable. But then a similar result applies also to H = G ∗ ρ ∗ Uσ ∗ U3 , which is simply a type of delayed renewal measure in which the initial ‘distribution’ G ∗ ρ ∗ Uσ has total mass 1 × n0 × (1 − σ)−1 , so that ∞ λ (H ∗ g)(t) → g(x) dx (t → ∞). 1−σ 0 Finally, since the density of A is continuous and vanishes outside a bounded set, we can take g(t) = a(t), in which case the left-hand side of the last equation reduces to u1G (t) and we obtain u1G (t) →

λ 1−σ

∞

a(x) dx = λ. 0

We have the following corollary (see Arjas, Nummelin and Tweedie, 1978). Corollary 4.4.V. If F is spread out and g ≥ 0 is bounded, integrable, and satisfies g(x) → 0 as x → ∞, then & & ∞ & & lim sup &&(UG ∗ f )(t) − λ f (x) dx&& → 0. (4.4.10) t→∞ |f |≤g

0

Proof. We consider separately the convolution of g with each of the two components in the decomposition (4.4.8) of UG . Taking first the a.c. component, and setting uG (x) = 0 for x < 0, we have & t & sup && u1G (t − x)f (x) dx − λ

|f |≤g

0

0

∞

& & f (x) dx&& ≤

∞

& & &u1G (t − x) − λ& g(x) dx.

0

Now u1G (t) → λ so it is bounded for sufficiently large t, |u1G (t) − λ| ≤ C say, for t > T , and we can write the last integral as 0

t−T

& & g(x) &u1G (t − x) − λ& dx +

T

& & &u1G (s) − λ& g(t − s) ds,

0

where the first integral tends to zero by dominated convergence because |u1G (t − x) − λ| is bounded, u1G (t − x) → λ for each fixed x, and g(x) is integrable, while the second tends to zero by dominated convergence since |u1G (s) − λ| has finite total mass over (0, T ) and by assumption g(t − s) → 0 for each fixed s.

90


Similarly, the integral against the second component is dominated for all |f | ≤ g by t g(t − x) dU2G (x), 0

where again the integrand is bounded and tends to zero for each fixed x, while U2G has finite total mass, so the integral tends to zero by dominated convergence. Corollary 4.4.VI. If F is spread out, then for each finite interval (0, M ) St UG − λM → 0. The version of the renewal theorem summarized by these results has the double advantage of not only strengthening the form of convergence but also replacing the rather awkward condition of direct Riemann integrability by the simpler conditions of Proposition 4.4.IV. Further variants are discussed in Exercise 4.4.4 and in the paper by Arjas et al. (1978). With further conditions on the lifetime distributions—for example, the existence of moments—it is possible to obtain bounds on the rate of convergence in the renewal theorem. For results of this type, see Stone (1966), Schäl (1971), and Bretagnolle and Dacunha-Castelle (1967); for a very simple case, see Exercise 4.4.5(a).

Exercises and Complements to Section 4.4 4.4.1 Conditions for direct Riemann integrability. Let z(x) be a measurable function defined on [0, ∞). Show that each of the following conditions is sufficient to make z(·) directly Riemann integrable (see also Feller, 1966). (a) z(x) is nonnegative, monotonically decreasing, and Lebesgue integrable. (b) z(x) is continuous, and setting αn = supn<x≤n+1 |z(x)|, Σαn < ∞. [Hint: z(x) is Riemann integrable on any finite interval, and the remainder term outside this interval provides a contribution that tends to zero.] (c) z(x) ≥ 0, z(x) is uniformly continuous and bounded above by a monotonically decreasing integrable function. 4.4.2 (a) If g is bounded and continuous and f is integrable, then their convolution product f ∗ g = R g(t − x)f (x) dx is uniformly continuous. (b) Extend this to the case where g is any bounded measurable function by approximating g by bounded continuous functions. In particular, therefore, f (t − x) dx is uniformly continuous whenever A is a measurable set. A (c) Let F have a.c. component f ; show from (b) that F ∗ F has an a.c. component f2 , which dominates a uniformly continuous function and hence a bounded function that vanishes outside a bounded set and is twice continuously differentiable. 4.4.3 Apply the key renewal theorem as around (4.4.5) to show that if F has density f with f k∗ bounded, and if λ−1 < ∞, then the renewal density u(x) exists and satisfies

2k−1

u(x) −

j=1

f j∗ (x) → λ.

4.4.

Renewal Theorems

91

2k−1

∞

f j∗ (x) satisfies the renewal equation [Hint: u(x) − j=1 f j∗ (x) = j=2k 2k∗ with z(x) = f (x), which is uniformly continuous and bounded above by an integrable function. Necessary and sufficient conditions for u(x) itself to converge are given in Smith (1962); see also Feller (1966, Section XI.4).] 4.4.4 Strong convergence counterexample. Let Gu denote the distribution of the forward recurrence time at t = u and G∞ its limit, if it exists, of a renewal process N (·) with lifetime distribution F with mean 1/λ. (a) Suppose that xF has discrete support but is nonlattice. Show that Gu (x) → G∞ (x) = λ 0 [1−F (u)] du, but that Gu −G∞ = 2 (all finite u). [Hence, Gu does not converge in variation norm · , i.e. strong convergence fails.] (b) Show that Gu − G∞ → 0 (u → ∞) when F is spread out. 4.4.5 Rate of convergence in renewal theorems. ∞ (a) Consider (4.1.8) with z(t) = λ t F (y) dy, where F (y) = 1 − F (y) and F has second moment σ 2 + µ2 . Deduce that Z, the solution of (4.1.8) with such z, equals φ(t) ≡ U (t) − λt. Use the key renewal theorem to conclude that for nonlattice F ,

t

∞

0 ≤ φ(t) = λ 0

F (v) dv dU (u) → 12 λ2 (σ 2 + µ2 )

(0 ≤ t → ∞).

t−u

(b) Let the r.v.s T1 , T2 be independent with Pr{T1 > t} = z(t) as in (a). Use the subadditivity of the renewal function U (·) to give, for all t ≥ 0, U (2t) ≤ 2EU (t + T1 − T2 ), and hence deduce from EU (t−T1 ) = λt (cf. Example 4.1(c) and Proposition 4.2.I) that 2λt ≤ U (2t) ≤ 2λt + λ2 σ 2 + 1. [See Carlsson and Nerman (1986) for details and earlier references.] (c) Suppose that the generator z(·) in the general renewal equation (4.1.8) is t positive and decreases monotonically. Show that J1 (t) ≡ 0 z(u)λ du → ∞

t

(t → ∞) if and only if J2 (t) ≡ 0 z(t − u) dU (u) → ∞ (t → ∞) and that then limt→∞ J1 (t)/J2 (t) = 1. that, when F (·) has infinite second moment, U (t) − λt ∼ ∞Deduce 2 λ min(v, t)F (v) dv ≡ G(t) (Sgibnev, 1981). 0 ∞ For an alternative proof, show that φ(t) ≤ 0 U (min(v, t))λF (v) dv ≡ GU (t) ≥ G(t) by the elementary renewal theorem. Use Blackwell’s theorem to show that lim supt→∞ GU (t)/G(t) ≤ 1. When F (·) has finite second moment and is nonarithmetic, show that limt→∞ [J1 (t) − J2 (t)] = 0. (d) Use the asymptotics of φ(·) to deduce that for a stationary orderly renewal process N (·), var N (0, t] ∼ (var λX)(λt) when t the lifetime d.f. has finite second moment, and var N (0, t] ∼ λ2 t2 − λ3 0 (t − v)2 F (v) dv otherwise. [Hint: First, find var N (0, t] from (3.5.2) and (3.5.6).]

92


4.5. Neighbours of the Renewal Process: Wold Processes The specification of a renewal process via independent identically distributed intervals raises the possibility of specifying other point processes via intervals that are one step removed from independence. In this section, we consider point processes for which the successive intervals {Xn } form a Markov chain so that the distribution of Xn+1 given Xn , Xn−1 , . . . in fact depends only on Xn . Such processes seem to have been considered first by Wold (1948); accordingly, we call them Wold processes. Example 4.5(a) A first-order exponential autoregressive process. Suppose that the family {Xn } of intervals satisfy the relation Xn+1 = ρXn + n

(4.5.1)

for some 0 ≤ ρ < 1 and family {n } of i.i.d. nonnegative random variables (note {Xn } is itself i.i.d. if ρ = 0). For the particular distribution given by Pr{n = 0} = ρ

and

Pr{n > y} = (1 − ρ)e−y

(y > 0),

taking Laplace transforms of (4.5.1) shows that if a stationary sequence of intervals is to exist, the common distribution F of the {Xn } must have its Laplace–Stieltjes transform F satisfy the functional equation F(ρs)(1 + ρs) F(s) = . 1+s The only solution of this equation for which F(0) = F(0+) = 1 is F(s) = (1 + s)−1 . Thus, a stationary version of the Markov chain exists and the marginal distribution for the intervals is exponential as for a Poisson process. The parameter ρ controls the degree of association between the intervals. For ρ > 0, a realization of the process consists of a sequence of intervals each one of which is an exact fraction of the preceding one, followed by an interval independently chosen from the same exponential distribution. The construction can be extended to more general types of gamma distribution and has been studied extensively by P.A.W. Lewis and co-authors: see, for example, Gaver and Lewis (1980). They have advocated its use as an alternative to the Poisson process, partly on the grounds of the very simple behaviour of the spectrum of the interval process. Other aspects are more intractable, however, and from a point process viewpoint its partly deterministic behaviour gives it a rather special character (see Exercises 4.5.2 and 4.5.9). In general, the interval structure of a Wold process is determined by a Markov transition kernel P (x, A); that is, a family {P (x, ·): 0 ≤ x < ∞} of probability measures in [0, ∞), and the distribution, P0 (·) say, of the initial interval X0 , with P (·, A) measurable for each fixed Borel set A ⊆ [0, ∞). When the chain {Xn } is irreducible [see e.g. Harris (1956), Orey (1971) or Meyn and

4.5.

Neighbours of the Renewal Process: Wold Processes

93

Tweedie (1993) for discussions of the precise meaning of irreducibility] and admits a stationary distribution, π(·) say, so that for all such Borel subsets A ∞ π(A) = P (x, A) π(dx), (4.5.2) 0−

an interval sequence {Xn } with a stationary distribution can be specified. The following construction then leads to a counting process N (·) that is stationary in the sense of Definition 3.2.I. First, let {X0 , X1 , . . .} be a realization of the Markov chain for which X0 has the initial distribution x π(dx) P0 (dx) ≡ Pr{X0 ∈ (x, x + dx)} = ∞ , u π(du) 0−

(4.5.3a)

where we suppose both π{0} = 0 and finiteness of the normalizing factor; i.e. ∞ ∞ λ−1 ≡ x π(dx) = π(u, ∞) du < ∞. (4.5.3b) 0−

0

Next, conditional on X0 , let X0 be uniformly distributed on (0, X0 ), and determine N by N (0, x] = #{n: Sn ≤ x}, where

S1 = X0 ,

Sn+1 = Sn + Xn

(n = 1, 2, . . .).

The relation (4.5.3), in conjunction with the definition of Sn , states that the origin is located uniformly at random within an interval selected according to the length-biased distribution with increment around x proportional to x π(dx). Since π{0} = 0, the normalizing constant λ is just the intensity of the process. Note that the distributions here are consistent with the relations found in Exercise 3.4.1 for the stationary distributions for the forward recurrence time and the length of the current interval. Indeed, the construction here can be rephrased usefully in terms of the bivariate, continuous-time Markov process X(t) = L(t), R(t) , (4.5.4) where L(t) is the length of the interval containing t and R(t) is the forward recurrence time at time t. The Markovian character of X(t) follows readily from that of the sequence of intervals. Moreover, it is clear that the process N (t) is uniquely determined by X(t) and vice versa. By starting the Markov process with its stationary distribution, we ensure that it remains stationary in its further evolution, and the same property then holds for the point process. An immediate point of contrast to the ordinary point process is that it is not necessary, in (4.5.2), to have R+ π(dx) < ∞. If the underlying Markov chain is null recurrent, a stationary regime can exist for the point process (though not for its intervals) in which, because of the dependence between the lengths

94


of successive intervals, long runs of very short intervals intervene between the occurrences of longer intervals; in such situations, divergence of R+ π(dx) can coexist with convergence of R+ x π(dx) (i.e. near the origin, π may integrate x but not 1). This leads to the possibility of constructing stationary Wold processes with infinite intensity but finite mean interval length. One such construction is given in Daley (1982); another is outlined in Exercise 4.5.1. With such examples in mind, it is evident that the problem of formulating analogues of the renewal theorems for the Wold process needs to be approached with some care. One possible approach is through the family of renewal measures U (A | x) = E[#{n: Sn ∈ A} | X0 = x] and their associated cumulative processes U (t | x) ≡ U ([0, t] | x). The latter functions satisfy the renewal-type equations ∞ U (t | x) = I{t≥x} (t) + U (t − x | y) P (x, dy). (4.5.5) 0

Unfortunately, these equations seem rather intractable in general. The analogy with the renewal equations of Section 4.4 becomes clearer on taking Laplace–Stieltjes transforms of (4.5.5) with respect to t. Introducing the integral operator Tθ with kernel tθ (dy, x) = e−θx P (x, dy), the transform versions of equation (4.5.5) become ∞ e−θt U (dt | x) = e−θx + (Tθ Uθ )(x) Uθ (x) ≡ 0

with the formal solution Uθ = (1 − Tθ )−1 eθ , where (eθ )(x) ≡ e−θx , which may be compared to equation (4.1.6). Example 4.5(b) Discrete Wold processes. Consider a simple point process ({0, 1}-valued process) on the lattice of integers {0, 1, . . .}; the kernel P (x, dy) here becomes a matrix pij and in place of the cumulative form#in (4.5.5) it is more$ natural to consider the renewal functions u(j | i) = Pr N {j} = 1 | X0 = i . Then ∞

u(j | i) = δij + pik u(j − i | k), k=1

taking the right-hand ∞side here to be zero for j < i. By introducing the transforms ui (z) = k=i z k u(k | i), these become ui (z) = z i +

∞

pik z i uk (z),

k=1

or in matrix-vector form u(z) = ζ + Pz u(z),

4.5.


95

where Pz = {pik z i }, u(z) = {ui (z)}, and ζ = (1, z, z 2 , . . .). The asymptotic behaviour of u(j | i) as j → ∞ is therefore related to the behaviour of the resolvent-type matrix (I − Pz )−1 as z → 1. When P is finite, this can be discussed in classical eigenvector/eigenvalue terms; see Exercise 4.5.4 and for further details Vere-Jones (1975). A particular question that arises relates to periodicity of the process: nonzero values of u(j | i) may be restricted to a sublattice of the integers. This phenomenon is not directly related to periodicity of the underlying Markov chain; again, see Exercise 4.5.4 for some examples. A more general approach, which can be extended to the denumerable case and anticipates the general discussion to be given below, is to consider the discrete version of the Markov chain X(t) in (4.5.4). When this bivariate chain is aperiodic and recurrent, returns to any given state pair—for example, time points at which an interval of specified length i0 is just commencing— constitute an imbedded renewal process for X(t) and allow standard renewal theory results to be applied. Example 4.5(c) Transition kernels specified by a diagonal expansion. Lancaster (1963) investigates the class of bivariate probability densities that can be represented by an expansion of the kind ! " ∞

f (x, y) = fX (x)fY (y) 1 + ρn Ln (x)Mn (y) , n=1

where fX (·), fY (·) are the marginal densities and Ln (x), Mn (y) are families of complete orthonormal functions defined with respect to the marginal distributions fX (·), fY (·), respectively. When fX and fY coincide (so Ln = Mn ), the bivariate density can be used to define the density of the transition kernel of a stationary Markov chain with specified stationary distribution fX (x): just put ! " ∞

f (x, y) = fX (y) 1 + ρn Ln (x)Ln (y) . p(x, y) = fX (x) n=1 For many of the standard distributions, this leads to expansions in terms of classical orthogonal polynomials (see e.g. Tyan and Thomas, 1975). In particular, when fX (x) and fY (y) are both taken as gamma distributions, fX (x) = xα−1 e−x /Γ(α),

say,

the Ln (x) become the Laguerre polynomials of order α. The bivariate exponential density of Example 4.1(e) is a case in point when α = 1 and ρn = ρn . The resulting Wold process then has exponential intervals, but in contrast to Example 4.5(a), the realizations have no deterministic properties but simply appear as clustered groups of small or large intervals, the degree of clustering being controlled by the parameter ρ. Lampard (1968) describes an electrical counter system that produces correlated exponential intervals. More gener-

96


ally, when α = 12 d, such correlated gamma distributions can be simulated from bivariate normal distributions with random variables in common; this leads to the possibility of simulating Wold processes with correlated gamma intervals starting from a sequence of i.i.d. normal variates (see Exercise 4.5.7). Even in such a favourable situation, the analytic study of the renewal functions remains relatively intractable. Lai (1978) studies the exponential case in detail and provides a perturbation expansion for the renewal function and (count) spectral density of the process in terms of the parameter ρ. As such examples illustrate, explicit computations for the Wold process are often surprisingly difficult. However, a useful and general approach to the asymptotic results can be developed by identifying a sequence of regeneration points within the evolution of the process and by applying to this sequence the renewal theorems of Section 4.4. It is by no means obvious that any such sequence of regeneration points exists, but the ‘splitting’ techniques developed for Markov chains with general state space by Nummelin (1978) and Athreya and Ney (1978) allow such a sequence to be constructed for a wide class of examples. The essence of this idea is to identify a particular set A0 in the state space and a particular distribution φ on A0 such that whenever the process enters A0 , it has a certain probability of doing so ‘according to φ’, when its future evolution will be just the same as when it last entered A0 ‘according to φ’. In effect, returns to A0 according to φ can be treated as if they are returns to a fixed atom in the state space and provide the regeneration points we seek. The following conditions summarize the requirements on the transition kernel for this to be possible (see Athreya and Ney, 1978). Conditions 4.5.I. (Regenerative Homing Set Conditions). For the Markov chain {Xn } on state space S ⊆ [0, ∞) ≡ R+ , there exists a homing set A0 ∈ B(R+ ), A0 ⊆ S, a probability measure φ on A0 , and a positive constant c such that for all x ∈ S, (i) Pr{Xn ∈ A0 for some n = 1, 2, . . . | X0 = x} = 1; and (ii) for every Borel subset B of A0 , P (x, B) ≥ cφ(B). The first of these conditions embodies a rather strong recurrence condition; indeed Athreya and Ney call a chain satisfying Condition 4.5.I ‘strongly aperiodic recurrent’ since the conditions imply aperiodicity as well as recurrence. The second condition is more akin to an absolute continuity requirement on the transition kernel. In particular, it is satisfied whenever the following simpler but more stringent condition holds. Condition 4.5.I . (ii) For all x ∈ A0 , P (x, B) has density p(x, y) on A0 with respect to φ such that p(x, y) ≥ c > 0 for all y ∈ A0 . Typically, A0 is a set with positive Lebesgue measure and φ the uniform distribution on A0 (i.e. a multiple of Lebesgue measure scaled to give A0 total mass unity). In the discrete case, 4.5.I(ii) is equivalent to the assumption that the matrix of transition probabilities has at least one positive diagonal element.

4.5.


97

Conditions 4.5.I are trivially satisfied in the independent (renewal) case if we take S to be the support of the lifetime distribution F and put A0 = S, φ = F , and c = 1. Under Conditions 4.5.I, Athreya and Ney (1978) show that the chain is recurrent in the sense of Harris (1956) and admits a unique finite invariant measure π(·). The important feature for our purposes is not so much the existence of the invariant measure as its relation to the sequence {νk } of ‘returns to A0 according to φ’. This aspect is made explicit in the following proposition [see Athreya and Ney (1978) and Nummelin (1978) for proof]. Proposition 4.5.II. Conditions 4.5.I imply that for the Markov chain {Xn }, (a) there exists a stopping time ν ≥ 1 with respect to the σ-fields generated by {Xn } such that for Borel subsets B of A0 Pr{Xν ∈ B | X0 · · · Xν−1 ; ν} = φ(B); (b) {Xn } has an invariant measure π(·) related to φ by ! ν−1 "

all B ∈ B(R+ ) , IB (Xn ) π(B) = Eφ

(4.5.6)

(4.5.7)

n=0

where Eφ refers to expectations under the initial condition that X0 has distribution φ on A0 , i.e. Pr{X0 ∈ B} = φ(B ∩ A0 ) for B ∈ B(R+ ). Equation (4.5.7) can be extended by linearity and approximation by simple functions to ! ν−1 "

f (x) π(dx) = Eφ f (Xn ) (4.5.8) R+

n=0

whenever f is Borel-measurable and either nonnegative or π-integrable. Special cases of (4.5.8) include Eφ (ν) = π(dx) (4.5.9a) R+

and Eφ (X0 + X1 + · · · + Xν−1 ) =

x π(dx).

(4.5.9b)

R+

n Now let Sn = i=1 Xi , and let {Tk } = {Sνk − 1} denote the sequence of times at which the process returns to A0 according to φ. These Tk form the regeneration points that we seek. If G(·) denotes the distribution function of the successive differences Tk − Tk−1 so that in particular G(u) = Eφ {ISν−1 ≤ u} = Prφ {Sν−1 ≤ u},

(4.5.10)

then the Tk form the instants of a renewal process with lifetime distribution G. We apply this fact, with the theorems of Section 4.4, to determine the asymptotic behaviour of the Wold process.

98


The results are stated for the renewal function Uφ (C × Tt B) = Eφ #{n: Xn ∈ C, Sn ∈ Tt B},

(4.5.11)

where Tt B is the translate of B through time t. If the process is started from a general distribution κ for X0 , we write Uκ (·) for the corresponding renewal function. The analogue of Blackwell’s renewal theorem for this function reads, for B = (0, h) and λ as in (4.5.3b), Uφ (C × Tt B) → λπ(C)(B). We approach these results through an extended version of the key renewal theorem, fixing a bounded measurable function h(x, y) with support in the positive quadrant x ≥ 0, y ≥ 0, and setting for t > 0 Z(t) = Eφ

! N (t)

" h(Xn , t − Sn )

∞ t

= 0

n=0

h(x, t − u) Uφ (dx × du). (4.5.12)

0

Considering the time T1 to the first return to A0 according to φ, we find that t Z(t) satisfies the renewal equation Z(t) = z(t) + 0 Z(t − u) dG(x), where z(t) = Eφ

! ν−1

"

h(Xn , t − Sn )

T

h(XN (u) , t − u) dN (u) . (4.5.13)

= Eφ 0

n=0

If then we can show that z(t) satisfies the condition of direct Riemann integrability (for Feller’s form of the key renewal theorem in 4.4.II) or the conditions in 4.4.III for the Breiman form of the theorem, we shall be able to assert that Z(t) → λ

∞

(t → ∞).

z(t) dt 0

To evaluate the integral, we make use of (4.5.8) so that formally

∞

∞

z(t) dt = 0

Eφ 0

" h(Xn , t − Sn ) dt

n=0

= Eφ

! ν−1

! ν−1

n=0

∞ ∞

∞

" h(Xn , t − Sn ) dt

Sn

h(x, t) π(dx) dt,

= 0

= Eφ

! ν−1

n=0

"

∞

h(Xn , u) du

0

(4.5.14)

0

the formal operations being justified by Fubini’s theorem whenever h ≥ 0 or h is (π × )-integrable.

4.5.


99

Direct Riemann integrability can be established directly in simple cases, to which we add the following general sufficient condition. For δ > 0, any α in 0 ≤ α < δ, and Ij (δ) ≡ (jδ, (j + 1)δ], define mδ (x, α) =

∞

sup h(x, t)

and

mδ (x) = sup mδ (x, α), 0≤α 0. If G is spread out, the same result holds for B any bounded Borel set. If π(·) is totally finite, these results hold without any further condition on A. We next extend the results to an arbitrary initial distribution, κ say, for X0 . If we denote the corresponding renewal functions by Uκ , Zκ , then Zκ satisfies t

Zφ (t − u) G(du)

Zκ (t) = zκ (t) +

(4.5.19)

0

with zκ (t) = Eκ

! ν −1

" h(Xn , t

−

Sn )

,

(4.5.20)

n=0

where X1 , Sn refer to the sequence of interval lengths and renewals for the process with initial distribution κ, and ν is the time of the first entry to A0 according to φ, again starting from X0 distributed according to κ. It follows

4.5.


101

from Condition 4.5.I(i) that this entry is certain, so ν is finite with probability 1. It then follows from (4.5.19) that Zκ (t) − Zφ (t) = zκ (t) − zφ (t) so that we need conditions to ensure the convergence of the right-hand side to zero. This will follow from (4.5.20) if Eκ (ν ) < ∞ and h is bounded and satisfies (4.5.16a). Corollary 4.5.V. Suppose that (4.5.17) holds for Uφ and that κ is an arbitrary initial distribution for X0 . Then (4.5.17) continues to hold with Uκ in place of Uφ if and only if zκ (t) − zφ (t) → 0, in particular if h is bounded and satisfies (4.5.16a), and Eκ (ν ) < ∞, Eφ (ν) = R+ π(dx) < ∞. Finally, we turn to the question of the weak convergence of the process X(t) in (4.5.4). It somewhat simplifies the algebraic details to work with the bivariate process Y(t) = (L(t), L(t) − R(t)), i.e. with the backward recurrence time L(t) − R(t) in place of the forward one. If then ξ(x, y) is any bounded continuous function of x, y in R+ × R+ , we consider ξ(Y(t)), which we may write in the form ∞ ξ Y(t) = h(Ln , t − Sn ), n=0

where

ξ(x, t) (0 ≤ t ≤ x), 0 (t > x), since in fact only the term with n = N (t) contributes to the sum. Suppose first that G is nonlattice, and define the modulus of continuity ω(x, δ) of h(·) by ω(x, δ) = sup sup |h(x, t) − h(x, t + u)|. h(x, t) =

0≤t≤x−δ 0≤u≤δ

Then, for the particular choice of h given above, mδ (x) − mδ (x) ≤ (x/δ)ω(x, δ)

so that δ

R+

[mδ (x) − mδ (x)] π(dx) ≤

x ω(x, δ) π(dx). R+

For each fixed x > 0, h(x, t) is continuous and nonvanishing on a finite closed interval so it is uniformly continuous, and hence ω(x, δ) → 0. Also, ω(x, δ) is uniformly bounded in x and δ, so by dominated convergence, the integral on the right converges to zero as δ → 0; that is, (4.5.15) holds. Also, & & |zκ (t)| ≤ Eκ &ξ Y(t) &; T > t ≤ CPκ {T > t}, where the last term tends to zero from the recurrence property assumed in Condition 4.5.I(i). Consequently, the conditions for Corollary 4.5.V hold. If, furthermore, G is spread out, then this result alone is sufficient to ensure the truth of the Riemann-type theorem. This means the continuity condition on ξ can be dropped, implying that the weak convergence of Y(t) to its limit can be replaced by convergence in variation norm.

102


Proposition 4.5.VI. Let Pκ,t denote the distribution of X(t) supposing X0 has initial distribution κ, and π∞ the stationary distribution for X(t) with elementary mass λ π(dx) dy over the region 0 ≤ y ≤ x < ∞. If G is nonlattice and λ−1 = R+ x π(dx) < ∞, then Pκ,t → π∞ weakly. If, furthermore, G is spread out, then Pκ,t → π∞ in variation norm. Throughout our discussion, we have assumed finiteness of the mean λ−1 [see (4.5.3b)]. When the mean is infinite, further types of behaviour are possible, some of which are sketched in Athreya, Tweedie and Vere-Jones (1980).

Exercises and Complements to Section 4.5 4.5.1 A Wold process with infinite intensity. Consider a symmetric random walk {Xn } with reflecting barrier at the origin, supposing the walk to have density and be null recurrent; for example, the single-step distribution could be N (0, 1). Then, the invariant measure for Xn is Lebesgue measure on (0, ∞). Now transform the state space by setting Yn = T (Xn ), where for y > 0 x = T −1 (y) = y −β (1 + y)−α

(α > 0, β > 0);

note that under T the origin is mapped into the point at infinity and vice versa. Then, the transformed process Yn is Markovian with invariant measure having density π(y), where near the origin π(y) ∼ y −(1+β) and near infinity π(y) ∼ ∞ y −(α+β+1) . Choose α and β so that 0 < β < 1, α+β > 1; then 0 y π(y) dy
0),

0

where 0,∞) min(x, 1) M (dx) < ∞. Show that there exists a stationary sequence {Xn }, satisfying the autoregressive equation Xn+1 = ρXn + n

( n independent of Xn )

and having marginal distribution with Laplace–Stieltjes transform ψ(θ), whenever M is absolutely continuous with monotonically decreasing density m(x), hence in particular whenever the Xn are gamma distributed. [Hint: If n is also infinitely divisible, its Laplace–Stieltjes transform, φ(θ) say, ∞ must satisfy φ(θ) = ψ(θ)/ψ(ρθ) = exp ( 0 (e−θx − 1) [M (dx) − M (ρ−1 dx)]).] 4.5.3 Let F (t; x, y) be the distribution function of the bivariate process Y(t) = (L(t), L(t)−R(t)), conditional on an event at the origin and L(0−) = s. Then, if F has a density f (t; x, y) ≡ f (t; x, y | s), it satisfies for 0 < y < min(x, t) ∂F ∂F + = ∂t ∂y

t

f (t; u, u)P (u, (0, x]) du − 0

y

f (t; u, u) du, 0

and if also the density function is sufficiently regular, then for the same x, y, t, ∂f ∂f + = 0. ∂t ∂y

4.5.


103

Argue on probabilistic grounds that f (t; x, y) = f (t−v; x, y−v) for 0 < y−v < min(x, t − v), so f (t; x, x) = f (t − x; x, 0+) for 0 < x < t, and that

t

f (t; x, 0+) = p(s, t)p(t, x) +

f (t; u, u)p(u, x) du.

(4.5.21)

0

When the p.d.f.s p(u, x) are independent of u, this reduces to the renewal density function equation. Assuming that the conditions for the limits of Theorem 4.5.III and its corollaries are satisfied, identify f (x, y) ≡ limt→∞ f (t; x, y) with the density function π(x) for the stationary measure π(·) of the theorem, and deduce the density version of equation (4.5.2) by taking the limit in (4.5.21). Now t let L(0−) ∞ be an r.v. with p.d.f. λsπ(s) with λ as in the theorem. Interpret 0 dx 0 yf (t; x, y | s)λsπ(s) ds as the density of the expectation function U (·) of the Wold process. [Lai (1978) has other discussion and references.] 4.5.4 Discrete Wold processes. (a) Suppose integer-valued intervals are generated by a finite Markov chain on {1, 2, 3} with transition matrices of the forms

⎧ ⎫ ⎧ 0 1 0⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ (i) P = ⎪ ; (ii) P = 0 0 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ 21 1

0

0

2

0 1 2 1 2

⎫ ⎧ 0 12 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0⎪ ; (iii) P = ⎪ 1 0 ⎪ ⎪ ⎭ ⎩ 1⎪

0

1

0

⎫ ⎪ ⎪ ⎪ 0⎪ . ⎪ ⎪ ⎭ 1 2

0

For which of these P do the corresponding Wold processes show lattice behaviour? What is the relation of periodicity of P to lattice behaviour of the associated Wold process? (b) Define mij (n) = Pr{interval of length j starts at n | X0 = i} and show that, for n ≥ 0, mij (n) = δij δ0n +

mik (n − k)pkj = δij δ0n +

k

pik mkj (n − i),

k

where we interpret mij (n) = 0 for n < 0. In matrix form, the p.g.f.s are given by % ) M(z) = {m ij (z)} ≡

∞

mij (n)z n

−1

= (I − H(z))

,

n=0

where H(z) = (hij (z)) ≡ (z i pij ). (c) If the Wold process is nonlattice and P is irreducible, (1 − z)[I − H(z)]−1 = λΠ + (1 − z)Q(z), where Π is the one-dimensional projection onto the null space of I − P and Q(z) is analytic within some disk |z| ≤ 1 + , > 0 (see Vere-Jones, 1975). 4.5.5 Denumerable discrete Wold processes. Consider the bivariate process X(n) = (L(n), R(n)) [or Y(n) = (L(n), L(n) − R(n))] as a Markov chain with an augmented space. Show that the Wold process is nonlattice if and only if this augmented chain is aperiodic, and that if the original Markov chain is

104

4. Renewal Processes positive recurrent with stationary distribution {πj }, having finite mean, the augmented chain X(n) is positive recurrent with stationary distribution

π(h, j) = Pr{Ln = j, Rn = h} = where λ−1 =

λπj

(h = 1, . . . , j),

0

otherwise,

jπj < ∞ as before.

4.5.6 Markov chains with kernels generated by a power diagonal expansion. (a) If {Xn } is generated by a kernel with the structure p(x, y) = f (y)

∞

ρn Ln (x)Ln (y)

n=1

for an orthogonal family of functions Ln (·), then the m-step transition kernel p(m) (x, y) is generated by a kernel with similar structure and ρ replaced by ρm = ρm . (b) In the particular case where f (·) is exponential and the {Ln (x)} are Laguerre polynomials, a key role is played by the Hille–Hardy formula ∞

Ln (x)Ln (y)ρn =

n=0

e−(x+y)ρ/(1+ρ) I0 1−ρ

2√xyρ 1−ρ

.

Use this to show the following [see Lai (1978) for details]: (i) Convergence to the stationary limit as m → ∞ is not uniform in x. h (ii) For every x > 0, the conditional d.f.s F (h | x) = 0 p(x, y) dy are bounded by a common function α(h), where α(h) < 1 for h < ∞. (iii) If A(θ) is the integral operator on L1 [0, ∞) with kernel p(x, y)e−θx , then for all θ with Re(θ) ≥ 0, θ = 0, A2 (θ) < 1, so the inverse [I − A(θ)]−1 exists and is defined by an absolutely convergent series of powers of A(θ). , Z1 , . . . be a 4.5.7 Simulation of Wold process with χ2 interval distribution. Let Z0 sequence of i.i.d. N (0, σ 2 ) variables; define successively Y1 = Z0 / 1 − ρ2 and Yi+1 = ρY1 + Zi (i = 1, 2, . . .). Then {Yi } is a stationary sequence of normal r.v.s with first-order autoregressive structure. Construct d independent realizations of such autocorrelated normal series, {Y1i , . . . , Ydi ; i = 1, 2, . . .} say, and generate a stationary sequence of autocorrelated gamma r.v.s {Xi } by setting Xi =

d

2 Yki

k=1 2

2

−1

so EXi = dσ /(1 − ρ ) ≡ λ , var Xi = 2dσ 4 /(1 − ρ2 )2 , and cov(Xi , Xi+1 ) = dσ 4 (1 + ρ2 )/(1 − ρ2 )2 . These Xi can be used as the intervals of a point process, but the process so obtained is not initially stationary: to obtain a stationary version, the length-biased distribution may be approximated by choosing T λ−1 , selecting a time origin uniformly on (0, T ) and taking the initial interval to be the one containing the origin so selected, and the subsequent intervals to be X1 , X2 and so on.

4.5.


105

4.5.8 Wold processes with intervals conditionally exponentially distributed. Let p(x, y) be of the form λ(x)e−λ(x)y . (a) When λ(x) = λx−1/2 , the marginal density π(x) can be found via Mellin transforms (Wold, 1948). (b) When λ(x) = λ + αx, the density π(x) is given by π(x) = c(λ + αx)−1 e−λx for finite c > 0 [see Cox (1955), Cox and Isham (1980, pp. 60–62), and Daley (1982); the model has a simple form of likelihood function and has been used to illustrate problems of inference for Poisson processes when the alternative is a Wold process, in particular of the type under discussion]. 4.5.9 Time-reversed exponential autoregression. Let the intervals Yn of a point process be stationary and satisfy Yn+1 = min(Yn /ρ, ηn ) for i.i.d. nonnegative ηn and 0 < ρ < 1. Show that when ηn is exponentially distributed, so also is Yn , with corr(Y0 , Yn ) = ρ|n| . Furthermore, {Yn } =d {X−n }, where Xn are as in Example 4.5(a) with Pr{ n > y} = (1 − ρ)e−y [see Chernick et al. (1988), where it is also shown that this identification of {Xn } as the time-reversed process of {Yn } characterizes the exponential distribution]. 4.5.10 Lampard’s reversible counter system [see Lampard (1968) and Takacs (1976)]. Consider a system with two counters, one of which is initially empty but accumulates particles according to a Poisson process of rate λ, the other of which has an initial content ξ0 + r particles and loses particles according to a Poisson process of rate µ until it is empty. At that point, the roles of the two counters are reversed; an additional r particles are added to the number ξ1 accumulated in the first counter, which then begins to lose particles at rate µ, while the second counter begins to accumulate particles again at rate λ. We take X0 , X1 , . . . to be the intervals between successive reversals of the counters. Then, the {Xi } form a Markov chain that has a stationary distribution if and only if µ > λ. 4.5.11 mth-order dependence. Suppose that the intervals {Xi } of a point process form an mth-order Markov chain. Then, in place of the process (L(t), R(t)), we may consider the process X(t) = (L−m+1 (t), . . . , L−1 (t), L(t), R(t)), where the state is defined as the set of m − 1 preceding intervals, the current interval, and the forward recurrence time. The regenerative homing set conditions can be applied to the discrete time vector process with state Un = (Xn−m+1 , . . . , Xn−1 , Xn ), which is Markovian in the simple sense. Establish analogues to Theorem 4.5.III and its corollaries. [See Chong (1981) for details.] 4.5.12 A non-Poisson process with exponentially distributed intervals. Let the intervals τ1 , τ2 , . . . of a point process on R+ be defined pairwise by i.i.d. pairs {(τ2n−1 , τ2n )}, n = 1, 2, . . . as follows. For each pair, the joint density function f (u, v) = e−u−v + f (u, v), where f (u, v) = 0 except for (u, v) in the

106

4. Renewal Processes set A = {0 < u < 2 and 2 < v < 4, or 0 < v < 2 and 2 < u < 4}, where it equals for u ∈ (0, 1) and v ∈ (2, 3); u ∈ (1, 2) and v ∈ (3, 4); v ∈ (0, 1) and u ∈ (3, 4); and v ∈ (1, 2) and u ∈ (2, 3); and f = − on the complement in A of these four unit squares. Check that τ2n−1 and τ2n are not independent, that each τi is exponentially distributed y with unit mean, and that every pair (τi , τi+1 ) has Pr{τi + τi+1 ≤ y} = 0 we−w dw. Conclude that for any k = 1, 2, . . . , the length of k consecutive intervals has the same distribution as for a Poisson process at unit rate and hence that N (a, b] for a < b is Poisson-distributed with mean b − a. [This counterexample to Theorem 2.3.II is due to Moran (1967).]

4.5.13 A stationary point process N with finite second moment is long-range dependent when var N (0, x] lim sup = ∞. x x→∞ (a) A renewal process is long-range dependent if and only if the lifetime distribution has infinite second moment (Teugels, 1968; Daley, 1999). (b) Construct an example of a stationary Wold process that is long-range dependent but for which the marginal distribution of intervals has finite second moment. [Daley, Rolski and Vesilo (2000) note two examples.]

4.6. Stieltjes-Integral Calculus and Hazard Measures The results in this section can be regarded as being a prelude to the general discussion of conditional intensities and compensators in Chapters 7 and 14. The simplest case concerns a renewal process whose lifetime distribution function F (·) is absolutely continuous with density f (·). An important role is played by the hazard function q(x) = f (x)/S(x) [see (1.1.3)], particularly in applications to forecasting because we can interpret q(x) as the risk of an event occurring in the next short time interval, given the time elapsed since the last renewal; that is, q(x) dt = Pr{event in t, t + dt | last event at t − x}. Example 4.6(a) Prediction of the time to the next event in a renewal process. Suppose a renewal process has hazard function q(·) as just described and that at time t the time back to the last event is observed to be x. Then, the distribution of the time to the next event has hazard function qx (y) = q(x + y)

(y ≥ 0),

corresponding to a d.f. with tail (i.e. conditional survivor function) Sx (y) = 1 − Fx (y) = exp

−

y

q(x + u) du 0

=

1 − F (x + y) . 1 − F (x)

4.6.

Stieltjes-Integral Calculus and Hazard Measures

107

Note that x here denotes an observation, and that for a stationary Poisson process, the risk qx (y) is everywhere constant. What of the nonabsolutely continuous case in this example? An appropriate extension of the hazard function is the hazard measure Q(·) in Definition 4.6.IV below. Our discussion of Q(·) is facilitated by two results for Lebesgue– Stieltjes integrals. The first is just the formula for integration by parts in the Lebesgue–Stieltjes calculus. The second is much more remarkable: it is the exponential formula, which has been used mainly in connection with martingale theory without its being in any sense a martingale result; it is in fact a straightforward (if unexpected) theorem in classical real analysis. Lemma 4.6.I (Integration-by-Parts Formula). Let F (x) and G(x) be monotonically increasing right-continuous functions of x ∈ R. Then b b F (x) dG(x) = F (b)G(b) − F (a)G(a) − G(x−) dF (x). (4.6.1) a

a

This is a standard result on Lebesgue–Stieltjes integrals; it can be proved directly from first principles or as an application of Fubini’s theorem (see e.g. Brémaud 1981, p. 336). Note that the last term of (4.6.1) contains the leftcontinuous function G(x−); also, recall the convention for Lebesgue–Stieltjes integrals that b ∞ u(x) dG(x) = I(a,b] (x)u(x) dG(x); −∞

a

if we wish to include the contribution from a jump of G at a itself, then we write the integral as b u(x) dG(x); similarly,

b− a

a−

u(x) dG(x) excludes the effect of any jump of G at b.

Lemma 4.6.II (Exponential Formula). Suppose F (x) is a monotonically increasing right-continuous function of x ∈ R and that u(x) is a measurable t function for which 0 |u(x)| dF (x) < ∞ for each t > 0. Let {xi } be the set of discontinuitiesof F in [0, ∞); set ∆F (xi ) = F (xi ) − F (xi −) and write Fc (x) = F (x) − 0<xi ≤t ∆F (xi ) for the continuous part of F (·). Then, the function t H(t) = H(0) exp 1 + u(xi )∆F (xi ) u(x) dFc (x) (4.6.2) 0

0<xi ≤t

is the unique solution in t ≥ 0 of the integral equation t H(t) = H(0) + H(x−)u(x) dF (x) 0

satisfying sup0≤s≤t |H(s)| < ∞ for each t > 0.

(4.6.3)

108


Proof. We outline a proof (see Brémaud, 1981, pp. 336–339; Andersen et al., 1993, Theorem II.6.1). Write 1 + u(xi ) ∆F (xi ) G1 (t) = H(0) 0<xi ≤t

and

G2 (t) = exp

t

u(x) dFc (x) .

0

Then, the relation between (4.6.2) and (4.6.3) is just an application of the integration-by-parts formula to obtain an expression for G1 (t)G2 (t), noting that G1 (·) increases by jumps only at the points t = xi , where in fact the jump is equal to G1 (xi ) − G1 (xi −) = 1 + u(xi ) G1 (xi −) − G1 (xi −) = u(xi )G1 (xi −). To show that (4.6.2) is the unique bounded solution to (4.6.3), let D(t) = H1 (t) − H2 (t) be the difference between any two bounded solutions. Then D(t) itself is bounded in every finite interval, and we can form the estimate, using (4.6.3) and for fixed finite s and t with 0 < s < t, s s |D(x−)| |u(x)| dF (x) ≤ M |u(x)| dF (x), |D(s)| ≤ 0

0

where M = sup0≤s≤t |D(s)|. Now feeding this estimate back into (4.6.3) yields s 2 s x M |D(s)| ≤ M |u(x)| dF (x) . |u(y)| dF (y) |u(x)| dF (x) ≤ 2 0 0 0 Evidently, this iteration may be continued and yields for general n ≥ 1 s n M |D(s)| ≤ |u(x)| dF (x) . n! 0 This last expression converges to zero as n → ∞, so D(s) ≡ 0. Corollary 4.6.III. Lemmas 4.6.I and 4.6.II remain true when the functions F and G are of bounded variation on finite intervals. Proof. For Lemma 4.6.I, use the fact that any function of bounded variation is the difference of two monotonically increasing right-continuous functions. For Lemma 4.6.II, observe that the argument depends only on the use of the formula for integration by parts and the estimate, for any bounded interval A, & & & & & &≤ u(x) dF (x) |u(x)| dVF (x), & & A

A

where VF is the total variation of F . We now specialize these results to the case where F is a distribution function of a positive random variable, so F (0+) = 0, F (∞) = limx→∞ F (x) ≤ 1.

4.6.

Stieltjes-Integral Calculus and Hazard Measures

109

Definition 4.6.IV. The hazard measure Q(·) associated with the distribution F on [0, ∞) is the measure on [0, ∞) for which Q(dx) =

F (dx) F (dx) = ; S(x−) 1 − F (x−)

in integrated form, the integrated hazard function (IHF) is the function t dF (x) . Q(t) = 0 1 − F (x−) In the case where F has a density f , we have simply t Q(t) = q(x) dx = − log S(t), 0

where q(x) = f (x)/S(x) is the hazard function and S(x) = 1 − F (x) the survivor function of F . However, this logarithmic relation holds only in the continuous case; in the discrete case, it must be replaced by a relation analogous to (4.6.2) (Kotz and Shanbhag (1980) or Andersen et al. (1993, Theorem II.6.6)]. Proposition 4.6.V. The IHF of a right-continuous d.f. F is monotonically increasing and right continuous, and at each discontinuity xi of F it has a jump of height ∆F (xi ) ≤ 1. ∆Q(xi ) = S(xi −) Conversely, any monotonically increasing right-continuous nonnegative function Q with discontinuities of magnitude < 1, except perhaps for a final discontinuity of size 1, can be the IHF of some d.f. F given by the inversion formula t S(t) = 1 − F (t) = 1 − ∆Q(xi ) exp − dQc (x) , (4.6.4) 0≤xi ≤t

0

where ∆Q(xi ) is the jump of Q at its discontinuity xi and Qc the continuous part of Q. Proof. Given a d.f. F on [0, ∞), observe first that when F has a jump ∆F (xi ) at the discontinuity xi , the corresponding jump in the IHF is ∆F (xi )/S(xi −) by Definition 4.6.IV. Since ∆F (xi ) = F (xi ) − F (xi −) ≤ 1 − F (xi −) = S(xi −) with equality if and only if F (xi ) = 1—that is, xi is a discontinuity of F and is the supremum of the support of F —we must have ∆Q(xi ) ≤ 1 with equality possible only for such xi . The inversion formula (4.6.4) is an immediate application of the exponential formula. To see this, we have from Definition 4.6.IV dF (xi ) = S(xi −) dQ(xi )

110


with

t

t

dF (x) = 1 −

S(t) = 1 − F (t) = 1 − 0

S(x−) dQ(x). 0

Taking u(x) = −1 in (4.6.3), S(·) is the unique solution of the equation t satisfying 0 |S(x)| dQ(x) < ∞ for t < ∞, so (4.6.4) holds. Corollary 4.6.VI. The d.f. F is uniquely determined by its IHF and conversely. This corollary is simply a formalization and extension of the fact that a renewal process is determined entirely by its lifetime d.f. The fact that the hazard measure is also the central concept in estimating the time to the next renewal has been shown already in Example 4.6(a) which we now continue but without any assumption of absolute continuity. Example 4.6(a) (continued). Recall the setting leading to the density qx (y) earlier. If the lifetime has a jump at x, then we should think of the risk as having a δ-function component at x, the weight associated with the δ-function being given by ∆Q(x) as above. Then, in place of the survivor function Sx (y) given earlier, we now appeal to the corresponding modification of (4.6.4), namely x+y Sx (y) = 1 − ∆Q(xi ) exp − dQc (u) . x≤xi ≤x+y

x

In a Wold process, the risk has to be conditioned not only by the time since the last event but also by the length of the most recently observed complete interval as in the following example. Example 4.6(b) Wold process with exponential conditional distributions (see Exercise 4.5.8). Wold (1948) and Cox (1955) both considered processes with Markov-dependent intervals, where the transition kernel has the form P (x, dy) = p(x, y) dy = λ(x) exp[−λ(x)y] dy

(x, y > 0),

corresponding to the assumption that, conditional on the length x of the last interval, the current interval is exponentially distributed with parameter λ(x). In this case, if we observe the process at time t and the length of the last completed interval is x, the risk is constant at λ(x) until the occurrence of the next event. As a stochastic process, the conditional risk appears as a step function, constant over intervals, the constant for any one interval being a function of the length of the preceding interval. Clearly, the ideas in these two examples can be generalized to situations where the dependence on the past extends to more than just the time since the last event or the length of the last completed interval. Such extensions and further examples are explored in Chapters 7 and 14.

CHAPTER 5

Finite Point Processes

The Poisson process can be generalized in many directions. We have already discussed some consequences of relaxing the independency assumptions while retaining those of stationarity and orderliness of a point process on the line. In this chapter we examine generalizations in another direction, stemming from the observation in Chapter 2 that, for a Poisson process, conditional on the total number of points in a bounded region of time or space, the individual points can be treated as independently and identically distributed over the region. This prompts an alternative approach to specifying the structure of point processes in a bounded domain or, more generally, of any point process in which the total number of points is finite with probability 1. Such a process is called a finite point process. Such finite point processes arise naturally as models for populations of animals, insects, and plants in the ecological field and as models for particle processes in physics, which was also the context of the first general theory of point processes given by Moyal (1962a) following earlier work by Yvon (1935), Bogoliubov (1946), Janossy (1950), Bhabha (1950) and Ramakrishnan (1950). More recently, spatial point processes have been extensively studied with an emphasis on finite models. Useful reviews can be found in Ripley (1981), Diggle (1983), Stoyan, Kendall and Mecke (1987, 1995), Baddeley and Møller (1989), Cressie (1991), Stoyan and Stoyan (1994), Baddeley et al. (1996), and Barndorff-Nielsen (1998), amongst others. In this chapter, we give a somewhat informal introduction to concepts and structure theorems for finite point processes, with a sketch of some of their applications. In contrast to the methods of the previous two chapters, the order properties of the real line here play no role in the discussion, and the theory can be developed as easily for a general state space as it can for the real line. In this sense, the present chapter serves as a precursor to the general theory developed more systematically in Volume Two. 111

112

5. Finite Point Processes

The approach we take is first to specify the distribution of the total number N of points, and then, given N , to specify the joint distribution of the N points over the region. This leads to a treatment of point process probabilities as probability measures over the space X ∪ introduced formally above Proposition 5.3.II and of the associated battery of Janossy measures, moment measures, cumulant measures, etc., all of which are recurrent themes in the development of the general theory. A special feature of the treatment of finite point processes is its dependence on combinatorial arguments. The reader may find it helpful to brush up on the definitions of binomial and multinomial coefficients and their relation to the number of ways of sorting a set of objects into various subsets. Closely related to these ideas are the results collected together in Section 5.2 concerning some basic tools for handling discrete distributions: factorial moments and cumulants and their relation with probability generating functions. The importance of this material for the theory of point processes would be hard to overemphasize. Most of the results of this chapter, and much of the general theory also, may be seen as extensions of the results for discrete distributions summarized in that section.

5.1. An Elementary Example: Independently and Identically Distributed Clusters We start with an elementary example that may help to illustrate and motivate the more general discussion. Let a random number N of particles be independently and identically distributed (i.i.d.) over a Euclidean space X according to some common probability measure F (·) on the Borel sets of X . Then, given N , the number of particles in any subregion A is found by ‘binomial sampling’: each particle, independently of the others, may fall in A with probability p = F (A), so, conditional on N , the number of particles in A has the binomial distribution N p(n; A | N ) = (F (A))n (1 − F (A))N −n . n Similarly, given any finite partition A1 , . . . , Ak of X , the joint distribution of the number of particles is given by the multinomial probability N (F (A1 ))n1 . . . (F (Ak ))nk . p(n1 , . . . , nk ; A1 , . . . , Ak | N ) = n1 · · · nk Unconditionally, the joint distribution of the numbers N (A1 ), . . . , N (Ak ) of particles in A1 , . . . , Ak is found by averaging over N : Pr{N (Ai ) = ni (i = 1, . . . , k)} =

∞

n=0

Pr{N = n} p(n1 , . . . , nk ; A1 , . . . , Ak | n).

5.1.

Independently and Identically Distributed Clusters

113

The procedure just outlined is most readily carried out in terms of probability generating functions (p.g.f.s). Let PN (z) = E(z N ), and write for convenience pi = F (Ai ). Then, the joint p.g.f. of N (Ai ) (i = 1, . . . , k) is N (A )

N (A )

P (A1 , . . . , Ak ; z1 , . . . , zk ) ≡ E(z1 1 · · · zk k ) = PN (p1 z1 + · · · + pk zk ).

(5.1.1)

More generally, for A1 , . . . , Ak just a set of mutually disjoint subregions, P (A1 , . . . , Ak ; z1 , . . . , zk ) = PN (p1 z1 +· · ·+pk zk +(1−p1 −· · ·−pk )); (5.1.2) in effect, we have introduced a further subset Ak+1 = (A1 ∪ · · · ∪ Ak )c and set zk+1 = 1 on Ak+1 . As special cases, when N is Poisson-distributed with parameter λ, the N (Ai ) are independent Poisson random variables with parameters λF (Ai ). In this case, (5.1.1) reduces to the identity ! * k +"

P (A1 , . . . , Ak ; z1 , . . . , zk ) = exp λ zi f (Ai ) − 1 i=1

=

k

exp[λF (Ai )(zi − 1)].

i=1

When N has a negative binomial distribution on {0, 1, . . .} so that PN (z) = (1 + µ(1 − z))−α for some µ, α > 0, {N (Ai )} is a set of mutually correlated binomial random variables with joint p.g.f. −α k

P (A1 , . . . , Ak ; z1 , . . . , zk ) = 1 + µ F (Ai )(1 − zi ) . i=1

In particular, from (5.1.2), the distribution of N (Ai ) itself has the p.g.f. P (Ai ; z) = [1 + µF (Ai )(1 − z)]−α and is again negative binomial with parameters µF (Ai ), α. It is not only the distributions of the N (Ai ) that may be of interest but also their moments. Consider, for example, the problem of finding the covariance of the number of points in two complementary subsets A1 , A2 = Ac1 . For any given N , we have from the binomial sampling property that E[N (A1 )N (A2 ) | N ] = N (N − 1)F (A1 )(1 − F (A1 )) = N (N − 1)F (A1 )F (A2 ). Hence, E(N (A1 )N (A2 )) = m[2] F (A1 )F (A2 )

(5.1.3)

cov(N (A1 ), N (A2 )) = c[2] F (A1 )F (A2 ),

(5.1.4)

and where m[2] is the second factorial moment, and c[2] the second factorial cumulant, of the total number N of points. In the Poisson case, the covariance is

114


zero, and in the negative binomial case it is positive; both contrast with the more familiar case of fixed N when the covariance is clearly negative. Note that both the second moment and the covariance have the form of a measure evaluated on the product set A1 × A2 . This is also the case in general and anticipates the introduction of the factorial moment and cumulant measures in Section 5.4.

5.2. Factorial Moments, Cumulants, and Generating Function Relations for Discrete Distributions Factorial moments and cumulants are natural tools for handling nonnegative integer-valued random variables, a characteristic they bequeath to their offspring, the factorial moment and cumulant measures, in the point process context. We begin by recalling some basic definitions. For any integers n and r, the factorial powers of n, written n[r] , may be defined by n(n − 1) · · · (n − r + 1) (r = 0, . . . , n), [r] n = 0 (r > n). We then have the following definition. Definition 5.2.I. For r = 0, 1, . . . , the rth factorial moment m[r] of the nonnegative integer-valued random variable N is m[r] ≡ E(N [r] ). Thus, when N has probability distribution {pn } = {Pr{N = n}}, m[r] =

∞

n[r] pn .

(5.2.1)

n=0

Consequently, when the distribution is concentrated on a finite range 0, 1, . . . , n0 , all factorial moments of order larger than n0 are zero. It is useful to be able to convert from factorial moments to ordinary moments and back again. The coefficients that arise in these conversions are the Stirling numbers of the first and second kinds, defined, respectively, as the coefficients arising in the expansion of x[r] and xr in powers or factorial powers of x, where, by analogy with the definition of n[r] , x[r] = x(x − 1) · · · (x − r + 1) for any real x and positive integer r. We follow the notation of David and Barton (1962) in denoting them by Dj,r and ∆j,r . Definition 5.2.II. The Stirling numbers of the first kind Dj,r and second kind ∆j,r are defined by the relations n[r] =

r

j=1

Dj,r (−1)r−j nj

(n ≥ r)

(5.2.2)

5.2.

Factorial Moments, Cumulants, and Generating Function Relations

and nr =

r

(n ≥ r).

∆j,r n[j]

115

(5.2.3)

j=1

Replacing n in (5.2.2) and (5.2.3) by the random variable N and taking expectations, we obtain the corresponding relations between moments: m[r] = mr ≡ E(N r ) =

r

j=1 r

Dj,r mj (−1)r−j ,

(5.2.4)

∆j,r m[j] .

(5.2.5)

j=1

It is clear that, for a nonnegative random variable, the rth factorial moment is finite if and only if the ordinary rth moment is finite. Some useful recurrence relations for the Stirling numbers are given in Exercise 5.2.1. For further properties, relation to Bernoulli numbers, and so on, see David and Barton (1962, Chapter 15) and texts on finite differences. The factorial moments of the random variable N are related to the Taylor series expansion of the p.g.f. P (z) = E(z N )

(|z| ≤ 1)

about z = 1 in much the same way as the ordinary moments arise in the expansion of the characteristic or moment generating function about the origin. Proposition 5.2.III. For a nonnegative integer-valued random variable N whose kth factorial moment is finite, the p.g.f. is expressible as P (1 + η) = 1 +

k

m[r] η r r=1

r!

+ o(η k )

(5.2.6)

for all η such that |1 + η| ≤ 1. The complete Taylor series expansion of the p.g.f., ∞

m[r] η r P (1 + η) = 1 + , (5.2.7) r! r=1 is valid for some nonzero η if and only if all moments exist and the series in (5.2.7) has nonzero radius of convergence in η; equivalently, if and only if the p.g.f. P (z) is analytic in a disk |z| < 1 + for some > 0. Equation (5.2.7) then holds for |η| < . Proof. To establish (5.2.6), write (1 + η)N = 1 +

k

N [r] η r r=1

r!

+ Rk (N, η)

(k = 1, 2, . . .)

for remainder terms Rk (N, η) that we now investigate. For k = 0, set R0 (N, η) = (1 + η)N − 1

116


and observe that |R0 (N, η)| ≤ 2 under the condition of the theorem that |1 + η| ≤ 1. For general k = 1, 2, . . . , repeated integration of R0 (N, ·) shows that &

& &Rk (N, η) η k & ≤ 2N [k] k! (|1 + η| ≤ 1). Since the left-hand side of this inequality → 0 (η → 0) for each fixed N and the right-hand side has finite expectation under the assumption of the theorem, it follows by dominated convergence that E Rk (N, η) = o(η k ), which is the result required. To establish (5.2.7), consider the binomial expansion (1 + η)N = 1 +

∞

N [r] η r r=1

r!

.

For η > 0, the finiteness of the expectation on the left is equivalent to requiring the p.g.f. to be analytic for |z| < 1 + η. When this condition is satisfied, it follows from Fubini’s theorem that for such η the expectation can be taken inside the summation on the right, leading to the right-hand side of (5.2.7). Conversely, suppose all moments exist and that the sum on the righthand side of (5.2.7) is at least conditionally convergent for some nonzero η0 . Then m[r] η0r /r! → 0 as r → ∞, and it follows from a standard power series argument that the series in (5.2.7) is absolutely convergent for |η| < |η0 | and so defines an analytic function of η there. Since each m[r] = E(N [r] ) is nonnegative, we can now take any positive η < |η0 | and use Fubini’s theorem to reverse the argument used earlier to deduce that because (5.2.7) holds for all 0 ≤ η ≤ |η0 |, P (z), being a power series with nonnegative coefficients, has its first singularity on the positive half-line outside |z| < 1 + |η0 |. In the sequel, we also require the version of Proposition 5.2.III in which the remainder term is bounded by a term proportional to the (k + 1)th moment. The proof, which is along similar lines, is left to the reader. An alternative approach is indicated in Exercise 5.2.2. A similar expansion holds for log P (1 + η), the coefficients of η r /r! being the factorial cumulants c[r] (r = 1, 2, . . .). If P (·) is analytic in a disk as below (5.2.7), then the infinite expansion log P (1 + η) =

∞

c[r] η r r=1

(5.2.8a)

r!

is valid, while under the more limited assumption that mk < ∞, we have the finite Taylor series expansion log P (1 + η) =

k

c[r] η r r=1

r!

+ o(η k )

valid for |1 + η| < 1; verification is left to the reader.

(η → 0),

(5.2.8b)

5.2.


117

The factorial cumulants are related to the factorial moments by the same relations as hold between the ordinary cumulants and moments. The first few relations between the ordinary cumulants cr , central moments mr , and factorial moments and cumulants are useful to list as below: c[1] = c1 = µ = m[1] ,

(5.2.9a)

c[2] = c2 − c1 = σ − µ = m[2] − 2

c[3] = c3 − 3c2 + 2c1 =

m3

m2[1]

,

(5.2.9b)

− 3σ + 2µ = m[3] − 3m[2] m[1] + 2

2m3[1] .

(5.2.9c)

Generally, the factorial moments and cumulants provide a much simpler description of the moment properties of a discrete distribution than do the ordinary moments. In particular, for the Poisson distribution {pn (λ)}, m[r] = λr ,

c[1] = λ,

c[r] = 0

(r = 2, 3, . . .).

This vanishing of the factorial cumulants of the Poisson distribution is reminiscent of the vanishing of the ordinary cumulants of the normal distribution and is perhaps one indication of why the Poisson process plays such an outstanding role in the theory of point processes. There are in fact four expansions of the p.g.f. of possible interest, according to whether we expand P (z) itself or its logarithm and whether the expansion is about z = 0 or z = 1. The expansions about z = 1 yield the factorial moments and factorial cumulants, and the expansion of P (z) about z = 0 yields the probability distribution {pn }. This leaves the expansion of log P (z) about z = 0, an expansion that, while rarely used, has an important interpretation in the case of an infinitely divisible (compound Poisson) distribution. Since the analogous expansion for the probability generating functional (p.g.fl.) of a point process is also important, again in the context of infinite divisibility, we now consider the last case in some detail. Proposition 5.2.IV. If p0 > 0, the p.g.f. P (·) can be written in the form log P (z) = −q0 +

∞

qn z n

(|z| < R)

(5.2.10)

n=1

where p0 = e−q0 and R is the distance from the origin to the nearest zero or singularity of P (z). When P (·) is the p.g.f. of a compound Poisson dis∞ tribution, the terms qn are nonnegative and q0 = n=1 qn , so the sequence {πn : n = 1, 2, . . .} ≡ {qn /q0 } can be interpreted as the probability distribution of the cluster size, given that the cluster is nonempty; in this case, (5.2.10) can be rewritten as ∞

log P (z) = −q0 πn (1 − z n ) (|z| < R). n=1

Proof. The structure of the compound Poisson distribution follows from analysis in Chapter 2 (see Theorem 2.2.II and Exercise 2.2.2). The other remarks are standard properties of power series expansions of analytic functions.

118


Example 5.2(a) Negative binomial distribution and generating functions. To illustrate these various expansions consider the p.g.f. of the negative binomial distribution, P (z) = [1 + µ(1 − z)]−α (µ > 0, α > 0, |z| ≤ 1). Putting z = 1 + η, we find P (1 + η) = (1 − µη)−α = 1 +

∞

α+r−1 r

r=1

µr η r

so that m[r] = α(α + 1) · · · (α + r − 1)µr . Taking logarithms, log P (1 + η) = −α log(1 − µη) = α

∞

µr η r r=1

r

,

and hence c[r] = (r − 1)! αµr . For the expansions about z = 0, we have −α ∞

µz α + n − 1 µz n 1 1 1 − = , P (z) = (1 + µ)α 1+µ (1 + µ)α n=0 1+µ n

so pn = and

µ n α+n−1 1 , (1 + µ)α 1 + µ n

µz log P (z) = −α log(1 + µ) − α log 1 − 1+µ ∞

n = −[α log(1 + µ)] 1 − πn z , n=1 −1

where πn = [n log(1 + µ)] [µ/(1 + µ)] . Clearly, these {πn } constitute a probability distribution, namely the logarithmic distribution, illustrating the well-known fact that the negative binomial is infinitely divisible and hence must be expressible as a compound Poisson distribution. n

Corresponding to the four possible expansions referred to above, there are twelve sets of conversion relations between the different coefficients. One of these, the expression for factorial moments in terms of the probabilities, is a matter of definition: what can be said about the others? Formally, either expansion about z = 1 can be converted to an expansion about z = 0 by a change of variable and expansion, for example, in (formally) expressing the probabilities in terms of the factorial moments via P (z) = 1 +

∞

m[r] (z − 1)r r=1

r!

;

5.2.


119

expanding (z − 1)r and equating coefficients of z n , we obtain

m[r] r pn = (−1)r−n r! n r≥n

or, in the more symmetrical form, n! pn =

∞

∞

(−1)r−n

r=n

m[r] m[n+r] = . (−1)r (r − n)! r=0 r!

(5.2.11)

This relation may be compared with its converse m[r]

∞

∞

Jr+n , = n pn = n! n=r n=0 [r]

(5.2.12)

where Jn+r = (n + r)! Pn+r . Thus, to display the symmetry in these (formal) relations to best advantage, we need to use the quantities Jn , which are analogues of the Janossy measures to be introduced in Section 5.3. Under what circumstances can the converse relation (5.2.11) be established rigorously? For the derivation above to be valid, we must be able to expand P (z) about z = 1 in a disk |z − 1| < 1 + for some > 0, requiring P (z) itself to be analytic at all points on the line segment (−, 2 + ). Since P (z) has nonnegative coefficients, its radius of convergence is determined by the first singularity on the positive real axis. Consequently, in order for (5.2.11) to hold for all r = 1, 2, . . . , it is sufficient that P (z) should be analytic in the disk |z| < 2 + for some > 0. A finite version of (5.2.11) with remainder term is due to Fréchet (1940); extensions are given in Takacs (1967) and Galambos (1975) (see also Daley and Narayan, 1980). We give a simple result in the theorem below, with some extensions left to Exercises 5.2.2–4. Proposition 5.2.V. If the distribution {pn } has all its moments finite and its p.g.f. P (z) is convergent in a disk |z| < 2 + for some > 0, then (5.2.11) holds. Without assuming such analyticity, the finiteness of m[k] ensures that for integers n = 0, 1, . . . , k − 1, k−1

m[r] (n) + Rk , (r − n)!

(5.2.13a)

≤ m[k] (k − n)! .

(5.2.13b)

If all moments are finite and for some integer n0 m[k] = o (k − n0 )! (k → ∞),

(5.2.14a)

n! pn =

(−1)r−n

r=n

where (n)

0 ≤ (−1)k−n Rk

then lim

k→∞

k

(−1)r−n m[r] (r − n)!

(5.2.14b)

r=n

exists for n = 0, 1, . . . , n0 and the formal relation (5.2.11) holds for such n.

120


Proof. When P (z) is analytic for |z| < 2 + , the expansion P (z) =

∞

m[r] (z − 1)r r=0

r!

is valid for |z − 1| < 1 + , within which region, and at z = 0 in particular, it can be differentiated n times, leading at once to (5.2.11). Under the weaker condition that m[k] < ∞, n-fold differentiation in the definition P (z) = E(z N ) is possible for all |z| ≤ 1 for n = 1, . . . , k, leading to P (n) (z) = E(N (n) z N −n ). Now P (n) (z) is (k − n) times differentiable in |z| ≤ 1, so the Taylor series expansion P (n) (z) =

k−n−1

r=0

(z − 1)r P (n+r) (1) (z − 1)k−n P (k) (1 + (z − 1)ν) + r! (k − n)!

holds for real z in |z| ≤ 1 for some ν ≡ ν(z) in (0, 1). In particular, (5.2.13a) results on putting z = 0 with (n)

Rk

= (−1)k−n

E(N (k) (1 − ν)N −k ) , (k − n)!

from which relation the inequalities in (5.2.13b) follow. When (5.2.14) holds, (n) Rk → 0 (k → ∞) for each fixed n, and hence (5.2.11) holds in the sense indicated. Special cases of (5.2.13) give the Bonferroni inequalities (see Exercise 5.2.5). Similar relations can be obtained between the factorial cumulants and the quantities πn of Proposition 5.2.IV. Thus, when log P (z) is analytic in a disk |z| < 1 + for some > 0, r-fold differentiation of (5.2.10) and then setting z = 1 yields ∞

c[r] = qn n[r] = q0 µ[r] , (5.2.15) n=r

where µ[r] in the case of a compound Poisson process is the rth factorial moment of the cluster-size distribution. Reversing the exercise, when log P (z) is analytic in the disk |z| < 2 + , we have [see the derivation of (5.2.11)] n! qn =

∞

r=n

(−1)r−n

c[r] . (r − n)!

(5.2.16)

The most difficult relations to treat in a general form are those between the moments and cumulants, or between the {pn } and the {qn }; these arise from taking exponentials or logarithms of a given series and expanding it by formal manipulation. The feature of these relations is that they involve partitions. For given positive integers j and k with j ≤ k, we define a j-partition of k as a partition of the set of k numbers {1, . . . , k} into j nonempty subsets.

5.2.


121

Let Pjk denote the collection of all such j-partitions and write T = {S1 (T ), . . . , Sj (T )} for an element of Pjk , noting that the order in which the subsets S. (T ) are labelled or written is immaterial. Thus, for example, the collection of sets {1, 2, 4}, {3, 5}, {6, 8}, {7} is a 4-partition of 8 and is the same as {1, 2, 4}, {6, 8}, {7}, {3, 5}. The following lemma is basic (see e.g. Andrews, 1976); in it, |Sj (T )| denotes the number of elements in Sj (T ) ⊂ {1, . . . , k}. ∞ Lemma 5.2.VI. Let {cj : j = 1, 2, . . .} be a sequence satisfying j=1 |cj |/j! < ∞. Then ! ∞ " ∞

cj z j dk z k exp = (all |z| ≤ 1), (5.2.17) j! k! j=1 k=0

where d0 = 1 and for k = 1, 2, . . . , dk =

j k

j=1 T ∈Pjk i=1

ck =

k

j=1

j−1

(−1)

c|Si (T )| ,

(j − 1)!

j

T ∈Pjk i=1

(5.2.18)

d|Si (T )| .

(5.2.19)

Proof. Establishing (5.2.18) and (5.2.19) is essentially a matter of counting 2 terms. For (5.2.18), consider the expansion /2!+· · · of the exponen∞ 1+Σ+Σ j tial function in (5.2.17) (here, Σ = j=1 cj z /j!), and concentrate attention on all the terms in a specified product of coefficients such as c3 c22 c1 . Observe first that such terms involve z to the power of the sum of the indices, here 3 + 2 + 2 + 1 = 8, and thus they contribute to the term d8 . Second, if we transfer the coefficient 1/k! of dk z k to the multiplier k! on the opposite side, each particular term c3 c22 c1 is then multiplied by the ratio of factorials 8!/3! 2! 2! 1! arising from the factorials associated with the cj and dk . Third, the number of such terms obtained from expanding Σ4 equals the multinomial coefficient 4!/1! 2! 1! , which on division by the factorial 4! from the expansion of exp(Σ) leaves the factor 1!/1! 2! 1! . Thus, altogether the contribution of the coefficient of c3 c22 c1 to d8 is 8!/{(3! 2! 2! 1!) (1! 2! 1!)}. On the other hand, in the expression asserted for dk in (5.2.18), we have to look at 4-partitions of 8 into subsets of sizes 3, 2, 2, 1. The number of such subsets is just 8!/3! 2! 2! 1! , which must be divided by 2! because there are two subsets of size 2. Thus, the coefficient of c3 c22 c1 is of the form implied by (5.2.18). Arguing this way in general establishes (5.2.18), and a similar kind of argument leads to (5.2.19). We remark that the advantage of working with j-partitions, rather than with additive partitions as in David and Barton (1962), is that the counting procedure automatically takes into account repeated terms without requiring explicit notation for the number of repetitions; such notation would make (5.2.18) and (5.2.19) appear much more cumbersome. Examples of full expansions are given in Exercises 5.2.6–8.

122


Corollary 5.2.VII. (a) Factorial moments m[k] and factorial cumulants c[k] are related as in (5.2.18) and (5.2.19) via the substitutions cj = c[j] and dk = m[k] . (b) In equation (5.2.10), the probabilities pn and qn are also related as at (5.2.18) and (5.2.19) with cj = j! qj /(− log p0 ) and dk = k! pk /p0 .

Exercises and Complements to Section 5.2 5.2.1 Recurrence relations for Stirling numbers. Use n[r+1] = (n − r)n[r] to show that ∆j,r+1 = j∆j,r + ∆j−1,r ,

∆1r = 1

(r ≥ 1),

Dj,r+1 = rDj,r + Dj−1,r ,

D0r = 0

(r ≥ 1),

∆j0 = 0

(j ≥ 1),

D11 = 1, Dj1 = 0 (j ≥ 2).

5.2.2 Show that when P (z) is any p.g.f. with finite first moment P (1), the function (1 − P (z))/P (1)(1 − z) is also a p.g.f. Use this fact in an induction argument to show that (see Proposition 5.2.III) when m[k] = P (k) (1) < ∞, the function mk (z) in the expansion P (z) = 1 +

k−1

(z − 1)r m[r]

r!

r=1

+

(z − 1)k mk (z) k!

equals m[k] times a p.g.f. Since mk (z) = m[k] + o(1) as z → 1 through values |z| ≤ 1, (5.2.6) follows, as well as the alternative version with remainder bounded by m[k] . Equations (5.2.13) can also be derived by n-fold differentiation of an expansion to k − n terms (e.g. Daley and Narayan, 1980). 5.2.3 Let the nonnegative integer-valued r.v. N have all factorial moments m[r] finite and lim supr→∞ (m[r] /r!)1/r = 1/ for some > 0. Show that the p.g.f. P (z) of N has radius of convergence 1 + , and hence deduce that the moments m[r] determine the distribution of N uniquely. Relate P (z) to a moment generating function and deduce that 1 + = exp , where 1/ ≡ lim supr→∞ (mr /r!)1/r . 5.2.4 (Continuation). By using an analytic continuation technique (see Takacs, 1965), show that when > 0 and for any nonnegative z > −2 − 1, pn =

∞

r r=n

r

1 r − n r−s m[s] (−1)s−n z . r+1 s! n (1 + z) s−n s=n

5.2.5 Bonferroni inequalities. Let the r.v. N count the number of occurrences amongst a given set of ν events A1 , . . . , Aν . Show that Sr ≡

Pr(Ai ∩ Aj ∩ · · ·) = E(N (r) )/r! ,

(r)

extends over all νr distinct subsets {i, j, . . .} of where the summation (r) size r from the index set {1, . . . , ν}. [Hint: Using indicator r.v.s, write N (r) = r!

(r)

I(Ai ∩ Aj ∩ · · ·),

5.3.

The General Finite Point Process: Definitions and Distributions

123

where the term r! arises from the r! ordered subsets of {1, . . . , ν} yielding the same (unordered) subset {i, j, . . .} containing r indices.] Deduce from (5.2.13) the Bonferroni inequalities

0 ≤ Sn −

n+1 Sn+1 + · · · + 1

n+k Sn+k − pn ≤ k

n+k+1 Sn+k+1 , k+1

where k is an even integer (see e.g. Moran, 1968, pp. 25–31). 5.2.6 For given positive integers j and k with j ≤ k, k) = {positive pdefine P(j, p integers {r1 , . . . , rp } and {π1 , . . . , πp } such that i=1 πi = j, i=1 πi ri = k} = set of all j-partitions of k. Write the series (5.2.7) in the form P = 1 + Σ so that log P (z) = Σ − Σ2 /2 + Σ3 /3 − · · · , and expand the series Σn as a multinomial expansion. By equating coefficients of z k , show formally that the factorial cumulants in (5.2.8) are given by

c[k] = k!

k

j=1

(−1)j−1 (j − 1)!

1 m[r ] π1 1 P(j,k)

π1 !

r1 !

···

1 m[rp ] πp . πp ! rp !

5.2.7 Apply Lemma 5.2.VI to show that c[4] = m[4] − 4m[3] m[1] − 3m2[2] + 12m[2] m2[1] − 6m4[1] , m[4] = c[4] + 4c[3] c[1] + 3c2[2] + 6c[2] c2[1] + c4[1] . 5.2.8 Investigate the use of Lemma 5.2.VI in deriving explicit expressions for probabilities of (i) the ‘doubly Poisson’ compound Poisson distribution with p.g.f. P (z) = exp{−µ[1 − exp(−λ(1 − z))]}; (ii) the Hermite distribution with p.g.f. P (z) = exp(az + bz 2 ) for appropriate constants a and b (see Milne and Westcott, 1993).

5.3. The General Finite Point Process: Definitions and Distributions We now drop any special assumptions and suppose only that the following conditions hold concerning a finite point process. Conditions 5.3.I. (a) The points are located in a complete separable metric space (c.s.m.s.) X , as, for example, X = Rd . . . .) is given determining the total number (b) A distribution {pn } (n = 0, 1, ∞ of points in the population, with n=0 pn = 1. (c) For each integer n ≥ 1, a probability distribution Πn (·) is given on the Borel sets of X (n) ≡ X × · · · × X , and it determines the joint distribution of the positions of the points of the process, given that their total number is n.

124


Such a definition is both natural and powerful. In particular, it provides a constructive definition that could be used to simulate the process: first, generate a random number N according to the distribution {pn } (and note that Pr{0 ≤ N < ∞} = 1), and then, supposing N = n and excepting the case n = 0 in which case there is nothing else to do, generate a random vector (x1 , . . . , xn ) according to the distribution Πn (·). At this stage, the distinction between ordered and unordered sets of points should be clarified. In talking of stochastic point processes, we make the tacit assumption that we are dealing with unordered sets of points: points play the role of locations at which a given set of particles might be found. We talk of the probability of finding a given number k of points in a set A: we do not give names to the individual points and ask for the probability of finding k specified individuals within the set A. Nevertheless, this latter approach is quite possible (indeed, natural) in contexts where the points refer to individual particles, animals, plants, and so on. Moreover, it is actually this latter point of view that is implicit in Conditions 5.3.I, for as yet there is nothing in them to prevent x1 , say—that is, the first point or particle named—from taking its place preferentially in some part of the space, leaving the other particles to distribute themselves elsewhere. To be consistent with treating point processes as a theory of unordered sets, we stipulate that the distributions Πn (·) should give equal weight to all n! permutations of the coordinates (x1 , . . . , xn ), i.e. Πn (·) should be symmetric. If this is not already the case in Condition 5.3.I(c), it is easily achieved by introducing the symmetrized form for any partition (A1 , . . . , An ) of X , Πsym n (A1 × · · · × An ) =

1 Πn (Ai1 × · · · × Ain ), n! perm

(5.3.1)

where the summation perm is taken over all n! permutations (i1 , . . . , in ) of the integers (1, . . . , n) and the normalizing factor 1/n! ensures that the resulting measure still has total mass unity. An alternative notation, which has some advantages in simplifying combinatorial formulae, utilizes the nonprobability measures

Jn (A1 × · · · × An ) = pn Πn (Ai1 × · · · × Ain ) perm (5.3.2) = n! pn Πsym (A × · · · × A ). 1 n n We follow Srinivasan (1969) in referring to these as Janossy measures after their introduction by Janossy (1950) in the context of particle showers. By contrast, Yvon (1935), Bogoliubov (1946) and Bhabha (1950) worked with the form (5.3.1), as have also Macchi (1975) and co-workers, who refer to quantities such as Πsym n (·) in (5.3.1) as exclusion probabilities. An important feature of Janossy measures is their simple interpretation when derivatives exist. If X = Rd and jn (x1 , . . . , xn ) denotes the density of

5.3.


125

Jn (·) with respect to Lebesgue measure on (Rd )(n) with xi = xj for i = j, then % ) there are exactly n points in the jn (x1 , . . . , xn ) dx1 · · · dxn = Pr process, one in each of the n distinct . infinitesimal regions (xi , xi + dxi ) This interpretation gives the Janossy densities a fundamental role in the structural description and likelihood analysis of finite point processes. Thus, they appear as likelihoods in Chapter 7, where they play a key role in the study of spatial point patterns (see also Chapter 15 and references there) and also in pseudolikelihoods. They are well adapted to describing the behaviour on observational regions, which, being finite, are typically bounded. Example 5.3(a) I.i.d. clusters (continued from Section 5.1). In this case, X = Rd and, assuming F (A) = A f (x) dx for some density function f (·), the joint density function for the ordered sequence of n points at x1 , . . . , xn is πn (x1 , . . . , xn ) = f (x1 ) · · · f (xn ), which is already in symmetric form. Here jn (x1 , . . . , xn ) = pn n! f (x1 ) · · · f (xn ), and it is jn (· · ·), not πn (· · ·), that gives the probability density of finding one particle at each of the n points (x1 , . . . , xn ), the factorial term giving the number of ways the particles can be allocated to these locations. Example 5.3(b) Finite renewal processes and random walks. Suppose X = R1 and that, given N = n, the points of the process are determined by the successive points S1 , . . . , Sn of a simple renewal process for which the common distribution of the lifetimes Sj − Sj−1 (where S0 ≡ 0 and j = 1, . . . , n) has a density function f (·). Then πn (S1 , . . . , Sn ) =

n

f (Sj − Sj−1 ).

(5.3.3)

j=1

In moving to the symmetrized form, some care is needed. For any x1 , . . . , xn , we have, formally, πnsym (x1 , . . . , xn ) =

1 f (xi1 )f (xi2 − xi1 ) · · · f (xin − xin−1 ). n! perm

Let x(1) , . . . , x(n) denote the set {x1 , . . . , xn } in ascending order. Then, at least one term in each product in perm will vanish (since f (x) = 0 for x < 0) unless we already have x1 , . . . , xn ordered; that is, xj = x(j) for j = 1, . . . , n. Hence, πnsym (x1 , . . . , xn ) =

1 f (x(1) )f (x(2) − x(1) ) · · · f (x(n) − x(n−1) ). n!

(5.3.4)

126


Comparing (5.3.3) and (5.3.4), 1/n! in the latter is seemingly a discrepant factor. The reconciliation lies in the fact that (5.3.3) vanishes outside the hyperoctant x1 < x2 < · · · < xn , whereas (5.3.4) repeats itself symmetrically in all n! hyperoctants. Finally, the Janossy densities are given by jn (x1 , . . . , xn ) = pn f (x(1) )f (x(2) − x(1) ) · · · f (x(n) − x(n−1) ),

(5.3.5a)

where as before pn is the probability that the process contains just n points. Again it is to be noted that (5.3.3) vanishes outside the first hyperoctant, whereas (5.3.5) gives positive measure to all hyperoctants. Once the unidirectional character of each step is lost, these simplifications do not occur. What is then available for a general random walk is confined to the forms (5.3.3) and the corresponding expression

f (xi1 )f (xi2 − xi1 ) · · · f (xin − xin−1 ). (5.3.5b) jn (x1 , . . . , xn ) = pn perm

The simplest renewal example occurs when f has an exponential density. The joint density (5.3.3) then reduces to n λ exp(−λxn ) (0 ≤ x1 ≤ xn ), πn (x1 , . . . , xn ) = 0 otherwise, or in terms of (5.3.5),

jn (x1 , . . . , xn ) = pn λn e−λx(n) .

Remarkably, the joint distribution depends only on the position of the extreme value x(n) ; given this value, the other points are distributed uniformly over (0, x(n) ). The simplest example of a symmetric random walk is probably that for which the individual steps are normally distributed N (0, 1). The successive Si are then the partial sums of a sequence of independent normal variates Si =

i

Zj

j=1

and for any given n are therefore jointly normally distributed with zero mean vector and covariance matrix having elements σij = min(i, j)

(1 ≤ i, j ≤ n).

No dramatic simplifications seem possible, but some further details are given in Exercise 5.3.1. Example 5.3(c) Gibbs processes: processes generated by interaction potentials. A fundamental class of point processes arising in statistical physics is described by means of forces acting on and between particles. The total

5.3.


127

potential energy corresponding to a given configuration of particles is assumed to be decomposable into terms representing the interactions between the particles taken in pairs, triples, and so on; first-order terms representing the potential energies of the individual particles due to the action of an external force field may also be included. This leads to a representation of the total potential energy for a configuration of n particles at x1 , . . . , xn by a series of the form n

U (x1 , . . . , xn ) = ψr (xi1 , . . . xir ), (5.3.6) r=1 1≤i1 σ, and no other interaction occurs. The second model is of intermediate type, approximating the behaviour of the hard-core model for large n. None of these models is easy to handle analytically, and special expansion techniques have been developed to approximate the partition functions. For the subsequent discussions, we use mainly the Janossy measures. In this formulation, the normalization condition pn = 1 takes the form ∞

Jn (X (n) ) =1 n! n=0

(5.3.9)

since we may interpret J0 (X (0) ) = p0 and, for n ≥ 1, we have

Πn (X (n) ) = pn n! . Jn (X (n) ) = pn perm

It is clear that from any family of symmetric measures Jn (·) satisfying (5.3.9), we can construct a probability distribution {pn } and a set of symmetric probability measures {Πsym n (·)} satisfying Conditions 5.3.I, and conversely.

5.3.


129

Either specification is equivalent to specifying a global probability measure P on the Borel sets A of the countable union (with X (0) interpreted as an isolated point) X ∪ = X (0) ∪ X (1) ∪ X (2) ∪ · · · ; (5.3.10) Moyal (1962a) takes (X ∪ , P) as the canonical probability space of a finite point process. Given such a measure P, the measure pn Πsym n , or equivalently, (n!)−1 Jn , appears as the restriction of P to the component X (n) . The situation is summarized in the following proposition. Proposition 5.3.II. Let X be a complete separable metric space, and let (n) BX be the product σ-field on X (n) i, with the added convention that the set X (0) denotes an ideal point such that X (0) × X = X × X (0) = X . Then, the following specifications are equivalent, and each suffices to define a finite point process on X : (i) a probability distribution {pn } on the nonnegative integers and a family (n) of symmetric probability distributions Πsym n (·) on BX , n ≥ 1; (n)

(ii) a family of nonnegative symmetric measures Jn (·) on BX , n ≥ 1, satisfying the normalization condition (5.3.9) and with J0 (X (0) ) = p0 ; (iii) a symmetric probability measure P on the symmetric Borel sets of the countable union in (5.3.10). There is one point of principle to be noted here concerning the canonical choice of state space for a finite point process. To be consistent with treating a point process as a set of unordered points, a realization with, say, k points should be thought of not as a point in X (k) but as a point in the quotient space of X (k) with respect to the group of permutations amongst the k coordinates. For example, when X = R and k = 2, then in place of all pairs (x1 , x2 ), with (x1 , x2 ) and (x2 , x1 ) being treated as equivalent, we should consider some representation of the quotient space such as the set {(x1 , x2 ): x1 ≤ x2 }. The difficulty with this approach in general is that it is often hard to find a convenient concrete representation of the quotient space (consider for example the case just cited with R replaced by the unit circle or sphere), with the attendant problems of visualizing the results and bringing geometric intuition to bear. We have therefore preferred the redundant representation, which allows a distinction between the points but then gives all permutations amongst the labelling of the points equal weight in the measure. It must be borne in mind that there is then a many–one relation between the points in the space X ∪ and the set of all totally finite counting measures. Another way of treating the same problem is to introduce the σ-algebra of symmetric sets in X (k) , that is, the sets invariant under permutations of the coordinate axes. A symmetric set in X ∪ is a set whose projections onto X (k) are symmetric for each positive integer k. Then, any event defined on the point process represents a symmetric set in X ∪ , and thus the natural σ-algebra to use in discussing point process properties is this σ-algebra of symmetric sets. We do not emphasize this

130


approach because our main development in Chapter 9 is given in terms of counting measures; we merely refer the reader seeking details to Moyal (1962a) and Macchi (1975) (see also Exercises 5.3.4–6). Now let us turn to the problem of expressing in terms of Janossy measures (or one of their equivalents) the probability distributions of the random variables N (Ai ). If (A1 , . . . , Ak ) represents a finite partition of X , the probability of finding exactly ni points in Ai (i = 1, . . . , k) can be written, with n1 + · · · + nk = n, as (n )

(n )

Jn (A1 1 × · · · × Ak k ) Pk (A1 , . . . , Ak ; n1 , . . . , nk ) = n1 ! · · · nk ! n (n1 ) (n ) Πsym = pn × · · · × Ak k ), n (A1 n1 · · · nk

(5.3.11)

where the multinomial coefficient can be interpreted as the number of ways of grouping the n points so that ni lie in Ai (i = 1, . . . , k). It is important in (5.3.11) both that the sets Ai are disjoint and that they have union X (i.e. they are a partition of X ). For any i for which ni = 0, the corresponding term is omitted from the right-hand side. From (5.3.11), it follows in particular that the probability of finding n points in A, irrespective of the number in its complement Ac , is given by n! P1 (A; n) =

∞

Jn+r (A(n) × (Ac ))(r) )

r!

r=0

.

(5.3.12)

Similarly, if A1 , . . . , Ak are any k disjoint Borel sets, C = (A1 ∪ · · · ∪ Ak )c , and n = n1 + · · · + nk , the probability of finding just ni points in Ai , i = 1, . . . , k, is given by n1 ! · · · nk ! Pk (A1 , . . . , Ak ; n1 , . . . , nk ) =

∞ (n ) (n )

Jn+r (A 1 × · · · × A k × C (r) ) 1

r=0

k

r!

.

(5.3.13) These probabilities are in fact the joint distributions of the random variables N (Ai ), i = 1, . . . , k. The fact that they do form a consistent set of finite-dimensional (fidi) distributions is implicit in their derivation, but it can also be verified directly, as we show following the discussion of such conditions in Chapter 9. An alternative approach, following Moyal (1962a), starts from the observation that each realization can be represented as a random vector Y ∈ X (n) for some n ≥ 0. Any such vector defines a counting measure on X , through N (A) = #{i: yi ∈ A}, where the yi are the components of the random vector Y . The random vector thus gives rise to a mapping from X (n) into the space NX# of all counting measures on X . It is easy to see that this mapping is measurable so it defines a point process (see Chapter 9). This being true for every n, the whole

5.3.


131

process is a point process, and since (5.3.13) are its fidi distributions, they are necessarily consistent. As Moyal pointed out, this approach to the existence of finite point processes can be extended to more general cases by considering the restrictions of the process to an increasing family of Borel sets (spheres, say) chosen so that they expand to fill the whole space but with probability 1 have only a finite number of points in each. The main difficulty with this approach from our point of view is that it does not extend readily to random measures, which we require for their own sake and for applications in later chapters. We conclude this section with a lemma that will play a useful role in simplifying the relations amongst various measures introduced in the sequel. It is needed in particular in checking that the distributions defined by (5.3.13) satisfy the consistency conditions of Chapter 9. Lemma 5.3.III. Let A be a Borel subset of X and S a symmetric measure defined on X (n) for some n > 0. Then, for any partition {A1 , . . . , Ak } of A,

n (j ) (j ) (n) S(A1 1 × · · · × Ak k ), (5.3.14) S(A ) = j1 · · · jk where the summation extends over all nonnegative integers j1 , . . . , jk for which j1 + · · · + jk = n. Proof. Equation (5.3.14) expresses the fact that the partitioning of A induces a partitioning of A(n) into k n subsets, which are grouped together into classes that are identified by vectors (j1 , . . . , jk ): within any given class, each constituent subset has Ai appearing as a coordinate or ‘edge’ ji times. The symmetry of S implies that all subsets in the same class have the same S measure; hence, (5.3.14) follows.

Exercises and Complements to Section 5.3 5.3.1 [see Example 5.3(b)]. For a finite random walk with normally distributed N (0, 1) steps, show that 2

π2sym (x, y) = and π3sym (x, y, z) = where f (x, y, z) = e−(x

2

(e−x

/2

2

+ e−y /2 )e−(x−y) 4π

2

/2

f (x, y, z) + f (y, z, x) + f (z, x, y) , 12π(2π)1/2

+(y−z)2 )/2

(e−(y−x)

2

/2

+ e−(z−x)

2

/2

).

5.3.2 Check Proposition 5.3.II in detail. 5.3.3 Show that, by a suitable choice of metric, X ∪ in (5.3.10) becomes a c.s.m.s. [Recall the assumption, made in Condition 5.3.I(a), that X is a c.s.m.s.] 5.3.4 Let A(k) denote the k-fold product A × · · · × A. Show that a symmetric measure on the Borel sets of X (2) is determined by its values on sets of the form A(2) but that the corresponding statement for X (k) with k ≥ 3 is false. [Hint: Consider first X = {1, 2} and k = 2, 3.]

132

5. Finite Point Processes (k)

5.3.5 (Continuation). Let Bsym be the smallest σ-algebra containing the sets A(k) (k) for Borel subsets A of X . Show that Bsym consists of all symmetric Borel (k) subsets of X and that any symmetric measure µ on B(k) is completely (k) determined by its values on Bsym . Show also that a symmetric measure µ on (k) B is completely determined by integrals of the form

X (k)

ζ(x1 ) · · · ζ(xk ) µ(dx1 × · · · × dxk )

for functions ζ in the class U of Definition 5.5.I. (n)

5.3.6 Let X0 denote the quotient space X (n) /Π(n) , where Π(n) is the permutation group over the coordinates of a point in X (n) . Prove that there is a one-to-one (n) correspondence between measures on the Borel subsets of X0 and symmetric ∞ (n) (n) measures on the Borel subsets of X . [Macchi (1975) uses n=0 X0 in place ∪ of X in (5.3.10) as the sample space for finite point processes.] 5.3.7 Let {jk (·): k = 1, 2, . . .} be a family of positive Janossy densities for an a.s. finite point process. Define functions ψ1 (x) = − log j1 (x), ψk (x1 , . . . , xk ) = − log jk (x1 , . . . , xk ) −

k−1

ψr (xi1 , . . . , xir ).

r=1 1≤i1 1 the ordinary powers inside the expectation in (5.4.3) are replaced by factorial powers: with Ai and ki as in (5.4.3), we set (k1 )

M[k] (A1

r) × · · · × A(k ) = E [N (A1 )][k1 ] · · · [N (Ar )][kr ] . r

(5.4.4)

As for Mk , the set function on the left-hand side of this defining relation is countably additive on rectangle sets in X (k) and can be interpreted as the expectation measure of a certain point process in X (k) . In this case, the realizations of the new process consist of all k-tuples of distinct points from the original process, still distinguishing the order within the k-tuple but not allowing repetitions. (Note that if the original process N has multiple points, each such point is to be enumerated according to its multiplicity: for example, a double point of N should be regarded as two distinct points having the same coordinates when constructing the k-tuples.) Then M[k] (A) represents (k) the expected number of such k-tuples falling in A ∈ BX . Proposition 5.4.I. If µk = E([N (X )]k ) < ∞, the set functions Mk and M[k] defined by (5.4.3) and (5.4.4) are countably additive on rectangle sets and have unique extensions to symmetric measures Mk and M[k] , respectively, (k) on BX .

134


Using the identities (5.2.2) and (5.2.3) that relate ordinary and factorial powers, it is possible to write down explicit expressions for Mk on certain sets in terms of {M[j] , j = 1, . . . , k} and for M[k] in terms of {Mj , j = 1, . . . , k}. Directly from (5.2.5), we have the important special case k

E [N (X )]k = Mk (A(k) ) = ∆j,k M[j] (A(j) ).

(5.4.5)

j=1

Such relations are particularly useful when the factorial moment measures are absolutely continuous so that the right-hand side of (5.4.5) can be expressed as a sum of integrals of the product densities introduced below Lemma 5.4.III. Note also relations such as M[2] (A × B) = E[N (A)N (B)] − E[N (A ∩ B)] = M2 (A × B) − M (A ∩ B)

(A, B ∈ BX )

(5.4.6)

(see Exercises 5.4.1–6 for a more systematic exposition of such relations). Applications of these moment measures appear in subsequent chapters; here we explore their relation to the Janossy measures and their interpretation in terms of product densities. Since (5.4.4) is simply the factorial moment of a fidi distribution, which can be expressed in terms of the Janossy measures by means of (5.3.11), we can obtain an expression for M[k] (·) in terms of Janossy measures. To examine this expression, we return to the case where A1 , . . . , Ar is a partition of X . Assuming E([N (X )][k] ) < ∞, we have directly from the definitions, when k1 + · · · + kr = k, that

(k ) [k ] r) M[k] (A1 1 × · · · × A(k )= j1 1 · · · jr[kr ] Pr (A1 , . . . , Ar ; j1 , . . . , jr ) r ji ≥ki , i=1,...,r

=

r)

Jj +···+j (A(j1 ) × · · · × A(j r ) 1 r 1 . r i=1 (ji − ki )!

ji ≥ki

To simplify the last sum, put ni = ji − ki and group together the terms for which n1 + · · · + nr = n. Setting k = k1 + · · · + kr , we obtain (k1 )

M[k] (A1

r) × · · · × A(k ) r ∞

1 = n! n=0

ni =n

n (k +n ) Jk+n (A1 1 1 × · · · × Ar(kr +nr ) ). n1 · · · nr

The inner sum can be reduced by Lemma 5.3.III, taking A = X and defining S by (k ) (n) r) S(B) = Jk+n (A1 1 × · · · × A(k × B) (B ∈ BX ), r thereby yielding the equation (k ) M[k] (A1 1

× ··· ×

r) A(k ) r

∞ (k ) (k )

Jk+n (A1 1 × · · · × Ar r × X (n) ) = . n! n=0

5.4.

Moment Measures and Product Densities

135

Using the countable additivity of both sides, this extends to the following elegant generalization of (5.2.12), Mk (B) =

∞

Jk+n (B × X (n) ) n! n=0

(k)

(all B ∈ BX ).

(5.4.7)

To obtain the inverse relation, suppose that all factorial moments µ[k] of N (X ) exist and that the p.g.f. P (1 + η) =

∞

µ[k] η k k=0

(5.4.8)

k!

is convergent in a disk |η| < 1 + for some > 0 [equivalently, that P (z) = E(z N (X ) ) is analytic in some disk |z| < 2+]. Then, the inverse relation (5.2.1) can be applied to yield, with the same notation as in (5.4.7) and following a parallel route, (k1 )

Jn (A1

r) × · · · × A(k )= r

∞

(k1 )

(−1)k

M[n+k] (A1

k=0

=

r

(kr )

× · · · × Ar k!

(j )

(−1)ji −ki

ji ≥ki i=1

× X (k) ) (j )

M[j1 +···+jr ] (A1 1 × · · · × Ar r ) (ji − ki )!

(n)

so that for general B ∈ BX , Jn (B) =

∞

(−1)k

k=0

M[n+k] (B × X (k) ) . k!

(5.4.9)

These results may be summarized for reference in the following theorem. Theorem 5.4.II. If the total population size has finite kth moment, then the kth factorial moment measure is defined and finite and can be represented in terms of the Janossy measures by (5.4.7). Conversely, if all moments are finite and for some > 0 the p.g.f. (5.4.8) is convergent for |η| < 1 + , then the Janossy measures can be represented in terms of the factorial moment measures by (5.4.9). Example 5.4(a) Avoidance function. To illustrate the application of Theorem 5.4.II, consider the set function P0 (A) ≡ Pr{N (A) = 0} = P1 (A; 0); that is, the probability of finding no points in a given subset A of X , or, equivalently, the probability that the support of N avoids A. Taking n = 0 in (5.4.9) and restricting X to A itself, we obtain immediately P0 (A) = J0 (A) =

∞

k=0

(−1)k

M[k] (A(k) ) . k!

(5.4.10)

136


An important feature of (5.4.10) is that it is not necessary to know anything about the nature of the moment measure outside A to determine the probability. In the case X = R and A equal to the interval (0, t], the result in (5.4.10) gives the survivor function for the forward recurrence time in terms of the moment measures on (0, t]. Of course, from another point of view, (5.4.10) is just a special case of equation (5.2.11) giving the probabilities of a discrete distribution in terms of the factorial moments. We now turn and consider densities for the moment measures, assuming X to be a real Euclidean space (or well-behaved subset thereof). Recall the standard result, which follows from Fubini’s theorem, that if a totally finite measure can be represented as the superposition of a finite or countably infinite family of component measures, then it is absolutely continuous with respect to a given measure if and only if each component is absolutely continuous, the density of the superposition being represented a.e. by the sum of the densities. Applied to the representation (5.4.7), this yields immediately the following lemma. Lemma 5.4.III. If the kth factorial moment measure M[k] (·) exists, then it is absolutely continuous if and only if the Janossy measures Jn (·) are absolutely continuous for all n ≥ k, in which case the densities m[k] (·) and jn (·) are related by the equations, for k = 1, 2, . . . , ∞

1 m[k] (x1 , . . . , xk ) = ··· jk+n (x1 , . . . , xk , y1 , . . . , yn ) dy1 · · · dyn . n! X X n=0 The inverse relation follows in a similar way: if all the factorial moment measures exist and are absolutely continuous, and if the series (5.4.9) is absolutely convergent, then the corresponding Janossy measure is absolutely continuous with density given by ∞

(−1)k jn (x1 , . . . , xn ) = · · · mn+k (x1 , . . . , xn , y1 , . . . , yk ) dy1 · · · dyk . k! X X k=0 (5.4.11) Historically, the introduction of factorial moment densities, also referred to as product densities in Bhabha (1950) and Ramakrishnan (1950) and as coincidence densities in Macchi (1975), considerably preceded the more general treatment as above using factorial moment measures. This is easily understood in view of the simple physical interpretation of the densities: equations (5.4.7) and (5.3.9) imply that if m[k] (x1 , . . . , xk ) is bounded in a neighbourhood of (x1 , . . . , xk ), then we can write ∞

Jk+n (dx1 × · · · × dxk × X (n) ) n! n=0 one particle located in each of the = Pr , infinitesimal subsets dxi (i = 1, . . . , k)

m[k] (x1 , . . . , xk ) dx1 · · · dxk =

(5.4.12)

where dxi denotes both the infinitesimal set (xi , xi + dxi ) and its Lebesgue

5.4.


137

measure. This interpretation may be contrasted with that for the density jk (x1 , . . . , xk ) dx1 · · · dxk exactly k points in realization, one in each = Pr . subset dxi (i = 1, . . . , k), and none elsewhere

(5.4.13)

From an experimental point of view, (5.4.12) can be estimated from the results of k observations at specific times or places, whereas the Janossy measure requires indefinitely many observations to determine the exact (total) number of occurrences. For this reason, the densities (5.4.12) are in principle amenable to experimental determination (through ‘coincidence’ experiments, hence the name coincidence densities) in a way that Janossy measures are not, at least in the context of counting particles. However, as Macchi (1975) has stressed, the Janossy measures, and hence the joint distributions, can be determined by the converse relations (5.4.9) and (5.4.11). Moment measures also have the important feature, in common with relations such as (5.4.10), that they are global in character, in contrast to the local character of the Janossy measures. We mean by this that the form of the moment measures is not influenced by the nature of the region of observations: if two observation regions overlap, the moment measures coincide over their common region. On the other hand, the Janossy measures depend critically on the observation regions: just as the number of points observed in the region depends on its size and shape, so also the Janossy measures are exactly tailored to the particular region. This feature lends further importance to the converse relations (5.4.9) and (5.4.11): knowing the moment densities, the Janossy densities for any observation region A can be calculated by taking X = A in (5.4.11), a remark that continues to have force even when the point process is not totally finite over the whole of X . Thus, the one set of moment measures suffices to determine the Janossy measures for as many observation regions as one cares to nominate. When the region of interest is indeed a bounded subset A of the space X where the point process is defined, we introduce the following definition. Definition 5.4.IV (Local Janossy Measures and Densities). Given any bounded Borel set A, the Janossy measures localized to A are the measures Jn (· | A) (n = 1, 2, . . .) satisfying, for locations xi ∈ A (i = 1, . . . , n), exactly n points in A at . Jn (dx1 × · · · dxn | A) = Pr locations dx1 , . . . , dxn When these measures have densities, they define the local Janossy densities. Such local functions have particular importance when the process is no longer a.s. finite-valued on the whole space X . For these local functions the identities in (5.4.9) and (5.4.11) continue to hold with X (k) replaced by A(k)

138


(and the local functions on the respective left-hand sides), as for example jn (x1 , . . . , xn | A) ∞

(−1)k = · · · mn+k (x1 , . . . , xn , y1 , . . . , yk ) dy1 · · · dyk . k! A A

(5.4.14)

k=0

What is remarkable about such a relation is that by merely changing the range of integration of a function defined globally, we can recover the local probabilistic structure when all the moments exist [see Example 5.5(b)]. Local Janossy densities jn (x1 , . . . , xn | A) feature prominently in the discussion of point process likelihoods in Section 7.1. The existence of densities is closely linked to the concept of orderliness, or more properly, simplicity, in the sense of Chapter 3, that with probability 1 there are no coincidences amongst the points. Suppose on the contrary that, for some population size n, the probability that two points coincide is positive. In terms of the measure Jn (·), the necessary and sufficient condition for this probability to be positive is that Jn (·) should allot nonzero mass to at least one (and hence all) of the diagonal sets {xi = xj }, where xi is a point in the ith coordinate space. Thus, we have the following proposition. Proposition 5.4.V. (a) A necessary and sufficient condition for a point process to be simple is that, for all n = 1, 2, . . . , the associated Janossy measure Jn (·) allots zero mass to the ‘diagonals’ {xi = xj }. (b) When X = Rd , the process is simple if for all such n the Janossy measures have densities jn (·) with respect to (nd)-dimensional Lebesgue measure. It is more convenient to frame an analogous condition in terms of the moment measures (assuming they exist). From the preceding result and the representation (5.4.7), we have immediately the following proposition. Proposition 5.4.VI. Suppose the second factorial moment measure M[2] (·) exists. Then, a necessary and sufficient condition for the point process to be simple is that M[2] (·) allots zero mass to the ‘diagonal’ set {xi = xj }. In particular, for X = Rd , the process is simple whenever M[2] (·) has a density m[2] (·) with respect to 2d-dimensional Lebesgue measure. An alternative approach to this proposition can be given in the context of random measures: for the stationary case, see Proposition 8.1.IV and its Corollary 8.1.V. In some applications, we may wish to verify that a given family of densities constitutes the product densities of some point process. The following result gives a simple sufficient condition, which, however, is far from necessary (see remarks after the proof). Proposition 5.4.VII. Let m[k] (·) on X (k) (k = 1, 2, . . .) be a family of symmetric nonnegative functions with finite total integrals µ[k] = m[k] (x) dx, X (k)

5.4.


139

∞ k and suppose that for some > 0 the series k=1 µ[k] z is convergent for |z| < 1 + . Then, a necessary and sufficient condition for the family {m[k] (·)} to be factorial moment densities of a finite point process is that the integrals in (5.4.11) should be nonnegative for every n = 1, 2, . . . and every vector x = (x1 , . . . , xn ). These factorial moment densities then determine the process uniquely. Proof. The integrals are convergent by assumption and clearly define a family of nonnegative symmetric functions. The only other requirement needed for them to form a set of Janossy functions is the normalization condition (5.4.9). On integrating (5.4.11) over x1 , . . . , xn , the required condition is seen to be equivalent to demanding that if we define {pn } by N ! pn =

∞

(−1)k

k=0

µ[k+n] , k!

then the {pn } should sum to unity. But this reduces to the condition µ[0] = m[0] = 1, which may be assumed without loss of generality. ∞ Remarks. The constraint that k=1 µ[k] /k! converges for |z| < 1+ is stronger than is needed: it is enough that lim supr→∞ (µ[r] /r!)1/r < ∞, but a more complicated definition of pn may then be needed (see Exercises 5.4.3–4). Also, for the product densities to define a point process that is not necessarily a finite point process, it is enough for the result to hold (with either the given or modified conditions on {µ[r] }) with the state space X replaced by a sequence {An } of bounded sets for which An ↑ X as n → ∞. Example 5.4(b) Moment densities of a renewal process (Macchi, 1971a). It is well known (see Chapter 4) that the moment properties of a renewal process are completely specified by the renewal function. Although the renewal process is not a finite point process, the machinery developed in this section can be carried over to give a particularly succinct formulation of this result in terms of the factorial moment densities, where for ease of exposition it is assumed that the renewal density exists, u(·) say. In these terms, and assuming stationarity, the renewal density is just a multiple of the second-moment density since for s < t and with m = M[1] ((0, 1]), m[2] (s, t) ds dt = Pr{renewals in (s, s + ds) and (t, t + dt)} = m ds u(t − s) dt. Similarly, exploiting the regenerative property, we have for t1 < · · · < tk that m[k] (t1 , . . . , tk ) dt1 · · · dtk = Pr{renewals in (ti , ti + dti ), 1 ≤ i ≤ k} = m dt1 u(t2 − t1 ) dt2 · · · u(tk − tk−1 ) dtk . (5.4.15) Thus, when the moment densities exist, a necessary condition for a point process to be a stationary renewal process is that the densities be expressible in the product form (5.4.15).

140


This condition is also sufficient. To see this, assume (5.4.15) holds for some constant m and some function u(·) for each k = 1, 2, . . . . From the cases k = 1, 2, first the constant m and then the function u(·) are identified in terms of first- and second-moment densities. From (5.4.11), we can obtain an expression for the density of the interval distribution by taking X = [0, t] and requiring exactly two events, one at 0 and one at t, thus yielding for the lifetime density f (·) the relation m f (t) = m

∞

(−1)k k=0

=m

∞

k=0

k!

· · · u(x1 )u(x2 − x1 ) · · · u(t − xk ) dx1 · · · dxk

[0,t](k)

k

(−1)

···

u(x1 )u(x2 − x1 ) · · · u(t − xk ) dx1 · · · dxk

0<x1 0 and ζ(xi ) = 0 for some i, and is unity if N = 0. We can get some feel for the p.g.fl. by taking A1 , . . . , Ar to be a measurable partition of X and setting r

ζ(x) = zi IAi (x), (5.5.2) i=1

where IA (x) is the indicator function of the set A and |zi | ≤ 1 for i = 1, . . . , r. The function ζ in (5.5.2) belongs to U, and substitution in (5.5.1) leads to " + ! r * r

N (A ) i , zi IAi (·) = E zi G i=1

i=1

which is just the multivariate p.g.f. of the number of points in the sets of the given partition. The case of a general function ζ ∈ U may be regarded as a limiting form of this result, where every infinitesimal region dx is treated as a separate set in a grand partition of X , and ζ(x) is the coefficient (z value) of the corresponding indicator function in (5.5.2). In this way, the p.g.fl. provides a portmanteau description of the p.g.f. of all possible finite or infinite families of counting r.v.s N (·). As in the case of an ordinary discrete distribution, the p.g.fl. provides a useful way of summarizing and illuminating the complex combinatorial results associated with the moments and a convenient formal tool for deriving relations between them. In further analogy to the univariate case, there are two useful expansions of the p.g.fl., the first about ζ ≡ 0 and the second about ζ ≡ 1. The first results directly from the definition (5.5.1) when the expectation is written out in terms of the elements {(pn , Πn )} of the point process or, equivalently, in terms of the Janossy measures Jn (·) [see Conditions 5.3.I and equation (5.3.11)]. For all ζ ∈ U, we have G[ζ] = p0 + = J0 +

∞

n=1 ∞

pn

1 n! n=1

X (n)

ζ(x1 ) · · · ζ(xn ) Πn (dx1 × · · · × dxn )

(5.5.3a)

ζ(x1 ) · · · ζ(xn ) Jn (dx1 × · · · × dxn ).

(5.5.3b)

X (n)

The second expansion can be derived as a generalization from the case where ζ has the particular form (5.5.2) when the p.g.fl. reduces to a multivariate

146


p.g.f., and the expansion can be expressed in terms the multivariate factorial of ∞ moments. Assuming as in (5.4.8) that the series k=0 µ[k] z k is convergent for |z| < for some > 0 and expressing the factorial moments of the counting r.v.s in terms of the factorial moment measures (5.4.4), we obtain + * + * r r

G zi IAi = G 1 + (zi − 1)IAi i=1

i=1

∞

1 =1+ k! k=1

k1 +···+kr =k

r k (k ) r) (zi − 1)ki M[k] (A1 1 × · · · × A(k ). r k1 · · · kr i=1

The final sum here can be identified with the integral with respect to M[k] (·) r of the product i=1 (zi − 1)ki IAi (xj ) so we have ∞

1 G[1 + η] = 1 + η(x1 ) · · · η(xk ) M[k] (dx1 × · · · × dxk ), (5.5.4) k! X (k) k=1

r

where η(x) = i=1 (zi − 1)IAi (x) in the special case considered. Since any Borel measurable function can be approximated by simple functions such as η, the general result follows by familiar continuity arguments,using the dominated convergence theorem and the assumed convergence of µ[k] z k in |z| < , supposing that |η(x)| < for x ∈ X . By taking logarithms of the expansions in (5.5.3) and (5.5.4), we can obtain expansions analogous to those in (5.2.10) and (5.2.8). The first of these takes the form, under the condition that J0 > 0, ∞

1 log G[ζ] = −K0 + ζ(x1 ) · · · ζ(xn ) Kn (dx1 × · · · × dxn ), (5.5.5) n! X (n) n=1 where J0 = exp(−K0 ) and the Kn (·) (n = 1, 2, . . .) are symmetric signed measures, which, following Bol’shakov (1969), we call Khinchin measures. This expansion is important when the point process is infinitely divisible and can be given a cluster interpretation generalizing that of the compound Poisson distribution (see Section 6.3). Here we note that in this case the measures Kn (·)/K0 can be identified asthe Janossy measures of the process charac∞ terizing the clusters, so K0 = n=1 Kn (X (n) )/n! , and the expansion can be rewritten in the form ∞

1 [ζ(x1 ) · · · ζ(xn ) − 1] Kn (dx1 × · · · × dxn ). (5.5.6) log G[ζ] = n! X (n) n=1 Taking logarithms of the expansions (5.5.4) leads to a development in terms of factorial cumulant measures C[k] , namely log G[1 + η] =

∞

1 η(x1 ) · · · η(xk ) C[k] (dx1 × · · · × dxk ). k! X (k)

k=1

(5.5.7)

5.5.

Generating Functionals and Their Expansions

147

This expansion converges under the same conditions as (5.5.4) itself, namely that the factorial moments µ[k] of the total population size should satisfy µ[k] k < ∞ for some > 0 or, equivalently, that the p.g.f. of the total population size should be analytic within a disk |z| < 1 + . Note that the scope of application of these results can be increased considerably by recalling that X itself can be deliberately restricted to a subspace such as a finite interval or rectangle of the original space in which the process may not even be finite. Relations between the factorial cumulant measures and factorial moment measures can be derived from the expansions (5.5.4) and (5.5.7) by formal substitution or by recalling that the measures appearing in those expansions are symmetric: without this restriction, they are not uniquely defined by integral representations such as (5.5.7). For example, by comparing the linear and quadratic terms of ζ, we have

ζ(x1 ) C[1] (dx1 ) =

ζ(x1 ) M[1] (dx1 ),

(5.5.8a)

X (2)

ζ(x1 )ζ(x2 ) C[2] (dx1 × dx2 ) =

X (2)

ζ(x1 )ζ(x2 ) M[2] (dx1 × dx2 ) −

X

ζ(x1 ) M[1] (dx1 )

X

ζ(x2 ) M[1] (dx2 ), (5.5.8b)

which can be abbreviated to C[1] (dx1 ) = M[1] (dx1 ),

(5.5.8c)

C[2] (dx1 × dx2 ) = M[2] (dx1 × dx2 ) − M[1] (dx1 ) M[1] (dx2 ). (5.5.8d) The latter statement follows because any Borel measure on X (2) is determined by its values on rectangles A × B, which in the case of a symmetric measure may be taken to be squares A × A for which the indicator functions have the form ζ(x1 )ζ(x2 ). In the sequel, we repeatedly use such infinitesimal notation to represent equality of measures on product spaces. Using this notation, the general relation between C[k] and the factorial moment measures M[j] for j ≤ k is most conveniently written in the form, analogous to (5.2.19), C[k] (dx1 × · · · × dxk ) =

k

j=1

(−1)j−1 (j − 1)!

j

M[|Si (T )|] (dxi1 × · · · × dxi,|Si (T )| ).

(5.5.9)

T ∈Pjk i=1

To check that (5.5.9) holds, apply Lemma 5.2.VI to the expansions (5.5.4) for the p.g.fl. and (5.5.7) for its logarithm. Note that in (5.5.9), unlike (5.2.19), here we must take explicit note of the elements xi1 , . . . , xi,|Si (T )| of each constituent set Si (T ) in each partition T in Pjk .

148


In practice, it is convenient to group together those partitions ∗ T in Pjk that have common numbers of elements in their subsets: using to denote summation over such groups, (5.5.9) then yields, for example when k = 4, C[4] (dx1 × · · · × dx4 ) = M[4] (dx1 × · · · × dx4 ) ∗ − M[1] (dx1 )M[3] (dx2 × dx3 × dx4 ) ∗ M[2] (dx1 × dx2 )M[2] (dx3 × dx4 ) − ∗ +2 M[1] (dx1 )M[1] (dx2 )M[2] (dx3 × dx4 ) − 6M[1] (dx1 ) · · · M[1] (dx4 ).

(5.5.10)

∗

Here, the first two terms come from P24 , with terms in the former sum four ∗ and three terms in the latter, while the other term comes from P34 and has six terms. This expression then compares immediately with the relation in Exercise 5.2.7. Inverse relations can be derived in the same way and take the form M[k] (dx1 × · · · × dxk ) =

j k

C[|Si (T )|] (dxi1 × · · · × dxi,|Si (T )| ).

j=1 T ∈Pjk i=1

(5.5.11) Just as with integer-valued r.v.s, expansions such as (5.4.9) and (5.5.11) can in principle be combined to provide expressions for the Janossy measures in terms of the factorial cumulant measures and vice versa. While they may appear to be too clumsy to be of any great practical value, when one or more of the entities concerned has a relatively simple structure, as occurs for example with the Poisson process, they can in fact provide a usable theoretical tool (see e.g. Proposition 7.1.III). Similar comments apply to the relations between the Khinchin measures and the factorial moment measures. For ease of reference, we give at the end of this section a summary of the various expansions of the p.g.fl. G[·] of an a.s. finite point process N , together with the corresponding relations between the associated families of measures. First, we illustrate uses of the p.g.fl. in three examples; for the third of these, concerning branching processes, it is convenient to present here a range of results needed later in the book. Example 5.5(a) I.i.d. clusters [continued from Section 5.1 and Example 5.3(a)]. Returning to our initial example, we see that equation (5.1.1) for the joint p.g.f. of this example is a special case of the general form for the p.g.fl. G[ζ] = PN X ζ(x) F (dx) , (5.5.12) where as before PN (·) is a p.g.f. of the cluster size and F (·) is the distribution of the individual cluster members about the origin. The case where PN (·) has the compound Poisson form (see Theorem 2.2.II) PN (z) = e−λ[1−Π(z)]

5.5.


149

and Π(·) is the p.g.f. of the compounding distribution, is of interest. Expanding log G[ζ], we have n ∞

πn ζ(x) F (dx) −1 ; log G[ζ] = λ Π X ζ(x) F (dx) − 1 = λ n=1

X

hence, K0 = λ and for n = 1, 2, . . . , Kn (dx1 × · · · × dxn ) = λπn n! F (dx1 ) · · · F (dxn ). This can be compared with the form for the Janossy measures for which J0 = e−λ and for n = 1, 2, . . . , Jn (dx1 × · · · × dxn ) = πn n! F (dx1 ) · · · F (dxn ), the interpretation being as follows. The process can be regarded as the superposition of ν i.i.d. nonempty subclusters, where ν has a Poisson distribution with mean λ, and for each subcluster, Kn (dx1 × · · · × dxn )/K0 is the probability that the subcluster consists of n points and that they are located at {x1 , . . . , xn }. The Janossy measure yields as Jn (dx1 × · · · × dxn ) the probability that the superposition of the ν subclusters results in n points in all, with these points being located at {x1 , . . . , xn }. In this particular case, the measures Jn (·) and Kn (·) for n = 1, 2, . . . differ only by a scale factor that depends on n: this is a consequence of the i.i.d. nature of the locations of the points. In the more complex examples studied in Chapters 6 and 10, this no longer need hold [see also Example 7.1(e)]. Example 5.5(b) P.g.fl. for the local process on A. Let V(A) denote the space of all measurable functions h on A satisfying 0 ≤ h ≤ 1, and for h ∈ V(A) extend h to all X by putting h∗ (x) = h(x)IA (x). Then, the p.g.fl. GA [h] of the local process on A is defined in terms of the global p.g.fl. G by the equation GA [h] = G[1 − IA + h∗ ]

(h ∈ V(A)).

(5.5.13)

This representation follows immediately from the interpretation of the p.g.fl. as the expectation * + * + GA [h] = E h(xi ) = E [1 − IA (xi ) + h∗ (xi )] . xi ∈A

xi ∈X

Thus, the local Janossy measures can be obtained from an expansion of the p.g.fl. about the function 1 − IA (·) rather than about 0. Specifically, GA [ρh] = G[1 − IA + ρh∗ ] ∞

ρn = p0 (A) + h(x1 ) · · · h(xn ) Jn (dx1 × · · · × dxn | A). n! A(n) n=1 (5.5.14)

150


A similar comment applies to the Khinchin measures arising from the expansion of the log p.g.fl. We can introduce local Khinchin measures, Kn (· | A) say, via the expansion [see equation (5.5.5)] of log GA [ρh] as log G[1 − IA + ρh∗ ]

∞

ρn = −K0 (A) − h(x1 ) · · · h(xn ) Kn (dx1 × · · · × dxn | A), n! A(n) n=1 (5.5.15) where p0 (A) = exp[−K0 (A)]. Example 5.5(c) General branching processes; multiplicative population chains. This basic model stimulated much of the early discussion of generating functionals and moment measures (see e.g. Bartlett and Kendall, 1951; Moyal, 1962a, b) and may be described as follows. A population evolves in discrete time or generations t = 0, 1, . . . . The members of each generation are characterized by both their total number and their locations in the state space X in such a way that the population consisting of the tth generation can be described by a finite point process on X . The fundamental multiplicative property of the process expresses the fact that the population at the (t + 1)th generation is built up as the sum or, more properly, the superposition of the contributing processes representing the offspring from each of the members of the tth generation. Here we shall assume that, given the number Zt and the locations {xti : i = 1, . . . , Zt } of the members of the tth generation, the contributing processes to the (t + 1)th generation are mutually independent and independent of both Zt and all generations prior to t. This relation is then expressible in the form Nt+1 (A) =

Zt

N (A | xti )

(A ∈ BX , t = 0, 1, . . .),

(5.5.16)

i=1

where the Zt finite point processes {N (· | xti ): i = 1, . . . , Zt } are mutually independent. The distributions of the contributing or offspring processes N (· | x) may depend on the location x of the parent. They can be specified by probability distributions {pn (x): n = 0, 1, . . .} and symmetric distributions Πn (· | x) as in Conditions 5.3.I with the additional requirement that, for fixed values of their other arguments, the pn (x) and Πn (· | x) are all assumed to be measurable functions of x for each n = 0, 1, . . . . Then, the offspring p.g.fl., G[ζ | x] say, will also be a measurable function, and the relation (5.5.16) can be expressed as Zt Gt+1 [ζ | Nt ] = G[ζ | xti ], (5.5.17) i=1

where the left-hand side represents the conditional p.g.fl. for the (t + 1)th generation given the number and locations of the members of the tth generation

5.5.


151

as specified by the point process Nt . It is clear that the right-hand side is a measurable function of {Zt , xti (i = 1, . . . , Zt )} and hence that the left-hand side is a measurable function of the finite process Nt . We may therefore take expectations over the left-hand side with respect to Nt , thus obtaining the relation Gt+1 [ζ] = Gt G[ζ | ·] , (5.5.18) where G[ζ | · ] is to be treated as the argument of Gt (note that G[ζ | · ] ∈ U whenever ζ ∈ U). Equation (5.5.18) is a far-reaching generalization of the functional iteration relation for the p.g.f.s of the number of offspring in successive generations of the Galton–Watson process (see also Exercise 5.5.3). Analogous formulae for the factorial moment measures can be established by similar conditioning arguments or else more formally by expanding the p.g.fl. in powers of ζ and equating like terms. We illustrate these procedures for the expectation measures, denoting by M (· | x) the expectation measure for the offspring process N (· | x) with a parent at x and by M(t) (·) the expectation measure for the population at the tth generation. Corresponding to (5.5.17), we have M(t+1) (A | Nt ) =

Zt

M (A | xti ) =

i=1

X

M (A | x) Nt (dx),

(5.5.19)

where again the measurability of M (A | x) as a function of x is clear from the assumptions. Taking expectations with respect to Nt , we then have M(t+1) (A) = M (A | x) M(t) (dx), (5.5.20) X

showing that the expectation measures for successive generations are obtained by operating on M(0) (·) by successive powers of the integral operator with kernel M (· | x). As in the case of a multitype Galton–Watson process (which indeed is the special case when the state space consists of a finite number of discrete points), this operator governs the asymptotic behaviour of the process. In particular, its maximum eigenvalue determines the asymptotic rate of growth (or decay) of the mean population size. These and many other properties are discussed in standard references on general branching processes (see e.g. Moyal, 1962b; Harris, 1963; Athreya and Ney, 1972; Jagers, 1975). Most attention has been given to the case where X is compact, which results in behaviour similar to that of the finite multitype case. New types of behaviour occur in the noncompact case: for example, M (A | ·) may be the kernel of a transient Markov chain, in which case the total mass is preserved but, in contrast to the compact case, the population need not necessarily become extinct—it may continue ‘moving’ indefinitely across the state space as a kind of population wave. Some further aspects and examples are taken up in the exercises [see also Chapter 12 of MKM (1978) and Liemant et al. (1988)]

152


For an alternative derivation of (5.5.20), write ζ = 1 + η in (5.5.18) and expand the two sides. We have

1+ X

η(x) M(t+1) (dx) + · · · = 1 + (G[1 + η(x)] − 1)M(t) (dx) + · · · X =1+ M(t) (dx) η(u) M (du | x) + · · · + · · · , X

X

where all terms omitted involve product terms in η. Equating the measures with respect to which η is integrated on each side of the equation, we obtain (5.5.20). This brief illustration is a typical example of the fact that the p.g.fl. acts as a portmanteau device for condensing a broad range of formulae (see also Exercise 5.5.4). We conclude this section with a summary of the various expansions of the p.g.fl. G[·] of an a.s. finite point process N , together with the corresponding relations between the associated families of measures. For brevity of notation, the latter are written in density form: they can easily be translated into measure notation [for example, equation (5.5.11) is an analogue of (5.5.28) both for measure notation and analogous expansions]. For point processes that are not a.s. finite, the expansions must be applied to the local process on A, N (· ∩ A) say, for any bounded A ∈ BX [see Example 5.5(b)]. Some statements below have already been proved; proofs of the rest are left to the reader. (I) G[h] Janossy measures

(II) G[1 + η] Factorial moment measures

(III) log G[h] Khinchin measures

(IV) log G[1 + η] Factorial cumulant measures

(A) Definitions, Ranges of Validity For suitable measurable functions h and family of measures {µn : n = 0, 1, . . .} with µ0 a constant and µn defined on B(X (n) ), write Y [h, {µn }] =

∞

1 h(x1 ) · · · h(xn ) µn (dx1 × · · · × dxn ), n! X (n) n=1

(5.5.21)

where V denotes the class of measurable functions h: X → [0, 1] such that h(x) = 1 for x outside some bounded Borel set. R denotes the radius of ∞ convergence of the p.g.f. P (z) = n=0 pn z n = E(z N (X ) ). Always, R > 1.

5.5.


153

(I) Janossy Measures {Jn }. G[h] = J0 + Y [h, {Jn }],

(5.5.22)

valid for h ∈ V and subject to {Jn } satisfying the normalizing condition ∞

Jn (X (n) ) . (5.5.23) n! n=1 ∞ (n) , with pn = {Jn (·)/n! } is a probability measure on X ∪ = n=0 X (n) Jn (X )/n! (n = 0, 1, . . . ).

1 = G[1] = J0 +

(II) Factorial Moment Measures {M[n] }. G[1 + η] = 1 + Y [η, {M[n] }],

(5.5.24)

valid for |1 + η| ∈ V for which |η(x)| < (all x) provided R ≥ 1 + > 1, imply that all M[n] (X (n) ) < ∞, M[0] = 1. (III) Khinchin Measures {Kn }. log G[h] = −K0 + Y [h, {Kn }],

(5.5.25)

valid for h ∈ V with K0 > 0 and {Kn } satisfying the normalizing condition K0 =

∞

Kn (X (n) ) . n! n=1

(5.5.26)

For n ≥ 1, Kn (·) need not necessarily be nonnegative; if every Kn (·) ≥ 0, then N is infinitely divisible. (IV) Factorial Cumulant Measures {C[n] }. log G[1 + η] = Y [η, {C[n] }],

(5.5.27)

valid for η as in (II), with R ≥ 1 + > 1 implying that |C[n] (X (n) )| < ∞ for all n, C[0] = 0.

(B) Relations Between Measures in Different Expansions The conditions given for validity are sufficient but not always necessary. (I) → (II). This is a matter of definition! For n such that M[n] (X (n) ) < ∞, ∞

1 m[n] (x1 , . . . , xn ) = jn+r (x1 , . . . , xn , y1 , . . . , yr ) dy1 · · · dyr . r! X (r) r=0 (5.5.28) (II) → (I). For R > 2, ∞

(−1)r jn (x1 , . . . , xn ) = m[n+r] (x1 , . . . , xn , y1 , . . . , yr ) dy1 · · · dyr . r! X (r) r=0 (5.5.29)

154


(I) → (III). K0 = −J0 (and hence needs J0 > 0) and R > 1. kn (x1 , . . . , xn ) =

n

r

(−1)r−1 (r − 1)!

T ∈Prn i=1

r=1

j|Si (T )| (xi1 , . . . , xi,|Si (T )| ). (5.5.30)

(III) → (I). J0 = exp(−K0 ) (and hence needs K0 < ∞) and R > 1. ! jn (x1 , . . . , xn ) = J0

n r

" k|Si (T )| (xi1 , . . . , xi,|Si (T )| ) .

(5.5.31)

r=0 T ∈Prn i=1

(III) → (IV) and (IV) → (III). These are the direct analogues of the relations between (I) and (II), noting that C[0] = 0. Valid for R > 2. (II) → (IV) and (IV) → (II). These are the direct analogues of the relations between (I) and (III), noting that M[0] = 1. Valid for R > 2.

Exercises and Complements to Section 5.5 5.5.1 [ Section 5.1 and Examples 5.3(a) and 5.5(a)]. Derive (5.1.1) from (5.5.12) by j z I (x), where {A1 , . . . , Aj } is a finite partition of X . putting ξ(x) = i=1 i Ai Put ξ = 1 + η to establish the formal relation

G[1 + η] = 1 +

∞

µ[k] k=1

k!

X

···

η(x1 ) · · · η(xk ) Π(dx1 ) · · · Π(dxk ), X

and hence, when µ[k] = E(N (k) ) < ∞, M[k] (dx1 × · · · × dxk ) = µ[k] Π(dx1 ) · · · Π(dxk ), of which the case k = 2 appears in (5.1.3). 5.5.2 For a Gibbs process as in Example 5.3(c), express the Khinchin densities in terms of the interaction potentials ψr (·). More generally, for finite point processes for which the Janossy densities exist, explore the relationship between Khinchin densities and the interaction potentials ψr (·) (see Exercise 5.3.7). 5.5.3 Branching process [continued from Example 5.5(c)]. Let Gt [ζ | x] denote the p.g.fl. for the point process Nt (· | x) describing the points that constitute the tth generation of the process of Example 5.5(c) starting from a single ancestor at x; so, G1 [ζ | x] = G[ζ | x]. Show that for all k = 1, . . . , t − 1, Gt [ζ | x] = Gt−k [Gk [ζ | · ] | x] = G(t) [ζ | x], where G(t) [ζ | x] is the tth functional iterate of G[ · | · ] [see (5.5.18)].

5.5.


155

5.5.4 (Continuation). Let qt (x) denote the probability of extinction within t generations starting from a single ancestor at x, so that qt (x) = Pr{Nt (X | x) = 0}. Show that for each fixed x ∈ X , {qt (x): t = 0, 1, . . .} is a monotonically decreasing sequence and that, for k = 1, . . . , t − 1, qt (x) = Gt−k [qk (·) | x], so, in particular, qt+1 (x) = G[qt (·) | x]. Deduce that the probability of ultimate extinction starting from an initial ancestor at x, q(x) say, is the smallest nonnegative solution of the equation q(x) = G[q(·) | x]. 5.5.5 (Continuation). Show that the first-moment measure M(t) (· | x) of Nt (· | x) (t) and the second factorial cumulant measure, C[2] (A × B | x) say, of Nt (· | x) satisfy the recurrence relations (with M ≡ M(1) )

M(t+1) (A | x) = (t+1)

C[2]

M(t) (A | y) M (dy | x),

X (A × B | x) =

X (2)

M(t) (A | y)M(t) (B | z) C[2] (dy × dz)

+

X

(t)

C[2] (A × B | y) M (dy | x).

[Hint: Use Nt+1 (A | X) =d Nt (A | xi ), where the {xi } denote the indixi viduals of the first generation; see also equations (6.3.3–5).] 5.5.6 (Continuation). Let Ht [ζ | x] denote the p.g.fl. for all individuals up to and including those in the tth generation starting from an initial ancestor at x. Show that these p.g.fl.s satisfy the recurrence relations Ht+1 [ζ | x] = ζ(x)G[Ht [ζ | · ] | x]. Show also that, if extinction is certain, the total population over all generations has p.g.fl. H[ζ | · ], which for 0 < ζ < 1 is the smallest nonnegative solution to the functional equation H[ζ | x] = ζ(x)G[H[ζ | · ] | x], and find equations for the corresponding first two moment measures. 5.5.7 Model for the spread of infection. Take X = Rd , and suppose that any individual infected at x in turn gives rise to infected individuals according to a Poisson process with parameter measure µ(· | x) = µ(· − x | 0) ≡ µ(· − x), where X µ(du) = ν < 1. Show that the total number N (X | 0) of infected individuals, starting from one individual infected at 0, is finite with probability 1 and that the p.g.fl. H[· | · ] for the entire population of infected individuals satisfies the functional equation

H[ζ | 0] = ζ(0) exp

−

(1 − H[ζ | u]) µ(du) , X

where H[ζ | u] = H[Tu ζ | 0] and Tu ζ(v) = ζ(v + u).

156

5. Finite Point Processes Deduce, in particular, the following: (i) The p.g.f. of N (X | 0) satisfies f (z) ≡ Ez N (X |0) = z exp[−ν(1 − f (z))]. (ii) The expectation measure M (· | 0) for the total population of infected individuals, given an initial infected individual at the origin, satisfies

M (A | 0) = δ0 (A) +

M (A − u | 0) µ(du) X

= δ0 (A) + µ(A) + µ2∗ (A) + · · · . (iii) The second factorial moment measure M[2] (A × B | 0) of N (· | 0) satisfies M[2] (A × B | 0) = M (A | 0)M (B | 0)

M[2] (A − u, B − u | 0) µ(du) − δ0 (A)δ0 (B).

+ X

(iv) The Fourier transforms for M (· | 0) and M[2] (· | 0) are expressible in terms of µ ˜(θ) = X eiθ·x µ(dx) thus:

(θ | 0) = M

eiθ·x M (dx | 0) = X

[2] (θ, φ | 0) = M

1 , 1−µ ˜(θ)

ei(θ·x+φ·y) M[2] (dx × dy | 0) =

(θ | 0)M (φ | 0) − 1 M . 1−µ ˜(θ + φ)

5.5.8 Age-dependent branching process. Let X = R, and suppose that an individual born at time u produces offspring according to a Poisson process with parameter measure µ(· | u) = µ(· − u | 0) ≡ µ(· − u) for some boundedly finite measure µ(·) that vanishes on (−∞, 0]. Let Gt [h | 0] denote the p.g.fl. for the ages of individuals present in the population at time t starting from a single newly born individual at time 0. (a) Show that Gt satisfies the equation

Gt [h | 0] = h(t) exp

t

−

(1 − Gt [h | u]) µ(du) , 0

where Gt [h | u] = Gt−u [h | 0] for 0 < u < t. (b) When µ(A) = µ(A ∩ R+ ), show that

Gt [h | 0] = h(t) 1 + µ

−1

t

[1 − h(u)]eµ(t−u) du

.

0

5.5.9 Equation (5.5.29) expresses Janossy densities in terms of factorial moment densities when R > 2. Investigate whether the relation in Exercise 5.2.4 has an analogue for densities valid when only R > 1.

CHAPTER 6

Models Constructed via Conditioning: Cox, Cluster, and Marked Point Processes

In this chapter, we bring together a number of the most widely used classes of point process models. Their common theme is the generation of the final model by a two-stage construction: first, the generation of an indexed family of processes, and then an operation applied to members of the family to produce the final process. The first two classes (Cox and cluster processes) extend the simple Poisson process in much the same way that the mixed and compound Poisson distributions extend the basic Poisson distribution. Independence plays a central role and leads to elegant results for moment and generating functional relationships. Both processes are used typically in contexts where the realizations are stationary and therefore define infinite collections of points. To deal with these issues, we anticipate the transition from finite to general point processes to be carried out in Chapter 9 and present in Section 6.1 a short review of some key results for more general point processes and random measures. The third class of processes considered in this chapter represents a generalization in a different direction. In many situations, events are characterized by both a location and a weight or other distinguishing attribute. Such processes are already covered formally by the general theory, as they can be represented as a special type of point process on a product space. However, marked point processes are deserving of study in their own right because of their wide range of applications, such as in queueing theory, and their conceptual importance in contexts such as Palm theory (see [MKM] especially).

6.1. Infinite Point Families and Random Measures Although the framework developed for finite point processes in Chapter 5 needs to be extended, it nevertheless contains the essential ingredients of the 157

158

6. Models Constructed via Conditioning

more general theory. We retain the assumption that the points are located within a complete, separable metric space (c.s.m.s.) X , and will generally interpret X as either R1 or R2 . The space X ∪ as in (5.3.10) is no longer the appropriate space for defining the realizations; instead we move to a description of the realizations in terms of counting measures, meaning measures whose values on Borel sets are nonnegative integers. The interpretation is that the value of the measure on such a set counts the number of points falling inside that set. A basic assumption, which really defines the extent of current point process theory, is that the measures are boundedly finite: only a finite number of points fall inside any bounded set (i.e. there are no finite accumulation points). In the martingale language of Chapters 7 and 14, this is equivalent to requiring the realizations to be ‘nonexplosive’. The space X ∪ is then replaced by the space1 NX# of all boundedly finite counting measures on X . A remarkable feature is that a relatively simple and natural distance between counting measures can be defined and allows NX# to be interpreted as a metric space in its own right. It then acquires a natural topology and a natural family of Borel sets B(NX# ) that can be used to define measures on NX# . We shall not give details here but refer to Chapter 9 and Appendix A2.6. Thus, the way is open to formally introducing a point process on X as a random counting measure on X , meaning technically a measurable mapping from a probability space (Ω, E, P) into the space (NX# , B(NX# )). Often, the latter space itself is taken as the canonical probability space for a point process on X . Every distinct probability measure on (NX# , B(NX# )) defines a distinct point process. As in the finite case, specific examples of point processes are commonly specified by their finite-dimensional distributions, or fidi distributions for short. These can no longer be defined globally, as was done through the Janossy measures for a finite point process, but are introduced by specifying consistent joint distributions Pk (A1 , . . . , Ak ; n1 , . . . , nk ) = Pr{N (A1 ) = n1 , . . . , N (Ak ) = nk }

(6.1.1)

for the number of points in finite families of bounded Borel sets. Indeed, this was the way we introduced the Poisson process in Chapter 2. Consistency here combines conditions of two types: first, the usual conditions (analogous to those for any stochastic process) for consistency of marginal distributions and invariance under simultaneous permutation of the sets and the numbers falling into them; second, conditions to ensure that the realizations are almost surely measures, namely that N (A ∪ B) = N (A) + N (B) 1

a.s.

and

N (An ) → 0

a.s.

(6.1.2)

# In this edition, we use M# X (and NX ) to denote spaces of boundedly finite (counting)

, X (and N, X ), respectively. measures on X where in the first edition we used M

6.1.

Infinite Point Families and Random Measures

159

for (respectively) all disjoint Borel sets A, B, and all sequences {An } of Borel sets with An ↓ ∅. These two conditions reduce to the requirements on the fidi distributions that, for all finite families of disjoint bounded Borel sets, (A1 , . . . , Ak ), n

Pk (A1 , A2 , A3 , . . . , Ak ; n − r, r, n3 , . . . , nk )

r=0

= Pk−1 (A1 ∪ A2 , A3 , . . . , Ak ; n, n3 , . . . , nk ),

(6.1.3)

and P1 (Ak ; 0) → 1

(6.1.4)

for all sequences of bounded Borel sets {Ak } with Ak ↓ ∅. Moreover, for point processes defined on Euclidean spaces, it is enough for these relationships to hold when the sets are bounded intervals. Example 6.1(a) Simple Poisson process on R. Recall equation (2.2.1): Pr{N (ai , bi ] = ni , i = 1, . . . , k} =

k [λ(bi − ai )]ni i=1

ni !

e−λ(bi −ai ) .

(6.1.5)

Consistency of the marginals means that if one of the variables, say N (a1 , b1 ], is integrated out (by summing over n1 ), the resulting quantity is the joint probability corresponding to the remaining variables. Invariance under permutations of the variables means that if the sets and the number of points falling into them are written down in a different order, the resulting probability is not affected. In the present example, both conditions are obvious from the product form of the joint distributions. The additivity requirement (6.1.3) comes from the additivity property of the Poisson distribution: for Poisson random variables N1 and N2 that are independent (as is implied here by the product form of the distributions), their sum again has a Poisson distribution. Finally, (6.1.4) follows from the property e−δn → 1 when δn → 0. Moment measures, factorial moment measures, and probability generating functionals can be defined as in Sections 5.4 and 5.5. The main differences are that in defining the moment measures we should restrict ourselves to bounded sets and that in defining the p.g.fl. we should confine ourselves to functions h in V(X ), the space of nonnegative, measurable functions bounded by unity and such that 1 − h(x) vanishes outside some bounded set. Within these constraints, the relations between generating functionals, moment measures, and all the various quantities derived from these in Chapter 5 hold much as they did there. A more detailed account, examining existence and convergence conditions, is given in Chapter 9. For many of the examples that we consider, the point processes will be defined on a Euclidean space and stationary, meaning that their fidi distributions are invariant under simultaneous shifts of their arguments: writing

160


A + u = {x + u, x ∈ A}, stationarity means that, for all real u, Pk (A1 , . . . , Ak ; n1 . . . nk ) = Pk (A1 + u, . . . , Ak + u; n1 , . . . , nk ).

(6.1.6)

The full consequences of this assumption are quite profound (see the foretaste in Chapter 3), but for the present it is enough to note the following. Proposition 6.1.I (Stationarity Properties). (i) A point process with p.g.fl. G[h] is stationary if and only if for all real u, G[(Su h)] = G[h], where (Su h)(x) = h(x − u). (ii) If a point process is stationary and the first-moment measure M1 exists, then M1 reduces to a multiple of the uniform measure (Lebesgue measure), M1 (dx) = m (dx) = m dx, say. (iii) If a point process is stationary and the second-moment measure M2 exists, then M2 reduces to the product of a Lebesgue component along the ˘ 2 (du) say, where u = x − y, diagonal x = y and a reduced component2 , M orthogonal to the diagonal. Proof. The fidi distributions as above are determined by the p.g.fl. and can be evaluated by taking h to be the sum of simple functions on disjoint sets; conversely, the fidi distributions determine the p.g.fl., which has the shift-invariance properties under stationarity. Property (ii) can be proved from Cauchy’s functional equation (see Section 3.6), while property (iii) is the measure analogue of the familiar fact that the covariance function of a stationary time series is a function of the difference in the arguments only: c(x, y) = c˘(x − y). Similar expressions for the moment densities follow from property (iii) whenever the moment measures have densities, but in general they have a singular component along the diagonal x = y, which reappears as an atom at the ˘ 2 (·) (see also Section 8.1). General routes to origin in the reduced measure M these reduced measures are provided by the factorization theorems in Section A2.7 or by the disintegration theory outlined in Section A1.4 (see Chapter 8 for further discussion and examples). Estimation of these reduced moment measures and their Fourier transforms (spectral measures) is a key issue in the statistical analysis of point process data and will be taken further in Chapter 8 and in more detail in Chapter 12. We shall also need the idea of a random measure, so we note some elementary properties. The general theory of random measures is so closely interwoven with point process theory that the two can hardly be separated. Point processes are indeed only a special class (integer-valued) of the former, 2

˘ 2 (·) and C ˘2 (·) to denote reduced second moment and covariance In this edition, we use M ,2 (·) and measures (and m ˘ and c˘ for their densities) where in the first edition we wrote M , etc. C(·),

6.1.


161

and much of the general theory runs in parallel for both cases, a fact exploited more systematically in Chapter 9. Here we provide just sufficient background to handle some simple applications. The formal definition of a random measure ξ(·) proceeds much as in the discussion for point processes given above. Once again, the realizations ξ(·) are required to be a.s. boundedly finite and countably additive, and their distributional properties are completely specified by their finite-dimensional distributions. Since the values of the measure are no longer integer-valued in general (although still nonnegative), these take the more general form Fk (A1 , . . . , Ak ; x1 , . . . , xk ) = Pr{ξ(Ai ) ≤ xi , i = 1, . . . , k}.

(6.1.7)

The moment measures are defined as for point processes, although the special role played by the factorial moment measures is not sustained, particularly when the realizations are continuous. In place of the p.g.fl., the most useful transform is the Laplace functional, defined for f ∈ BM+ (X ), the space of all nonnegative f ∈ BM(X ), by L[f ] ≡ Lξ [f ] = E exp − X f (x) ξ(dx) .

(6.1.8)

[We sometimes write Lξ as a reminder of the random measure ξ to which the Laplace functional L relates and f dξ as shorthand for the integral in (6.1.8).] Of course, the Laplace functional can also be defined for point processes and is therefore the natural tool when both are discussed together. Although Lξ defines (the fidi distributions of) a random measure ξ uniquely, via appropriate inversion theorems, there is no easy counterpart to the expansion of the p.g.fl. about the zero function as in equations (5.5.3). There is, however, a Taylor series expansion for the Laplace functional about f ≡ 0, corresponding to the p.g.fl. expansion about h ≡ 1. It takes the form s2 f (x1 )f (x2 ) M2 (dx1 × dx2 ) − · · · f (x) M1 (dx) + 2! X (2) X (−s)r + f (x1 ) . . . f (xr ) Mr (dx1 × · · · × dxr ) + · · · . (6.1.9) r! X (r)

L[sf ] = 1 − s

This expression is just the expectation of the expansion of the ordinary Laplace transform of the linear functional Y = X f (x) ξ(dx). Its validity depends first on the existence of all moments of the random measure ξ and second on the convergence, typically in a disk around the origin s = 0 with radius determined by the length of the largest interval (0, r) within which the Laplace transform is analytic. Finite Taylor series expansions, when just a limited number of moment measures exist, are possible for imaginary values of s, corresponding to the use of the characteristic functional, and are set out in Chapter 9.

162


Example 6.1(b) Gamma random measures (stationary case). Suppose that the random variables ξ(Ai ) in (6.1.7) are independent for disjoint Borel sets Ai in Rd and have the gamma distributions with Laplace–Stieltjes transforms E(e−sξ(Ai ) ) = ψ(Ai , s) = (1 + λs)−α(Ai )

(λ > 0, α > 0, Re(s) ≥ 0), (6.1.10) where (·) denotes Lebesgue measure. By inspection, ψ(Ai , s) → 1 as s → 0, showing that ξ(A) is a.s. finite for any fixed bounded set A. Then, since X is separable, it can be represented as a denumerable union Ai of such sets and ∞

Pr{at least one ξ(Ai ) is infinite} ≤ Pr{ξ(Ai ) = ∞} = 0. i=1

As in the case of a Poisson process, additivity of ξ is a consequence of independence and the additivity property of the gamma distribution. Also, ψ(Ai , s) → 1 as (Ai ) → 0, implying the equivalent of (6.1.4), which guarantees countable additivity for ξ and is equivalent to stochastic continuity of the cumulative process ξ((0, t]) when the process is on R1 . The Laplace functional of ξ can be found by extending (6.1.10) to the case where f is a linear combination of indicator functions and generalizing: it takes the form log[1 + λf (x)] α (dx) . L[f ] = exp − X

Expanding this expression as in (6.1.9) and examining the first and second coefficients, we find E ξ(dx) = λα (dx), (6.1.11) E ξ(dx) ξ(dy) = λ2 α2 (dx) (dy) + δ(x − y)λ2 α (dx). Thus, the covariance measure for ξ(·) vanishes except for the diagonal component along x = y, or, equivalently, the reduced covariance measure is just an atom of mass λ2 α at the origin. These features are consequences of the independence of the increments and the purely atomic nature of the sample paths ξ(·), equivalent when X = R1 to the pure jump character of the cumulative process (see Section 8.3 for further discussion). From these results, we can also confirm the expressions for the moments as follow directly from (6.1.10), namely Eξ(A) = λα (A)

and

var ξ(A) = λ2 α (A).

Exercise 6.1.1 gives a more general version of a gamma random measure. Example 6.1(c) Quadratic random measure. Let Z(t) be a Gaussian process with a.s. continuous trajectories, and consider, for any Borel set A, the set function ξ(A) = Z 2 (u) du. A

6.1.


163

Since Z is a.s. continuous, so is Z 2 , so the integral is a.s. well defined and is additive on disjoint sets. In particular, when Z has zero mean, each value Z 2 (t) is proportional to a chi-square random variable, so ξ(A) for suitably ‘small’ sets A is also approximately a chi-square r.v. Generally, ξ(A) can be defined (being an integral) as a limit of linear combinations of Z 2 (ti ) for points ti that become dense in A, and this is quadratic in the Z, hence the name. The random measure properties of ξ are discussed in more detail in Chapter 9. See Exercise 6.1.3 for the first two moments of ξ. The next example has a long history. It was originally introduced in early work by Campbell (1909) to describe the properties of thermionic noise in vacuum tubes. Moran (1968, pp. 417–423) gives further details and references. In his work, Campbell developed formulae for the moments, such as E g(x) N (dx) = g(x) M (dx), which led Matthes et al. (1978) to adopt the term Campbell measure for the concept that underlies their treatment of moments and Palm distributions (see also Chapter 13). Since that time, the ideas have appeared repeatedly in applications [see e.g. Vere-Jones and Davies (1966), where the model is referred to as a ‘trigger process’ and used to describe earthquake clustering]. Here we introduce it as a prelude to the major theme of this chapter. It is, like the other models in the chapter, a two-stage model, for which we consider here only the first stage. Example 6.1(d) Intensity of a shot-noise process. A model for a shot-noise process is that the observations are those of a Poisson point process with a random intensity λ(·) with the following structure. A stochastic process λ(t) is formed as a filtered version of a simple stationary Poisson process N (·) on R at rate ν with typical realization {ti }, the filtering being effected by (1) a nonnegative function g that integrates to unity and vanishes on (−∞, 0], and (2) random ‘multiplier’ effects, {Yi }, a series of i.i.d. nonnegative random variables with common distribution F (·). We then define λ(t) by ∞

Yi g(t − ti ) = Y (u)g(t − u) N (du), (6.1.12) λ(t) = 0

i:ti 0. (c) Show that ξ has as its Laplace functional

L[f ] = exp

−

log(1 + λf (x)) α(dx)

(f ∈ BM+ (X )).

X

[Hint: See Chapter 9 for more detail, especially parts (b) and (c).]

168


6.1.2 Stable random measure. Consider a random measure ξ for which E(e−sξ(A) ) = (1 + [exp(−sα )])−Λ(A) for some fixed measure Λ(·) and that has independence properties as in Example 6.1(a). Verify that for 0 < α < 1, there is a welldefined random measure with marginal distributions as stated. 6.1.3 Let ξ be the quadratic random measure of Example 6.1(c) in which the Gaussian process Z is stationary with zero mean, variance σ 2 and cov(Z(s), Z(t)) = c(s − t). Show that for bounded Borel sets A and B, E[ξ(A)] = σ 2 (A),

cov(ξ(A), ξ(B)) = 2 A

c2 (u − t) du dt.

B

6.1.4 Random measure and shot noise. Denote by {xi } the points of a stationary Poisson process on R with rate parameter ν, and let {Yj : j = 0, ±1, . . .} denote a sequence of i.i.d. r.v.s independent of {xj }. Let the function g be as in Example 6.1(d). Investigate conditions under which the formally defined process

Yj g(t − xj ) Y (t) = xj ≤t

is indeed well defined (e.g. by demanding that the series is absolutely convergent a.s.). Show that sufficient conditions are that (a) E|Y | < ∞, or else (b) g(·) is nonincreasing on R+ and there is an increasing nonnegative function ∞ g˜(·) with g˜(t) → ∞ as t → ∞ such that 0 g˜(t)g(t) dt < ∞ and whose −1 −1 g (|Y |) < ∞ [see also Daley (1981)]. inverse g˜ (·) satisfies E˜ 6.1.5 Write down conditions, analogous to (6.1.13), for a measurable family of random measures, and establish the analogue of Proposition 6.1.II for random measures. Frame sufficient conditions for the existence of a two-stage process similar to those in Lemma 6.1.III and Corollary 6.1.IV but using the Laplace functional in place of the p.g.fl. 6.1.6 Let ξ be a random measure on X = Rd . For a nonnegative bounded measurable function g, define G(A) = A g(x) (dx) (A ∈ BX ), where denotes Lebesgue measure on Rd , and

G(A − x) ξ(dx).

η(A) = X

(a) Show that η(A) is an a.s. finite-valued r.v. for bounded A ∈ BX and that it is a.s. countably additive on BX . Then, the existence theorems in Chapter 9 can be invoked to show that η is a well-defined random measure. (b) Show that if ξ has moment measures up to order k, so does η, and find the relation between them. Verify that the kth moment measure of η is absolutely continuous with respect to Lebesgue measure on (Rd )(k) . (c) Denoting the characteristic functionals of ξ and η by Φξ [·] and Φη [·], show that, for f ∈ BM+ (X ),

f (y)g(y − x) dy

h(x) = X

is also in BM+ (X ), and Φη [f ] = Φξ [h].

6.2.

Cox (Doubly Stochastic Poisson) Processes

169

6.1.7 (Continuation). By its very definition, η is a.s. absolutely continuous with respect to Lebesgue measure, and when ξ is completely random, its density

g(t − x) ξ(dx)

Y (t) ≡ X

is called a linear process. [The shot-noise process noted in (6.1.12) is an example; for other references, see e.g. Westcott (1970).] Find the characteristic functional of Y when ξ is a stationary gamma random measure.

6.2. Cox (Doubly Stochastic Poisson) Processes The doubly stochastic Poisson process—or, more briefly, the Cox process, so named in recognition of its appearance in a seminal paper of Cox (1955)—is obtained by randomizing the parameter measure in a Poisson process. It is thus a direct generalization of the mixed Poisson process in Example 6.1(e). We first give a definition, then discuss the consequences of the structural features it incorporates, and finally in Proposition 6.2.II give a more mathematical definition together with a list of properties. Definition 6.2.I. Let ξ be a random measure on X . A point process N on X is a Cox process directed by ξ when, conditional on ξ, realizations of N are those of a Poisson process N (· | ξ) on X with parameter measure ξ. We must check that such a process is indeed well defined. The probabilities in the Poisson process N (· | ξ) are readily seen to be measurable functions of ξ; for example, P (A; n) = [ξ(A)]n e−ξ(A) /n! is a measurable function of ξ(A), which in turn is a measurable function of ξ as an element in the metric space M# X of boundedly finite measures on X ; hence, we can apply Corollary 6.1.IV(a) and take expectations with respect to the distribution of ξ to obtain a well-defined ‘mixed’ point process on X . The finite-dimensional (i.e. fidi) distributions are easily obtained in terms of the distributions of the underlying directing measure ξ and are all of mixed Poisson type. Thus, for example, ∞ k [ξ(A)]k −ξ(A) x −x P (A; k) = Pr{N (A) = k} = E = e e FA (dx), k! k! 0 (6.2.1) where FA is the distribution function for the random mass ξ(A). The factorial moment measures of the Cox process turn out to be the ordinary moment measures of the directing measure; this is because the factorial moment measures for the Poisson process are powers of the directing measure. Thus, denoting by µk and γk the ordinary moment and cumulant measures for ξ, we have for k = 2, M[2] (A × A) = E E[N (A)(N (A) − 1) | ξ] = E [ξ(A)]2 = µ2 (A × A) ,

170


and similarly for the covariance measures C[2] (A × A) = γ2 (A × A) . The algebraic details are most easily handled via the p.g.fl. approach outlined in Corollary 6.1.IV(b). As a function of the parameter measure ξ, the p.g.fl. of the Poisson process can be written, for h ∈ V(X ), as (6.2.2) G[h | ξ] = exp − X [1 − h(x)] ξ(dx) . For fixed h, this is a measurable function of ξ as an element of MX . Thus, the family of p.g.fl.s (6.2.2) is a measurable family in the sense of Corollary 6.1.IV(b), which implies that we can indeed construct the p.g.fl of a point process by taking expectations in (6.2.2) with respect to any probability measure for ξ in MX . The expectation [1 − h(x)] ξ(dx) , E exp − X

however, can be identified with the Laplace functional [see (6.1.8)] of the random measure ξ, evaluated at the function [1 − h(x)]. This establishes the first part of the proposition below. The remaining parts are illustrated above for particular cases and are left for the reader to check in general. Proposition 6.2.II. Let ξ be a random measure on the c.s.m.s. X and Lξ its Laplace functional. Then, the p.g.fl. of the Cox process directed by the random measure ξ is given by G[h] = E exp X [h(x) − 1] ξ(dx) = Lξ [1 − h]. (6.2.3) The fidi distributions of a Cox process are of mixed Poisson type, as in (6.2.1); its moment measures exist up to order n if and only if the same is true for ξ. When finite, the kth factorial moment measure M[k] for the Cox process equals the corresponding ordinary moment measure µk for ξ. Similarly, the kth factorial cumulant measure C[k] of the Cox process equals the corresponding ordinary cumulant measure γk for ξ. Note that this last result implies that the second cumulant measure of a Cox process is nonnegative-definite (see Chapter 8). Also, for bounded A ∈ BX , var N (A) = M[1] (A) + C[2] (A × A) = M[1] (A) + var ξ(A) ≥ M[1] (A) = EN (A), so a Cox process, like a Poisson cluster process, is overdispersed relative to the Poisson process. Example 6.2(a) Shot-noise or trigger process [see Example 6.1(d) and Lowen and Teich (1990)]. We continue the discussion of this example by supposing the (random) function

λ(t) = Yi g(t − xi ) (6.2.4) i:xi κ0 } (κ), K µ(du) = I{u>0} (u) du, (c + u)1+p F (dκ) = βe−β(κ−κ0 ) I{κ>κ0 } (κ) dκ. These choices are dictated largely by seismological considerations: thus, the mark distribution cited above corresponds to the Gutenberg–Richter frequency–magnitude law, while the power-law form for µ follows the empirical Omori Law for aftershock sequences. The free parameters ∞ are β, α, c, A and p. K = p cp is a normalizing constant chosen to ensure 0 µ(du) = 1. In this case, sufficient conditions for a stationary process are that p > 0,

β > α,

and

ρ = Aβ/(β − α) < 1.

The last condition in particular is physically somewhat unrealistic since it is well known that the frequency–magnitude distribution cannot retain the pure exponential form indefinitely, but must drop to zero much more quickly for very large magnitudes. An important extension involves adding locations to the description of the offspring so that the branching structure evolves in both space and time. Then, one obvious way of extending the model is to have the ground process include both space and time coordinates, retaining the same mark space K. From the computational point of view, however, and especially for the conditional intensity and likelihood analyses to be described in Chapter 7, there are advantages in keeping the ground process to the set of time points and regarding the spatial coordinates as additional dimensions of the mark. The weight (magnitude) component of the mark retains its unpredictable character (so the weights are i.i.d. given the past), but we allow the spatial component of the mark to be affected by the spatial location of its ancestor. No matter which of these descriptions we adopt, the cluster structure evolves over both space and time, offspring events occurring at various distances away from the initial ancestor, just as they follow it in time. When the branching structure is spatially homogeneous, the infectivity measure µ(dt × dx) depends both on the time delay u = t − t0 and the displacement y = x − x0 from the time and location of the ancestor (t0 , x0 ).

6.4.

Marked Point Processes

205

Various branching mechanisms of this type have been proposed in the literature [see e.g. Ogata (1998) for a review]. Thus, Vere-Jones and Musmeci (1992) suggests a space–time diffusion with infectivity density βe−βu 1 y2 z2 µ(du × dy × dz) = du dy dz, exp − + 2 2πuσy σz 2u σy2 σz whereas Ogata’s space–time ETAS model uses a simpler product form for the space and time terms. Many choices are possible for the components of the model without affecting the underlying cluster character. In some applications, the assumption of spatial homogeneity may not be appropriate, so the infectivity or mark distribution may depend on the absolute location of the offspring as well as its separation from the ancestor. In all of this wide diversity of models, the basic sufficient condition for the existence of a stationary version of the model, essentially the subcriticality of the offspring branching process, is affected only insofar as the integral of the infectivity measure needs to be extended over space as well as time. We conclude this section with a preliminary foray into the fascinating and also practically important realm of stochastic geometry. Marked point processes play an important role here as models for finite or denumerable families of random geometrical objects. The objects may be of many kinds: triplets or quadruplets of points (then, the process would be a special case of a cluster process), circles, line segments, triangles, spheres, and so on. Definition 6.4.VIII (Particle process). A particle process is a point process with state space ΣX equal to the class of nonempty compact sets in X . Thus, a typical realization of a particle process is a sequence, ordered in some way, of compact sets {K1 , K2 , . . .} from the c.s.m.s. X . An underlying difficulty with such a definition is that of finding a convenient metric for the space ΣX . One possibility is the Hausdorff metric defined by ρ(K1 , K2 ) = inf{: K1 ⊆ K2 and K2 ⊆ K1 }, where K is the halo set x∈K S (x) (see Appendix A2.2); for further references and discussion, see Stoyan et al. (1995), Stoyan and Stoyan (1994), and Molchanov (1997), amongst others. In special cases, when the elements are more specific geometrical objects such as spheres or line segments, this difficulty does not arise, as there are many suitable metrics at hand. Very often, interest centres on the union set or coverage process Ξ= Si (see Hall, 1988), which is then an example of a random closed set in X . Now let us suppose that X = Rd and that for each compact set S ⊂ X we can identify a unique centre y(S), for example its centre of gravity. Then, we

206


may introduce an equivalence relation among the sets in ΣX by defining two compact sets to belong to the same equivalence class if they differ only by a translation. The sets in Σo ≡ ΣoX , the compact subsets of X with their centres at the origin, index the equivalence classes so that every set S ∈ ΣX can be represented as the pair (y, S o ), where y ∈ X and S o ∈ Σo , and S = y + S o (set addition). This opens the way to defining the particle process as an MPP {yi , Si }, where the {yi } form a point process in X and the marks {Si } take their values in Σo . Once again, there is the problem of identifying a convenient metric on Σo , but this point aside, we have represented the original particle process as an example of a so-called germ–grain model in which the {yi } are the germs and the {Si } are the grains. The next example illustrates one of the most straightforward and widely used models of this type. Example 6.4(e) Boolean model. This is the compound Poisson analogue for germ–grain models. We suppose that the locations {yi } form a Poisson process in X and that the compact sets Sio are i.i.d. and independent of the location process; write Si = yi +Sio . Two derived processes suggest themselves for special attention. One is the random measure Υ(·) formed by superposing the compact sets Si . With the addition of random weights Wi , this gives the bounded set A the (random) mass

Wi (A ∩ Si ) (A ∈ BX ), (6.4.13) Υ(A) = i

where (·) is the reference measure on X (e.g. Lebesgue measure, or counting measure on a lattice). The other is the localized measure of the union set Ξ described above, which gives the bounded set A the (random) mass # $ Ψ(A) = (A ∩ Ξ) ≡ (6.4.14) i (A ∩ Si ) . For example, (6.4.13) might represent the total mass of ejected material falling within the set A from a series of volcanic eruptions at different locations; then (6.4.14) would represent the area of A covered by the ejected material. In both cases, the processes can be represented in terms of densities forming random processes (random fields) on X . Thus, (6.4.13) and (6.4.14) have respective densities

υ(x) = Wi ISi (x) (6.4.15) and

i

ψ(x) = I{∪i Si } (x).

(6.4.16)

Many aspects of these and related processes are studied in the stochastic geometry literature such as Mathéron (1975), Stoyan et al. (1995) and Molchanov (1997). Here we restrict ourselves to a consideration of the mean and covariance functions of (6.4.15) and (6.4.16) under the more explicit assumptions that X = R2 , that the location process Ng of centres {y(Si )} = {yi } is a simple Poisson process with constant intensity λ, and that each Si is a

6.4.

Marked Point Processes

207

disk of random radius Ri and has weight Wi that may depend on Ri but that the pairs (Ri , Wi ) are mutually independent and independent also of the centres {yi }. Consistent with our earlier description, we thus have an MPP on R2 , with mark space R+ × R+ , and hence a point process N on R2 × R2+ . The mean and covariance function for υ(x) can be found by first conditioning on the ground process Ng as in earlier examples. Thus, writing υ(x) as υ(x) = R2 ×R2+

wI{r≥y−x} (r, y) N (dy × dr × dw)

(6.4.17)

and taking expectations, the independence assumptions coupled with the stationarity of the Poisson process yield E[υ(x)] = λ E W R2 2

I{R≥y} (R, y) dy = λ E W 0

R

2π

r dr dθ 0

= λ π E(W R ) . The second moment E[υ(x1 )υ(x2 )] can be found similarly by first conditioning on the {yi }. Terms involving both pairs of distinct locations and coincident locations (arising from the diagonal term in the second-moment measure of the location process) are involved. However, as for Poisson cluster processes, we find that the covariance cov[υ(x1 ), υ(x2 )] depends only on the term involving coincident locations: it equals E R2 ×R+ ×R+

w2 I{r≥y−x1 ,r≥y−x2 } (r, y) N (dy × dr × dw)

2 = λE W

I{R≥max(y−x1 ,y−x2 )} (R, y) dy = 2λE W 2 R2 arcos(u/R) − u R2 − u2 I{R≥u} (R) , R2

where u = 12 x1 − x2 . Note that the first moment is independent of x and the covariance is a function only of x1 − x2 , as we should expect from the stationary, isotropic character of the generating process. Note also that if the radius R is fixed, the covariance vanishes for x1 − x2 > 2R. The resemblance of these formulae to those for Poisson cluster processes is hardly coincidental. From a more general point of view, the process is a special case of LeCam’s precipitation model in Exercise 6.3.1, where the Poisson cluster structure is generalized to cluster random measures. Some details and extensions are indicated in Exercise 6.4.6. The corresponding formulae for the union process present quite different and, in general, much harder problems since we lose the additive structure for the independent contributions to the sum process. The first moment E[ψ(x)] represents the volume fraction of space (in this case area) occupied

208


by the union set Ξ. It can be approached by the following argument, which is characteristic for properties of the Boolean model. First, note that 1 − E[ψ(x)] = 1 − Pr{Ξ x} = Pr{Ξ x} = E [1 − ISi (x)] . i

Conditioning on the locations {yi } (i.e. on the ground process Ng ), we can write Pr{Ξ x | Ng } = Pr{Ri < x − yi } = h(yi ; x) , i

i

say, where h(y; x) = E[I[0,y−x) (R)] and R has the common distribution of the i.i.d. radii Ri . Removing the conditioning, we have h(yi ; x) = Gg [h(· ; x)] = exp − λ 1 − E[ψ(x)] = E [1 − h(y; x)] dy . R2

i

Substituting for h(y; x) and simplifying, we obtain for the mean density the constant 2 p∗ ≡ E[ψ(x)] = 1 − e−λ E(πR ) . (6.4.18) For the second product moment, using similar reasoning, we have m2 (x1 , x2 ) = E[ψ(x1 )ψ(x2 )] = Pr{Ξ x1 , Ξ x2 } = Pr{Ξ x1 } + Pr{Ξ x2 } − [1 − Pr{Ξ x1 or x2 }] = E[ψ(x1 )] + E[ψ(x2 )] − [1 − Pr{Ξ x1 , Ξ x2 }] = 2p∗ − 1 + Gg [h(· ; x1 , x2 )], say, where h(y; x1 , x2 ) = E[I[0,min(y−x1 ,y−x2 )] (R)]. Substituting for the p.g.fl. of the Poisson ground process, putting u = 12 x1 − x2 and simplifying, we find that m(x1 , x2 ) equals u 2p∗ −1+exp −λE πR2 (1+I{R µE(W ), then there is positive probability that ruin will never occur. (iv) In the latter case, if the Laplace–Stieltjes transform E(e−sW ) is an entire ∗ function of s, then there exists positive real s∗ such that E(e−s W ) = 1. ∗ (v) The sequence {ζn } = {exp(−s Un )} constitutes a martingale for which the time of ruin is a stopping time. (vi) Let pM denote the probability that ruin occurs before the accumulated reserves reach a large number M . Deduce from the martingale property that pM E[exp(s∗ ∆0 ) | 0] + (1 − pM )E[exp(−s∗ ∆M ) | M ] = exp(−s∗ U0 ), where −∆0 and ∆M are the respective overshoots at 0 and M . (vii) Hence, obtain the Cramér bound for the probability of ultimate ruin p = lim pM ≤ exp(−s∗ U0 ) . M →∞

6.4.4 Find first and second factorial moment measures for the ground processes of the marked and space–time Hawkes processes described in Example 6.4(c). [Hint: Use the cluster process representation much as in Example 6.3(c).] 6.4.5 Study the Laplace functional and moment measures for the random measure ξ for a Hawkes process with unpredictable marks. [Hint: Use the cluster representation to get a general form for the p.g.fl. of the process as a process on X × K. From it, develop equations for the first and second moments.] Are explicit results available? 6.4.6 Formulate the process Υ(A) in (6.4.13) as an example of a LeCam process (see Exercise 6.3.1). Show that in the special case considered in (6.4.17), when the random sets are spheres [= disks in R2 ] with random radii we can write

Lξ [f | x] = E exp

−W

R2

f (y) I{R≥ x−y } (y) dy

.

Derive expressions for the mean and covariance functions of υ(x) as corollaries.

210


6.4.7 Higher-order moments of the union set. In the context of the union set Ξ of the Boolean model of Example 6.4(e), show that the kth product moment E[ψ(x1 ) · · · ψ(xk )] = Pr{Ξ xj (j = 1, . . . , xk )}, for k distinct points x1 , . . . , xk in X = R2 , equals

1+

k

(−1)r

r

q(xj1 , . . . , xjr ),

r=1

denotes the sum over all distinct r-tuplets of the set {x1 , . . . , xk }, where q(x1 , . . . , xr ) = Gg [h(· ; x1 , . . . , xr )], and the function h(y ; x1 , . . . , xr ) = Pr{R < min1≤j≤r { xj − y }}. [Hint: The relation arises from taking expectations in the expansion of products of indicator random variables I{Ξ all xj } =

j

=1+

I{Ξ xj } =

k r=1

j

(1 − I{Ξ xj })

r r

(−1)

r

=1

I{Ξ xj }

and

r =1

I{Ξ xj } =

r =1

i

I{Si xj } =

r i

=1

I{Si xj },

and the conditional expectation of the last product, given the locations {yi }, equals h(yi ; xj1 , . . . , xjr ), as indicated.]

CHAPTER 7

Conditional Intensities and Likelihoods

A notable absence from the previous chapter was any discussion of likelihood functions. There is a good reason for this absence: the likelihood functions for most of the processes discussed in that chapter are relatively intractable. This difficulty was a block to the application of general point process models until the late 1960s, when a quite different approach was introduced in papers on filtering theory pioneered by the electrical engineers: see for example Yashin (1970), Snyder (1972), Boel, Varaiya and Wong (1975), Snyder(1975, 2nd ed. Snyder and Miller, 1991), and Kailath and Segall (1975). This approach led to the concept of the conditional intensity function. Once recognised, its role in elucidating the structure of point process likelihoods was soon exploited. General definitions of the conditional intensity function were given in Rubin (1972) and especially by Brémaud (1972), in whose work conditional intensity functions were rigorously defined and applied to likelihood and other problems (see also Brémaud, 1981). Even earlier, Gaver (1963) had introduced what is essentially the same concept through his notion of a random hazard function. Many of these ideas came together in the 1971 Point Process Conference (Lewis, 1972), as a result of which the links between likelihoods, conditional intensities, the theoretical work of Watanabe (1964) and Kunita and Watanabe (1967), and the more practical approaches of Gaver, Hawkes (1971a, b) and Cox (1972a) became more evident. Later, Liptser and Shiryayev (1974, 1977, 1978; 2nd ed. 2000) gave a comprehensive theoretical treatment, while Brémaud (1981) gave a more accessible account that emphasises applications to queueing theory; this same emphasis is in Baccelli and Brémaud (1994). The last two decades have seen the systematic development and application of these ideas to applied problems in many fields, perhaps especially in conjunction with techniques for simulating and predicting point processes. Throughout this chapter runs the theme of delineating classes of models for which the conditional intensity function, and hence the likelihood, has a rel211

212

7. Conditional Intensities and Likelihoods

atively simple form. A key requirement is that the point process should have an evolutionary character: at any time, the current risk—which is just informal terminology for the conditional intensity function—should be explicitly expressible in terms of the past of the process. Many simple point processes in time, including stationary and nonstationary Poisson processes, renewal and Wold processes, and Hawkes processes, fall into this category. So too do many marked point processes in time and also space–time processes, provided that the current distributions of the marks and spatial locations, as well as the current risk, are explicitly expressible in terms of the past. Purely spatial processes—so-called spatial point patterns—cannot be handled so readily this way because they lack a time-like, evolutionary dimension. Nor can processes such as the Neyman–Scott cluster process, in which estimation of the current risk requires averaging over complex combinations of circumstances. However, in some cases of this type, filtering and related iterative techniques can sometimes provide a route forward; they are discussed further in Chapters 14 and 15 alongside the more careful theoretical analysis required to handle conditional intensity functions in a general context. This chapter provides an informal treatment of these issues. We start with a brief introduction to point process likelihoods for a.s. finite point processes, based on the Janossy densities introduced in Chapter 5. In principle the methods can be applied to observations on a general point process observed within a bounded observation region, but in practice the usefulness of this approach is severely curtailed by the difficulty of writing down the Janossy densities for the process within the observation region in terms of a global specification of the process. In Section 7.2, we move to the representation of the likelihood of a simple point process evolving in time. Here the technique of successive conditionings on the past, as the process evolves in time, reduces the difficulty above to that of specifying initial conditions for the process. It leads to a simple and powerful representation of the likelihood in terms of the conditional intensity function. Then, in Section 7.3 we examine the extension of these ideas to marked and space–time point processes, where the process retains an evolutionary character along the time axis. Section 7.4 is devoted to the discussion of intensity-based random time changes, which have the effect of reducing a general initial process to a simple or compound Poisson process. The time changes are motivated by their applications to goodness-of-fit procedures based on the technique of ‘residual point process analysis’. The concluding Sections 7.5 and 7.6 are concerned with uses of the conditional intensity for testing, simulating, and forecasting such processes, and with the links between point process entropy and the evaluation of probability forecasts.

7.1. Likelihoods and Janossy Densities In the abstract at least, there are no special difficulties involved in the notion of a point process likelihood. Granted a realization (x1 , . . . , xn ) in some subset

7.1.

Likelihoods and Janossy Densities

213

A of the state space X , we require the joint probability density of the xi with respect to a convenient reference measure, which when X = Rd is commonly the n-fold product of Lebesgue measure on Rd . As usual, the likelihood should be considered as a function of the parameters defining the joint density and not as a function of the xi and n, which are taken as given. The density here is for an unordered set of points; it represents loosely the probability of finding particles at each of the locations xi and nowhere else within A, and so it is nothing other than the local Janossy density (Definition 5.4.IV) jn (x1 , . . . , xn | A) for the point process restricted to A. These considerations are formalized in the following two definitions. Definition 7.1.I. (a) Given a bounded Borel set A ⊆ Rd , a point process N on X = Rd is regular on A if for all integers k ≥ 1 the local Janossy measures Jk (dx1 × · · · × dxk | A) of Section 5.4 are absolutely continuous on A(k) with respect to Lebesgue measure in X (k) . (b) It is regular if it is regular on A for all bounded A ∈ B(Rd ). Proposition 5.4.V implies that a regular point process is necessarily simple. Definition 7.1.II. The likelihood of a realization x1 , . . . , xn of a regular point process N on a bounded Borel set A ⊆ Rd , where n = N (A), is the local Janossy density LA (x1 , . . . , xn ) = jn (x1 , . . . , xn | A).

(7.1.1)

For convenience, we often abbreviate LA to L. When the whole point process is a.s. finite, and the set A coincides with the space X , the situation is particularly simple. In many cases, the likelihood can be written down immediately from the definition; some examples follow. Example 7.1(a) Finite inhomogeneous Poisson process in A ⊂ Rd . Suppose the process has intensity measure Λ(·) with density λ(x) with respect to Lebesgue measure on Rd . It follows from the results in Section 2.4 that the total number of points in A has a Poisson distribution with mean Λ(A) and that conditional on the number N of such points, the points themselves are i.i.d. on A with common density λ(x)/Λ(A). Suppose we observe the points {x1 , . . . , xn } within A, with n = N (A). In this case, we may assume X = A without any effective loss of generality, as the complete independence property ensures that the behaviour within A is unaffected by realization of the process outside A. Then, taking logs of the Janossy density gives for the log likelihood the formula n

log L(x1 , . . . , xn ) = log λ(xi ) − λ(x) dx, (7.1.2) i=1

A

of which (2.1.9) is the special case X = R. This example continues shortly.

214


Equation (7.1.2) is basic to the likelihood theory of evolutionary processes. As we shall see in the next section, it extends to a wide range of such processes, provided the rate λ(t) is interpreted in a sufficiently broad manner. Another important use for the likelihood in (7.1.2) is as a reference measure for the more general concept of the likelihood ratio. Let N , N be two point processes defined on a common state space X and with probability measures P, P , respectively, on some common probability space (Ω, E). By a mild abuse of language, we shall say that N is absolutely continuous with respect to N , denoting it N N , if P is absolutely continuous with respect to P . In talking about a finite point process on a bounded Borel subset A of Rd , the appropriate probability space is A∪ [see (5.3.8)], and an appropriate reference measure is that of a Poisson process on A with constant intensity. In this context, we have the following result. Proposition 7.1.III. Let N , N be point processes defined on the c.s.m.s. X = Rd , and let A be a bounded Borel set ⊂ Rd . Then N N on A if and only if for each k > 0 the local Janossy measures Jk (· | A) and Jk (· | A) associated with N and N , respectively, satisfy Jk (· | A) Jk (· | A). In particular, if N is the Poisson process with constant intensity λ > 0, then N N if and only if N is regular on A. Proof. If N vanishes identically on A, the conclusion is trivial, so we suppose this is not the case. Recall from the discussionaround Proposition 5.3.II ∞ that an event E from A∪ has the structure E = 0 Sk , where each Sk is a (k) symmetric set; i.e. an element of Bsym (A) (see Exercise 5.3.5). To establish the absolute continuity N N on A, we have to show that if P, P are the probability measures induced on A∪ by N, N , then P(E) = 0 whenever P (E) = 0. Since N is not identically zero, P (E) = 0 only if S0 = ∅ and P (Sk ) = 0 for all k > 0. It is enough here to suppose that Sk is the symmetrized form of a product set A1 × . . . × Ak , where the Ai form a partition of A, since product sets of this form generate the symmetric sets in A(k) . Then, from the definition of the local Janossy measures, k! P(Sk ) = Jk (A1 × . . . × Ak | A) = Jk (Sk | A). Similarly, k! P (Sk ) = Jk (A1 × . . . × Ak | A). Thus, if P (E) = 0, then for each k, Jk (Sk | A) = 0, and if Jk (· | A) Jk (· | A), then Jk (Sk | A) = P(Sk ) = 0 as well, so P(E) = 0. The same equivalences establish the converse relation. If, in particular, N is the Poisson process on A with constant intensity λ, then k Jk (Sk | A) = k! P (Sk ) = λ(Ai ) e−λ(A) , i=1

7.1.


215

where is Lebesgue measure in Rd . Thus, each local Janossy measure Jk (· | A) is proportional to Lebesgue measure in (Rd )k , so Jk (· | A) Jk (· | A) for all k > 0 if and only if N is regular. When densities are known explicitly for both processes, the likelihood ratio for a realization {x1 , . . . , xn } within A is the ratio of the two Janossy densities of order n for the process on A. When the reference measure is that of a Poisson process with unit intensity, P # say, this can be written (A) jn (x1 , . . . , xn | A). LA /L# A =e

(7.1.3a)

In other words, it is directly proportional to the Janossy measure itself. Alternatively, (7.1.3a), or more properly the collection of such expressions for all integers n, can be regarded simply as the density of the given point process on A∪ relative to the Poisson process measure as a reference measure. Written out in full, the Radon–Nikodym derivative for the two measures on A∪ takes the form (see Exercise 5.3.8) ∞

dP λn λ(A) J (7.1.3b) (ω) = e I + (x , . . . , x )I j 0 N (A)=0 n 1 n N (A)=n . dP n! 1 We look again at the inhomogeneous Poisson process example in this light. Example 7.1(a) (continued). As in (7.1.2), PA denotes the distribution associated with an inhomogeneous Poisson process with intensity λ(x). Then, the log likelihood ratio relative to the unit-rate Poisson takes the form log(LA /L# A) =

N

log λ(xi ) −

[λ(x) − 1] dx. A

i=1

One further manipulation of this equation is worth pointing out. Suppose that λ(x) has the form λ(x) = Cφ(x), where C is a positive scale parameter and φ(x) is normalized so that A φ(x) dx = 1. Then (7.1.3) becomes log(LA /L# A ) = N log C +

N

log φ(xi ) − C + (A).

i=1

Differentiation with respect to C yields the maximum likelihood estimate , = N, C and it is clear that here N is a sufficient statistic for C. Moreover, substituting ˆ A , say, and the ratio becomes this value back into the likelihood yields L

ˆ A /L# ) = N log N − N + (A) + log(L log φ(xi ). A

216


Apart from a constant term, this is the same expression as would be obtained by first conditioning on N , when the likelihood reduces to that for N independent observations on the distribution with density φ(xi ). Clearly, in this situation, estimates based on Poisson observations with variable N yield the same results as estimates obtained by first conditioning on N , a statement that is not true with other distributions even asymptotically. Finally, consider the model with constant but arbitrary (unknown) rate C, so that λ(x) = C/(A) with likelihood L0A , say. We find as a special case of the above ˆ 0 /L# ) = N log N − N + (A) − N log (A), log(L A A from which ˆ A /L ˆ0 ) = log(L A

log φ(xi ) + N log (A).

Thus, the term on the right-hand side is the increment to the log likelihood ratio achieved by fitting a model with density proportional to φ(x) over a model with constant density. This elementary observation often provides a useful reduction in the complexity of numerical computations involving Poisson models. The next three examples form some of the key models in representing spatial point patterns within finite regions. Although the likelihoods can be given in more or less explicit form, explicit analytic forms for other characteristics of the process—moment and covariance densities, for example—are not easy to find, mainly because of the intricate links between the numbers and locations of particles within a given region. Another major problem is that, in many important examples, the characteristics of the process are not given directly in terms of the local Janossy measures for the process on A but in terms of global characteristics from which the local characteristics have to be derived. If the process is defined directly in terms of the local Janossy measures, then it is assumed, either tacitly or otherwise, that any effects from points outside the observation region A have been incorporated into the definitions or ignored. If this is not the case—if, for example, one wishes to fit a stationary version of a process with specified interaction potentials—the situation becomes considerably more complex. Allowing for the influence exerted in an average sense by points outside A amounts to nothing less than a generalized version of the Ising problem, where the issue was first posed in the context of magnetized particles in a one-dimensional continuum. The issue is discussed further around Example 7.1(e) and in Chapter 15. In the next three examples, this difficulty is avoided by assuming that the process is totally finite on X and that X = A. Example 7.1(b) Finite Gibbs processes on X ; pairwise interaction systems [see Example 5.3(c)]. An important class of examples from theoretical physics

7.1.


217

was introduced in Example 5.3(c), with Janossy densities and hence likelihoods of the form L(x1 , . . . , xn ) = C(θ) exp[−θU (x1 , . . . , xn )] ,

(7.1.4)

where U can be expressed as a sum of interaction potentials, and the partition function C(θ) is chosen to satisfy the normalization condition of equation (5.3.7). In the practically important case of pairwise interactions, only firstand-second order interaction terms are present, and U takes the form U (x1 , . . . , xn ) =

n

ψ1 (xi ) +

i

n

ψ2 (xi , xj ).

j

Although such models have a valuable flexibility in modelling different types of spatial interactions, their initial attractiveness is somewhat countered by the difficulty of expressing the partition function C(θ) in terms of the other parameters of the model. In fact, exact expressions for the likelihood do not seem to be available in any cases where the second-order term is nontrivial. Ogata and Tanemura (1981) advocate using the approximations (virial expansions) developed by physicists for this purpose, but even so the computations are laborious and their accuracy uncertain. Diggle et al. (1994) compares different numerical approximations. More recent work has focussed on Markov chain Monte Carlo (MCMC) approximations, where the equilibrium solution is obtained numerically as a long-term average of simulations of a Markov chain having the required distribution as its stationary distribution (see e.g. H¨ aggstrøm et al., 1999; Andersson and Britton, 2000, Chapter 11). By judicious choice of the Markov chain transition probabilities, the normalizing constant can be made to disappear from the estimates (e.g. Exercise 7.1.7). Another technique that obviates the need to explicitly evaluate the normalizing constant is to replace the true likelihood L by the pseudolikelihood L† defined by n jn (x1 , . . . , xn ) L† (x1 , . . . , xn ) = . jn−1 ({x1 , . . . , xn } \ xk ) k=1

Since this involves a ratio of Janossy densities, the normalizing constant disappears. It is very much easier, therefore, to derive the pseudolikelihood estimates for a model of this kind than it is to derive the true maximum likelihood estimates. On the other hand, the properties of estimates obtained by maximizing the pseudolikelihood, for example their consistency or asymptotic normality, are currently only partially resolved. In practice, they behave in much the same way as standard maximum likelihood estimates, and it seems likely that in time the theory of both will be subsumed under a more general umbrella. See Baddeley (2001) for examples and further discussion. Example 7.1(c) Strauss processes; hard-core models (Strauss, 1975; Kelly and Ripley, 1976). Strauss processes are the special cases of the model above

218


when ψ1 is a constant α and ψ2 (xi , xj ) has a fixed value β within the range xi − xj < R, for some fixed R < ∞, and is zero outside it. In this case, the Janossy density takes the form jn (x1 , . . . , xn ) = C(α, β, R) αn β m , where m = m(x1 , . . . , xn ) is the number of distinct pairs xi , xj for which xi − xj < R. The Janossy density is constant on hypercylinders around the diagonals xi = xj and their intersections in X (n) . For the process to be well defined, the sum of the Janossy measures must converge [see equation (5.3.9)], which occurs if and only if either β < 1 or β = 1 and α ≤ 1 (cf. Exercise 7.1.8). The condition β < 1 implies some degree of repulsion between points, implying underdispersion relative to the Poisson process. In particular, the choice β = 0 corresponds to a so-called hard-core model, in which points cannot come closer than within a distance R of each other. Other examples of hard-core models appear in Section 8.3. For other values of α and β, the series of Janossy measures diverges so that they no longer correspond to a well-defined finite point process. Thus, the process cannot be used directly to model clustering, but modified Strauss processes with β > 1 can be produced by weighting the Janossy densities with a sequence of constants, wn say, chosen to ensure convergence of the Janossy measures. The most extreme case, corresponding to setting wn = 1 for some selected value of n and to 0 otherwise, corresponds to conditioning on an outcome of fixed size n. See Kelly and Ripley (1976) and Exercise 7.1.8 for details. Example 7.1(d) Markov point processes (Ripley and Kelly, 1977). In order to introduce some concept of Markovianity into the unordered context of spatial point processes, Ripley and Kelly first assume the existence of a relationship ∼ among the points {xi } of a realization. When xi ∼ xj , the points (xi , xj ) are said to belong to the same clique or neighbourhood class. Given any realization of the process, the points may be uniquely divided up into cliques, where a point xi forms a clique by itself if there are no other points xj in the realization for which xi ∼ xj . Let ϕ: X ∪ → R+ be a function defined on cliques V and taking real positive values. Then, a finite point process is said to be a Markov point process if the Janossy density for a realization with a total of N points coming from V cliques Vk with Nk points in Vk takes the form V jN (x1 , . . . , xN | A) = C ϕ(Vk ), (7.1.5)

k=1

where N = k Nk and C is a normalization constant chosen to ensure the Janossy measures satisfy condition (5.3.7). This is equivalent to requiring that the density relative to a unit-rate Poisson process is always proportional V to the product k=1 ϕ(Vk ) no matter how many points the realization may contain.

7.1.


219

A common choice is to take xi ∼ xj if ||xi − xj || < R. We leave the reader to verify that this leads to a well-defined equivalence relation and that if 0 if N (V) ≥ 2, ϕ(V) = α otherwise, then we recover the hard-core version of the Strauss model. Many other important examples of spatial point processes may be put into this form, although the appropriate definitions of clique and the function φ may take some teasing out. A more extended discussion of Markov point processes is given in Chapter 10. In some examples, it is possible to take advantage of a simple expression for the log p.g.fl.; this generally leads to simple expressions for the Khinchin measures, which can then be used to construct the Janossy measures via the combinatorial formulae (5.5.31). The simplest example is the Poisson process, for which only the first Khinchin measure is nonzero, so in the notation of Exercise 5.5.8 we have, say, K0 = − log p0 (A) = λ(x) dx = Λ(A), A

k1 (x | A) = λ(x).

n Then, from (5.5.31) we have jn (x1 , . . . , xn ) = p0 (A) i=1 λ(xi ) as used in (7.1.3a). The next most complicated example of this type is the Gauss–Poisson process described in detail in Example 6.3(d) for which just the first two of the Khinchin measures are nonzero. At this point, we meet an example of the difficulty referred to in the discussion preceding Example 7.1(b). The defining quantities for the Gauss–Poisson process are the measures Q1 (dx) and Q2 (dx1 × dx2 ) described in Proposition 6.3.IV. If the process is observed on a bounded set A, then we have to determine whether these quantities are given explicitly for the process on A or quite generally for the process on the whole of R. In the former case the analysis can proceed directly and is outlined in Example 7.1(e)(i) below. In the latter case, however, and specifically in the case where we want to fit a model with densities q1 (x) ≡ q1 , q2 (x1 , x2 ) = q(x1 − x2 ) corresponding to a stationary version of the process, it is not clear how to allow for the interactions with points of the process lying outside of A and hence unobserved. It turns out that, for this particular model, explicit corrections for the average influence of such outside points can be made and amount to modifying the parameters for the process observed on A. This discussion is outlined in Example 7.1(e)(ii). Example 7.1(e) (i) Gauss–Poisson process on a bounded Borel set A. From (6.3.30) or Exercise 6.3.12, we know that the log p.g.fl. of a Gauss–Poisson process defined on a bounded Borel set A as state space has the expansion [1 − h(x)h(y)] K2 (dx × dy). − log G[h] = [1 − h(x)] K1 (dx) + A

A(2)

220


Assume that K1 (dx) = µ(x) dx and K2 (dx × dy) = 12 q(x − y) dx dy for some function µ(·) and some symmetric function q(·). Then, the Khinchin densities kr are given by k2 (x, y) = q(x − y),

k1 (x) = µ(x),

kr (·) = 0

and

(all r = 3, 4, . . .),

K0 = − log p0 (A) =

µ(x) dx + A

=

1 2

k1 (x) dx + A

1 2

q(x − y) dx dy A A k2 (x, y) dx dy. A A

We turn to the expansion of the Janossy densities in terms of Khinchin densities given by equation (5.5.31), namely jn (x1 , . . . , xn | A) = exp(−K0 )

n r

r=1 T ∈Prn i=1

k|Si (T )| (xi1 , . . . , xi,|Si (T )| ),

where the inner summation is taken over all partitions T of x1 , . . . , xn into i subsets as described above Lemma 5.2.VI. The only nonzero terms arising in this summation are those relating to partitions into sets of sizes 1 and 2 exclusively. This leads to the form for the Janossy densities jn (x1 , . . . , xn | A)

∗

[n/2]

= p0 (A)

µ(xi1 ) · · · µ(xin−2k ) q(xi1 − xi2 ) · · · q(xi2k−1 − xi2k ),

(7.1.6)

k=0

∗ where the summation extends over the n!/[(n − 2k)! 2k ] distinct sets of k pairs of different indices (i1 , i2 ), . . . , (i2k−1 , i2k ) from {1, . . . , n} satisfying i2j−1 < i2j (j = 1, . . . , k) and i1 < i3 < · · · < i2k−1 , and {i1 , . . . , in−2k } is the complementary set of indices. Given a realization x1 , . . . , xn of a Gauss–Poisson process on a set A, its likelihood is then jn (x1 , . . . , xn | A), which is in principle computable but in practice is somewhat complex as soon as n is of moderate size. Newman (1970) established (7.1.6) by an induction argument. (ii) Stationary Gauss–Poisson process. In the specific case of a stationary (translation-invariant) Gauss–Poisson process, we can proceed as follows. The global process is defined by two global parameters, a mean density, say m, and a factorial covariance measure C˘[2] , which we shall assume to have density q(x − y). From these we can obtain obtain versions of the local Khinchin densities from equations, analogous to (5.4.11), k1 (x | A) = c[1] (x) +

∞

(−1)j i=1

j!

A(j)

c[1+j] (x, y1 , . . . , yj ) dy1 · · · dyj ,

7.1.


which here reduces to

221

k1 (x | A) = m −

q(x − y) dy ≡ µ(x)

(x ∈ A),

A

and k2 (x1 , x2 | A) = q(x1 − x2 )

(x1 , x2 ∈ A),

while all higher-order Khinchin measures vanish. Since these two densities define the two measures Q1 , Q2 characterizing a Gauss–Poisson process [see Example 6.3(d)], we see firstly that the process on A is still a Gauss–Poisson process and secondly that its defining measures, unlike the moment measures, depend explicitly on the locations within the observation set A. In other words, although the local process on A is still a process of correlated pairs, its properties are no longer constant across A but depend in general on the proximity to the boundary of A. From this discussion, we see that there is no loss of generality in assuming that X = A, although to obviate the need for edge corrections we shall have to assume that the defining measures are not stationary, even though the global process may be so (see also Brix and Kendall, 2002). In principle, it is possible to write down expressions even more complicated than (7.1.6) for cluster processes with up to 3, 4, . . . points in each cluster. Baudin (1981) developed an equivalent systematic procedure for writing down the likelihood of a Neyman–Scott cluster process, but again it is of substantial combinatorial complexity: see Exercises 7.1.5–6 for details (see also Baddeley, 1998). The difficulty of finding the local Janossy measures in terms of global parameters of the model varies greatly with the model. In a few simple cases, such as the Poisson and Gauss–Poisson examples just considered, explicit expressions may be obtained. In other examples, finding exact solutions raises difficulties of principle as much as technical difficulty. Only the evolutionary processes, considered in the later sections of this chapter, provide a substantial class of models for which a ready solution exists and then only by taking special advantage of the order properties of the time-like dimension. Further discussion of the general problem is deferred until Chapter 15. At the practical level, the difficulty can be alleviated to some extent by the use of so-called plus sampling or minus sampling. This consists of either adding to (‘plus’) or subtracting from (‘minus’) the original sampling region A a buffer region in which the points contribute indirectly to the likelihood by virtue of their effects on the probability density of the points in the inner region but are not included as part of the realization as such. Of course, the points in the buffer region do not play their full weight in the analysis, and the corrections so obtained are only approximate. There is clearly some delicacy in choosing the buffer region large enough to improve accuracy by reducing bias (arising from edge effects) but not so large that the improvement is offset by the loss of information due to not making full use of the data points in the buffer region. Edge effects are discussed again at the end of Section 8.1.

222


Another possible strategy is to introduce ‘periodic boundary effects’, essentially by wrapping the time interval around a circle, in the case of a onedimensional problem, or, for a rectangular region in the plane, by repeating the original region (with the original data) at all contiguous positions in a rectangular tiling of the plane with the original region as base set. The rationale behind the procedure is that the missing data in a neighbourhood of the original observation will be replaced by data that may be expected to have similar statistical properties in general terms. Further discussion of these and similar techniques can be found in the texts by Ripley (1981), Cressie (1991), and Stoyan and Stoyan (1994). Example 7.1(f) Fermion and boson processes [see Examples 5.4(c) and 6.2(b)]. Each of these processes is completely specified by a global covariance function c(x, y), and the local Janossy densities appear as either determinants [for the fermion process: see (5.4.19)] or permanents [for the boson process: see (6.2.11)]. In each case, the densities are derived from a resolvent kernel of the integral equation on A with kernel c(· , ·). As for the Gauss–Poisson process, the resulting explicit expressions for the Janossy densities (and thus the likelihoods) incorporate requisite adjustments for boundary effects. We conclude this section with an excursion into the realm of hypothesis testing; it has the incidental advantage of illustrating further the role of the Khinchin density functions. A commonly occurring need in practice is to test for the null hypothesis of a Poisson process against some appropriate class of alternatives, and it is then pertinent to enquire as to the form of the optimal or at least locally optimal test statistic for this purpose. This question has been examined by Davies (1977), whose general approach we follow. The locally optimal test statistic is just the derivative of the log likelihood function, calculated at the parameter values corresponding to the null hypothesis. Davies’ principal result is that this quantity has a representation as a sum of orthogonal terms, containing contributions from the factorial cumulants of successively higher orders. The formal statement is as follows (note that we return here to the general case of an observation region A ⊂ X = Rd ). Proposition 7.1.IV. For a bounded Borel subset A of Rd , let the distributions {Pθ } correspond to a family of orderly point processes on Rd indexed by a single real parameter θ such that (i) for θ = 0 the process is a Poisson process with constant intensity µ, and (ii) for all θ in some neighbourhood V of the origin, all factorial moment and cumulant densities m[k] and c[k] exist and are differentiable functions of θ and are such that for each s = 1, 2, . . . the series ∞

1 ··· c[k+s] (x1 , . . . , xs , y1 , . . . , yk ; θ) dy1 · · · dyk k! A A

k=1

(7.1.7)

7.1.


223

is uniformly convergent for θ ∈ V , and the series ∞

(1 + δ)k ··· c[k] (y1 , . . . , yk ; θ) dy1 · · · dyk k! A A

(7.1.8)

k=1

converges for some δ > 0. & Then, the efficient score statistic ∂ log L/∂θ&θ=0 can be represented as the sum & ∞

∂ log L && = Dk , (7.1.9) D≡ ∂θ &θ=0 k=1

where, with I(y1 , . . . , yk ) = 1 if no arguments coincide and = 0 otherwise and Z(dy) = N (dy) − µ dy, 1 ··· I(y1 , . . . , yk )c[k] (y1 , . . . , yk ; 0) Z(dy1 ) · · · Z(dyk ). Dk = k µ k! A A (7.1.10) Under the null hypothesis θ = 0 and j > k ≥ 1, E(Dk ) = E(Dk Dj ) = 0, 1 · · · [c[k] (y1 , . . . , yk ; 0)]2 dy1 · · · dyk . var Dk = k µ k! A A

(7.1.11a) (7.1.11b)

Proof. We again use the machinery for finite point processes starting with the expression for the likelihood L ≡ Lθ = jn (x1(1)n ; θ) of the realization {x1 , . . . , xn } ≡ {x1(1)n } on the set A in the form [see (5.5.31)] L = exp(−K0 (θ))

j n

k|Si (T )| (xi,1 , . . . , xi,|Si (T )| ; θ),

(7.1.12)

j=1 T ∈Pjn i=1

where the kr (·) denote Khinchin densities and the inner summation extends over the set Pjn of all j-partitions T of the realization {x1(1)n }. Because θ = 0 corresponds to a Poisson process, K0 (0) = µ(A) and kr (y1(1)n ; 0) = 0 unless r = 1 when k1 (y; 0) = µ. Consequently, (7.1.12) for θ = 0 reduces to L0 = µn exp(−µ(A)), as it should. This fact simplifies the differentiation of (7.1.12) because, assuming (as we justify later) the existence of the derivatives & ∂ kr (y1(1)r ; 0) ≡ kr (y1(1)r ; θ)&θ=0 , ∂θ in differentiating the product term in (7.1.12), nonzero terms remain on setting θ = 0 only if at most one set Si (T ) has |Si (T )| > 1 and all other j − 1 sets have |Si (T )| = 1. Thus, & n

∗ kn−j+1 (xr1 , . . . , xrn−j+1 ; 0) ∂ log L && j−1 (log L) ≡ = −K (0) + µ 0 & ∂θ θ=0 µn j=1 =

−K0 (0)

+

n

i=1

µ−i

∗

ki (xr1 , . . . , xri ; 0),

224


∗ where the summation extends over all distinct selections of size i from the set {x }. Since this set is a realization of the process N (·) over A, the 1(1)n ∗ sum is expressible as the integral 1 ··· I(y1(1)i ; 0) N (dy1 ) · · · N (dyi ), i! A A where the factor I(y1(1)i ) avoids repeated indices and division by i! compensates for the i! recurrences of the same set of indices in different orders. This leads to the representation ∞

1 (log L) = −K0 (0) + · · · I(y1(1)i )k (y1(1)i ; 0) N (dy1 ) · · · N (dyi ) i i! µ A A i=1 (7.1.13) now valid on an infinite range for i as the sum terminates after N (A) terms. When the Khinchin measures are known explicitly, (7.1.13) can be used directly. Otherwise, use the expansion akin to (5.5.29) of k(·) in terms of factorial cumulant densities ∞

(−1)j ··· c[i+j] (y1(1)i , u1(1)j ; θ) du1 · · · duj , ki (y1(1)i ; 0) = j! A A j=0 which, in view of the assumption in (7.1.7), both shows that the ki (·) are differentiable as assumed earlier and justifies term-by-term differentiation. Because of (7.1.12), the same is also true of Lθ . Also, since by (5.5.26) K0 (θ) is a weighted sum of all other Khinchin measures, substitution for ki (·) yields ∞ ∞

1 (−1)j ··· K0 (θ) = i! A j! A i=1 j=0 × ··· c[i+j] (y1(1)i , u1(1)j ; θ) du1 · · · duj dy1 · · · dyi , A

A

which on replacing j by j − i, inverting the order of summation, and using j j−i (−1) /[i! (j − i)!] = −(−1)j /j! gives for θ = 0 i=1 K0 (0) = −

∞

(−1)j j=1

j!

A

···

c[j] (u1(1)j ; 0) du1 · · · duj .

A

Similar substitution after differentiation into (7.1.13), rearrangement of the order of summation, and substitution for −K0 (0) yields j ∞

1 (−µ)j−i j! (log L) = µj j! i=0 i! (j − i)! j=1 × ··· c[j] (y1(1)i , u1(1)j−i ; 0) N (dy1 ) · · · N (dyi ) du1 · · · duj−i .

A

A

7.1.


225

Here j we recognize that the inner sum can arise from an expansion of i=1 [N (dvi ) − µ(dvi )], the symmetry of the densities c[j] (·) implying equality of their integrals with respect to any reordering of the indices in a differential expansion such as N (dv1 ) · · · N (dvi ) dvi+1 · · · dvj . Inserting this product form leads to (7.1.9) and (7.1.10). Verification of equations (7.1.11a) and (7.1.11b) under the null hypothesis is straightforward. Example 7.1(g) Poisson cluster processes with bounded cluster size. Suppose the size of the clusters is limited to M so that only the first M terms are present in the expansions in terms of Khinchin or cumulant densities; the Gauss–Poisson case of Example 7.1(e) corresponds to M = 2. Then, for θ > 0, we may define the process as the superposition of a stationary Poisson process with parameter µ and a Poisson cluster process with clusters of size 2, . . . , M with Khinchin measures with densities θkj (y1 , . . . , yj ) taken from the p.g.fl. representation (6.3.32) (i.e. kj is the density of the measure Kj there). Then, the Khinchin densities in the resultant process have the form (identifying the state space X with the set A) K0 (θ) = θ(A) + θ k1 (x; θ) = µ + θk1 (x),

M

1 ··· kj (x1 , . . . , xj ) dx1 · · · dxj , j! A A j=1 kj (x1 , . . . , xj ) = θkj (x1 , . . . , xj )

(j = 2, . . . , M ).

From (7.1.21), we have the expansion & M

∂ log L && 1 I(y1 , . . . , yj ) kj (y1 , . . . , yj ) = · · · N (dy1 ) · · · N (dyj ) & j ∂θ θ=0 j=1 µ A j! A =

M

1 ∗ kj (xr1 , . . . xrj ). µj j=1

& This expression exhibits the efficient score ∂ log L/∂θ&θ=0 as the sum of first-, second-, . . . , M th-order statistics in the observed points x1 , . . . , xN . In the Gauss–Poisson case, only the first- and second-order terms are needed. The derivation here implies that the form of the cluster process, up to and including the detailed specification of the Kj , is known a priori. The situation if the structure is not known is much more complex but would in effect involve taking a supremum over an appropriate family of functions Kj . An alternative representation is available through (7.1.9) and (7.1.10). This has the advantage that the cumulant densities can be specified globally so that no implicit assumptions about boundary effects are needed. It follows from (6.3.32) (see Exercise 6.3.17) that only the first M factorial cumulant densities c[j] need be considered and (since the c[j] are derived from linear combinations

226


of the kj ) that the same kind of structure holds for the c[j] , namely c[1] (x; θ) = µ + θc[1] (x), c[j] (x1 , . . . , xj ; θ) = θc[j] (x1 , . . . , xj )

(j = 2, . . . , M ).

Then (7.1.9) leads to a similar expansion in terms of linear, quadratic, . . . statistics, namely Dk =

1 k! µk

···

A

I(y1 , . . . , yk ) c[k] (y1 , . . . , yk ) Z(dy1 ) · · · Z(dyk ). A

For further examples, asymptotic behaviour in the stationary case, and the possibility of representing the Dk in terms of spectral measures, see Davies (1977) and Exercises 7.1.8–10.

Exercises and Complements to Section 7.1 7.1.1 Let N1 , N2 be two finite Poisson processes with intensity measures Λ1 , Λ2 , respectively. Show that N1 N2 if and only if Λ1 Λ2 (see above Proposition 7.1.III for N1 N2 ). 7.1.2 Exercise 2.1.9 discusses the likelihood of a cyclic Poisson process with rate parameter µ(t) = exp[α + β sin(ω0 t + θ)], though the parametric form is different: eα here equals λ/I0 (κ) there. The derivation of maximum likelihood estimators given there assumes ω0 is known; here we extend the discussion to the case where ω0 is unknown. (a) Show that the supremum of the likelihood function in general is approached by a sequence of arbitrarily large values of ω0 for which sin ω0 ti ≈ constant and cos ω0 ti ≈ constant for every ti of a given realization. A global maximum of the likelihood is attainable if the parameters are constrained to a compact set. (b) Suppose the observation interval T → ∞, and constrain ω0 to an interval [0, ωT ], where ωT /T 1− → 0 (T → ∞) for some > 0. Then, the sequence of estimators ω ,0 (T ) is consistent. [See Vere-Jones (1982) for details.] 7.1.3 Another cyclic Poisson process model assumes µ(t) = α + β[1 + sin(ω0 t + θ)]. Investigate maximum likelihood estimators for the parameters [see earlier references and Chapter 4 of Kutoyants (1980, 1984)]. 7.1.4 Suppose that the density µ(·) of an inhomogeneous Poisson process on the bounded Borel set A such as the unit interval (or rectangle or cuboid, etc.) can be expanded as a finite series of polynomials orthogonal with respect to some weight function w(·) so that

µ(x) = αw(x) 1 +

r

j=1

βj vj (x)

≡ αw(x)ψ(x),

7.1.


227

where A w(x) dx = 1, A w(x)vj (x) dx = 0, A w(x)vj (x)vk (x) dx = δjk (j, k = 1, . . . , r). Show that the problem of maximizing the log likelihood ratio log(L/L0 ), where L0 refers to a Poisson process with density w(x), is N equivalent to the problem of maximizing log ψ(xi ) subject to the coni=1 straint that ψ(x) ≥ 0 on A. This maximization has to be done numerically; the main difficulty arises from the nonnegativity constraint. 7.1.5 Use the relations in equation (5.5.31) between the Janossy and Khinchin densities to provide a representation of the likelihood of a Poisson cluster process in terms of the Janossy densities of the cluster member process. [Hint: Suppose first that the process is a.s. totally finite. Expand log G[h] = (G[h | y] − 1) µc (dy) (h ∈ V(X )) and obtain X

jn (x1 , . . . , xn | y) µc (dy).

kn (x1 , . . . , xn ) = X

In the general case, proceed from the p.g.fl. expansion of the local process on A as in (5.5.14) and (5.5.15).] 7.1.6 (Continuation). When the cluster structure is that of a stationary Neyman– Scott process with µc (dy) = µc dy as in Example 6.3(a) so that G[h | y] =

∞

j

pj

h(y + u) F (du)

≡Q

h(y + u)f (u) du ,

say,

X

j=0

deduce that the Janossy densities for the local process on A are given by

jn (x1 , . . . , xn | A) = exp µc −1

2 n

×

b∈B01 i=1

*

µc

X (|ai |)

Q X

[Q(1 − F (A − y)) − 1] dy (1 − F (A − y))

n

+b(ai )

[f (xj − y)]

aij

dy

,

j=1

where ai = (ai1 , . . . , ain ) is the binary expansion of i = 1, . . . , 2n − 1, |ai | = #{j: aij = 1}, and B01 is the class ofall {0, 1}-valued functions b(·) defined on {ai : i = 1, . . . , 2n − 1} such that b(ai )ai = (1, . . . , 1). [Thus, any b(·) i has b(a) = 0 except for at most n subsets of a partition of {1, . . . , n}, and is here equivalent to in (5.5.31). Baudin (1981) used a b i j T x combinatorial lemma in Ammann and Thall (1979) to deduce the expression above and commented on the impracticality of its use for even a moderate number of points!] 7.1.7 Suppose that for each n the function U ≡ Un of (7.1.4) satisfies Un (x1 , . . . , xn ) ≥ −cn for some finite positive constant c. Show that a distribution is well defined (i.e. that a finite normalizing constant exists). 7.1.8 Clustered version of the Strauss process. In the basic Strauss model of Example 7.1(c), if β > 1, the Janossy densities, and hence also their integrals over the observation region, will tend to increase as the number of points in

228

7. Conditional Intensities and Likelihoods the region increases. Suppose that the densities are taken proportional to wn αn β m(n) , where m(n) is as defined in the example. Then, the integrals are dominated by the quantities Cwn αn β n(n−1) , and a sufficient condition for the process to be well defined is that

wn αn β n(n−1) < ∞.

Show that this condition is not satisfied if wn ≡ 1, and investigate conditions on the wn to make it hold. Note that such modifications will not affect the sampling patterns for fixed n but only the probabilities pn controlling the relative frequency of patterns with different numbers of events. See Kelly and Ripley (1976) for further discussion. 7.1.9 (a) For a stationary Gauss–Poisson process [see Example 7.1(e)] for which c[1] (u) = µ + θ and c[2] (u, v) = θγ(u − v) for some symmetric p.d.f. γ(·) representing the distribution of the signed distance between the points of a two-point cluster, show that its efficient score statistic D (see Proposition 7.1.IV) is expressible as D = D1 + D2 , where D1 = N (A) − µ(A) ≡ Z(A),

γ(x − y) Z(dx) Z(dy).

D2 = A

A

(b) In practice, µ ˆ is estimated by N (A)/(A), so D1 vanishes, and in the second , = N (·) − µˆ(·). Davies (1977) shows that the term, Z is replaced by Z(·) asymptotic results remain valid with this modification, so the efficiency of other second-order statistics can be compared with the locally optimum form D2 . Write the variance estimator in the form (r − 1)

r

[N (∆j ) − µ ˆ(∆j )]2 ,

j=1

where ∆1 ∪· · ·∪∆r is a partition of the observation region A into subregions of equal Lebesgue measure, in a form similar to D2 , and investigate the variance-to-mean ratio as a test for the Gauss–Poisson alternative to a Poisson process. [Davies suggested that the asymptotic local efficiency is bounded by 23 .] 7.1.10 (Continuation). In the case of a Neyman–Scott process with Poisson cluster size distribution, all terms Dk in the expansion in (7.1.9) are present, and D2 dominates D only if the cluster dimensions are small compared with the mean distance between cluster centres. 7.1.11 When the Poisson cluster process of Example 7.1(g) for X = R is stationary and A = (0, t], Dj ≈

1 ··· φj (l1 /t, . . . , lj /t) gj (λ1 , . . . , λj ; t), tj+1 j! µj l1 +···+lj =0

7.2.

Conditional Intensities, Likelihoods, and Compensators where

···

φj (λ1 , . . . , λj ) = R

kj (t1 , . . . , tj ) exp 2πi

R

j

229

λr tr

dt2 · · · dtj

r=1

with λ1 + · · · + λj = 0 and gj (λ1 , . . . , λj ; t) equals

t

··· 0

t

I(t1 , . . . , tj ) exp 2πi 0

j

λr tr

Z(dt1 ) · · · Z(dtj ).

r=1

[Hint: Use Parseval-type relations to show that t−1 E(|Dj − Dj |2 ) → 0 as t → ∞. See also Theorem 3.1 of Davies (1977).]

7.2. Conditional Intensities, Likelihoods, and Compensators If the discussion in the previous section suggests that there are no easy methods for evaluating point process likelihoods on general spaces, it is all the more remarkable, and fortunate, that in the special and important case X = R there is available an alternative approach of considerable power and generality. The essence of this approach is the use of a causal description of the process through successive conditionings. A full development of this approach is deferred to Chapter 14; here we seek to provide an introduction to the topic and to establish its links to representations in terms of Janossy densities. For simplicity, suppose observation of the process occurs over the time interval A = [0, T ] so that results may be described in terms of a point process on R+ . Denote by {t1 , . . . , tN (T ) } the ordered set of points occurring in the fixed interval (0, T ). As in the discussion around equation (3.1.8), the ti , as well as the intervals τi = ti − ti−1 , i ≥ 1, t0 = 0, are taken to be well-defined random variables. Suppose also that the point process is regular on (0, T ), so that the Janossy densities jk (·) all exist (recall Definition 7.1.I). We suppose that if there is any dependence on events before t = 0, it is already incorporated into the Janossy densities. For ease of writing, we use jn (t1 , . . . , tn | u) for the local Janossy density on the interval (0, u), and J0 (u) for J0 ((0, u)). Now introduce the conditional survivor functions Sk (u | t1 , . . . , tk−1 ) = Pr{τk > u | t1 , . . . , tk−1 } and observe that these can be represented recursively in terms of the (local) Janossy functions through the equations S1 (u) = J0 (u) S2 (u | t1 )p1 (t1 ) = j1 (t1 | t1 + u) S3 (u | t1 , t2 )p2 (t2 | t1 ) = j2 (t1 , t2 | t2 + u)

(0 < u < T ), (0 < tt < t1 + u < T ), (0 < t1 < t2 < t2 + u < T ),

and so on, where p1 (t), p2 (t | t1 ), . . . are the probability densities corresponding to the survivor functions S1 (u), S2 (u | t1 ), . . . . The fact that these densities exist is a corollary of the assumed regularity of the process. This can be

230


seen more explicitly by noting identities such as (for S1 (·)) T ∞

1 T ··· jk (u1 , . . . , uk | T ) du1 · · · duk , J0 (t) = J0 (T ) + k! t t k=1

from which p1 (t) = j1 (t | T ) +

∞

k=2

1 (k − 1)!

T

T

··· t

jk (t, u2 , . . . , uk | T ) du2 · · · duk , t

an expression that is actually independent of T for T > t. Similarly, for S2 we find (for t1 < t < T ) p1 (t1 )S2 (t | t1 ) = j1 (t1 | T ) +

∞

k=2

1 (k − 1)!

T

··· t

T

jk (t1 , u2 , . . . , uk | T ) du2 · · · duk t

= j1 (t1 | t), from which it follows that p1 (t1 )p2 (t | t1 ) equals T T ∞

1 j2 (t1 , t | T ) + ··· jk (t1 , t, u3 , . . . , uk | T ) du3 · · · duk , (k − 2)! t t k=3

again establishing the absolute continuity of S2 (t | t1 ). Further results follow by an inductive argument, the details of which we leave to the reader. Together they suffice to establish the first part of the following proposition. Proposition 7.2.I. For a regular point process on X = R+ , there exists a uniquely determined family of conditional probability density functions pn (t | t1 , . . . , tn−1 ) and associated survivor functions t Sn (t | t1 , . . . , tn−1 ) = 1 − pn (u | t1 , . . . , tn−1 ) du (t > tn−1 ) tn−1

defined on 0 < t1 < · · · < tn−1 < t such that each pn (· | t1 , . . . , tn−1 ) has support carried by the half-line (tn−1 , ∞), and for all n ≥ 1 and all finite intervals [0, T ] with T > 0, J0 (T ) = S1 (T ), jn (t1 , . . . , tn | T ) ≡ jn (t1 , . . . , tn | (0, T )) = p1 (t1 )p2 (t2 | t1 ) · · · pn (tn | t1 , . . . , tn−1 ) × Sn+1 (T | t1 , . . . , tn ),

(7.2.1a)

(7.2.1b)

where 0 < t1 < · · · < tn < T can be regarded as the order statistics of the points of a realization of the point process on [0, T ]. Conversely, given any such family of conditional densities for all t > 0, equations (7.2.1a) and (7.2.1b) specify uniquely the distribution of a regular point process on R+ .

7.2.

Conditional Intensities, Likelihoods, and Compensators

231

Proof. Only the converse requires a brief comment. Given a family of conditional densities pn , both J0 (T ) and symmetric densities jk (· | T ) can be defined by (7.2.1), and we can verify that they satisfy T ∞

1 T J0 (T ) + ··· jn (t1 , . . . , tn | T ) dt1 · · · dtn n! 0 0 n=1 ∞

··· = J0 (T ) + jn (t1 , . . . , tn | T ) dt1 · · · dtn = 1. n=1 0 0 and ‘periods of recovery’ when X(t) < 0, the terminology being suggested by the analogy of a collective risk model. Then, an argument similar to that used for M/G/1 queue and analogous storage problems can be used to show that the reduced covariance density c˘[2] (u) has Laplace transform of the form c∗[2] (s) = [1 + ω(s)]−1 , where ω(s) is the unique solution in Re(θ) > 0 of the equation θ − s = σ[1 − j ∗ (θ)] and j ∗ is the Laplace transform of the jump density. 7.2.11 Renewal process compensators. (a) By integrating the conditional intensity function in (7.2.14), show that when the lifetime distribution of a renewal process has a density f , the compensator has the form Λ∗ (t) = −

N (t) n=1

log S(Tn − Tn−1 ) − log S(t − TN (t) ),

where S(·) is the survivor function for the lifetime d.f. with density f . (b) Verify directly that Λ∗ (t) as defined makes N (t) − Λ∗ (t) a martingale. (c) Show that (b) continues to hold for a general renewal process whose lifetime r.v.s are positive a.s., provided the log survivor function is replaced by the integrated hazard function (IHF).

7.3. Conditional Intensities for Marked Point Processes The extension of conditional intensity models to higher dimensions is surprisingly straightforward provided that a causal, time-like character is retained for the principal dimension. When this is present, as in space–time processes, the development of conditional intensities and likelihoods can proceed along

7.3.

Conditional Intensities for Marked Point Processes

247

much the same lines as was developed for one-dimensional simple point processes in the preceding sections. When it is absent, as in purely spatial point patterns, analysis is still possible in the finite case (compare the discussions in Chapter 5 and Section 7.1) but raises major problems for nonfinite cases such as occur for homogeneous processes in the plane. In this section, we examine the extension of the ideas of Section 7.2 to MPPs in time and space– time point processes. A more general and rigorous discussion of conditional intensities and related topics, for both simple and marked point processes in time, is given in Chapter 14. An approach to likelihood methods for spatial processes, based on the Papangelou intensity, is in Chapter 15. The ground work for the material in the present section was laid in the basic paper by Jacod (1975); among many other references, Karr (1986) gives both a review of inference procedures for MPPs and a range of examples and applications. Consider then an MPP on [0, ∞) × K, where, as in Section 6.4, K denotes the mark space, which may be discrete (for multivariate point processes), the positive half-line (if the marks represent weights or energies), two- or three-dimensional Euclidean space (for space–time processes), or more general spaces [e.g. for the Boolean model of Example 6.4(d)]. In order to define likelihoods for MPPs, we need first to fix on a measure in the mark space (K, BK ) to serve as a reference measure in forming densities. We shall denote this reference measure by K (·), using (·) to denote Lebesgue measure on Rd . When K is also some Euclidean space, it will often be convenient to take K to be Lebesgue measure on that space but not always so; for example, in some situations it may be simpler to take K to be a probability measure on K. Similarly, when the mark space is discrete, it will often be convenient to take the reference measure to be counting measure, but in some situations it may again be more convenient to choose the reference measure to be a probability measure. Once the reference measure K has been fixed, we can extend the notion of a regular point process from simple to marked point processes. As in Definition 7.1.I, we shall say that an MPP on X = Rd × K is regular on A for a bounded Borel set A ∈ BX if for all n ≥ 1 the Janossy measure Jn is absolutely continuous with respect to the n-fold product of × K and regular if it is regular on A for all bounded A ∈ BX . Thus, when the MPP is regular on A, for every n > 0 there exists a well-defined Janossy density jn (· | A × K) with the interpretation jn (x1 , . . . , xn , κ1 , . . . , κn | A × K) dx1 . . . dxn K (dκ1 ) . . . K (dκn ) = Pr{points around (x1 , . . . , xn ) with marks around (κ1 , . . . , κn )}. The following equivalences extend to MPPs the discussion around Proposition 7.1.III. Proposition 7.3.I. Let N (·) be an MPP on Rd × K, let denote Lebesgue measure on (Rd , BRd ) and K the reference measure on (K, BK ), and let A be a bounded set in BRd . Then, conditions (i)–(iv) below are equivalent.

248


(i) N (·) is regular on A. ∪ (ii) The probability measure induced by N (·) on ZA , where ZA = A × K, is absolutely continuous with respect to the measure induced by × K on ∪ ZA . (iii) The ground process Ng (·) is regular on A, and for each n > 0 the conditional distribution of the marks (κ1 , . . . , κn ), for a given realization (x1 , . . . , xn ) of the locations within A, is absolutely continuous with re(n) spect to K with density fA,n (κ1 , . . . , κn | x1 , . . . , xn ), say. (iv) If Π(·) is a probability measure equivalent to K on (K, BK ), then N (·) is absolutely continuous with respect to the compound Poisson process N0 (·) for which the ground process N0g has positive intensity λ on A and the marks are i.i.d. with common probability distribution Π. Proof. The four statements are just alternative ways of stating the fact that the Janossy measures Jn (·) in the proposition have appropriate densities on all components of X ∪ . When any one of the conditions is satisfied, the Radon–Nikodym derivative of the probability measure P for N with respect to the probability measure P0 of the compound Poisson process N0 in (iv) has the form [see (7.1.3b)] dP e−λ(A) = J0 I{N (T )=0} dP0 ∞

j g (x1 , . . . , xn | A) fA,n (κ1 , . . . , κn | t1 , . . . , tn ) , + I{N (T )=n} n λn π(κ1 ) · · · π(κn ) n=1 (7.3.1a) in which π(κ) = (dΠ/dK )(κ) and is itself a portmanteau expression of the statements that, given a realization (t1 , κ1 ), . . . , (tn , κn ) with N (T ) = n, the likelihood ratio of N with respect to N0 is given by L/L0 = jng (x1 , . . . , xn | A)fn (κ1 , . . . κn | x1 , . . . , xn )/[λn π(κ1 ) · · · π(κn )]. (7.3.1b) Much as in the discussion leading to Proposition 7.2.I, we now rewrite the Janossy densities in a way that takes advantage of the directional character of time. Thus, the Janossy densities for the first few pairs may be represented in the form J0 (T ) = S1 (T ), j1 (t1 , κ1 | T ) = p1 (t1 , κ1 ) = p1 (t1 ) f1 (κ1 | t1 ) (0 < t1 < T ), j2 (t1 , t2 , κ1 , κ2 | T ) = p1 (t1 ) f1 (κ1 | t1 ) p2 (t2 | (t1 , κ1 )) f2 (κ2 | (t1 , κ1 ), t2 ) (0 < t1 < t2 < T ), where the pi (·) refer to the densities, suitably conditioned, for the locations in the ground process, and the fi (·) refer to the densities, again suitably conditioned, for the marks. There is a subtle difference in the conditioning incorporated into the conditional densities fn (κn | (t1 , κ1 ), . . . , (tn−1 , κn−1 ), tn ) that appear in the equations above and those that appear in the proposition. In the equations above we condition the distribution of the current mark, as

7.3.


249

time progresses, on both marks and time points of all preceding events; in the proposition, we condition on the full set of time points in (0, T ), irrespective of the marks and of their relative positions in time. Once again, the dependence of the left-hand side on T is illusory, and the densities for the locations can be expressed in terms of corresponding hazard functions. The conditioning in the hazard functions may now include the values of the preceding marks as well as the length of the current and preceding intervals. All this information is collected into the internal history H ≡ {Ht : t ≥ 0} of the process so that the amalgam of hazard functions and mark densities can be represented as a single composite function for the MPP, namely ⎧ h1 (t)f1 (κ | t) (0 < t ≤ t1 ), ⎪ ⎪ ⎪ .. ⎪ ⎪ ⎨ . ∗ λ (t, κ) = hn t | (t1 , κ1 ), . . . , (tn−1 , κn−1 ) × ⎪ ⎪ ⎪ fn κ | (t1 , κ1 ), . . . , (tn−1 , κn−1 ), t (tn−1 < t ≤ tn , n ≥ 2), ⎪ ⎪ ⎩ .. . (7.3.2) where h1 (t) is the hazard function for the location of the initial point, h2 (t | (t1 , κ1 )) the hazard function for the location of the second point conditioned by the location of the first point and the value of the first mark, and so on, while f1 (κ | t) is the density for the first mark given its location, and so on. Definition 7.3.II. Let N be a regular MPP on R+ × K. The conditional intensity function for N , with respect to its internal history H, is the representative function λ∗ (t, κ) defined piecewise by (7.3.2). Predictability is again important in that the hazard functions refer to the risk at the end of a time interval, not at the beginning of the next time interval, so left-continuity should be preferred where there is a jump in the conditional intensity. Similarly, the conditional mark density refers to the distribution to be anticipated at the end of a time interval, not immediately after the next interval has begun. More formal and more general discussions of predictibility in the MPP context will be given in Chapter 14. It is often convenient to write λ∗ (t, κ) = λ∗g (t) f ∗ (κ | t) ,

(7.3.3)

where λ∗g (t) is the H-intensity of the ground process (i.e. of the locations {ti } of the events), and f ∗ (κ | t) is the conditional density of a mark at t given Ht− (the reader will note that we use the ∗ notation as a reminder that the ‘functions’ concerned are also random variables dependent in general on the random past history of the process). The two terms in (7.3.3) correspond to the first and second factors in (7.3.2). Heuristically, equations (7.3.2) and (7.3.3) can be summarized in the form λ∗ (t, κ) dt dκ ≈ E[N (dt × dκ) | Ht− ] ≈ λ∗g (t) f ∗ (κ | t) dt dκ .

(7.3.4)

250


Notice that the H-intensity λ∗g (t) is not in general the same as the conditional intensity λg (t) of the ground process with respect to its own internal history Hg : H incorporates information about the values of the marks, whereas Hg does not. The example below illustrates the difference in a simple special case. Example 7.3(a) Bivariate Poisson process [see Example 6.3(e)]. We consider a bivariate Poisson process initiated at time 0 rather than the stationary version considered earlier. We consider also just the process of linked pairs, in which the points {ti } of component I form the ‘parents’ and arrive according to a simple Poisson process with rate λ while the points {sj } of component II represent the process of ‘offspring’. We assume each parent has just one offspring, delayed by nonnegative random times {τi } forming an i.i.d. sequence, independent also of the times {ti }, with common exponential distribution 1 − e−µτ . We shall treat this process as a special case of an MPP with mark space having two discrete points, corresponding to components I and II. The internal history, H, for the full process records the occurrence times and marks for both types of events but does not record which event in component II is associated with which event in component I. Suppose that, at time t, NI (t) = n, NII (t) = m, where necessarily m ≤ n. The full H-intensity is given by λ (κ = I), λ∗ (t, κ) = (n − m)µ (κ = II). Let HI , HII , and Hg denote the internal histories of the component I process, the component II process, and the ground process. The HI -intensity of component I is clearly equal to its H-intensity λI ≡ λ. To find the HII intensity of component II, we have to average over the n ≥ m points of component I. For a given value of n, the locations ti may be treated as n i.i.d. variables uniformly distributed over (0, t). The probability that any one such point produces an offspring that appears only after time t is given by t ds 1 − e−µt p(t) = e−µ(t−s) = . t µt 0 The k = n − m parent points that fail to produce offspring in the interval (0, t) then form a ‘thinned’ version of the original, Poisson-distributed number n of the component I points in (0, t), the selected and nonselected points forming two independent streams. Independently of the number m of successes, the expected number of points with offspring still pending is thus λtp(t) and we obtain for the HII -intensity of the component II process λII (t) = E[(n − m)µ | NII (t) = m] = µλt(1 − e−µt )/(µt) = λ(1 − e−µt ). This is a nonrandom function of t, and we recognize it as the conditional intensity of a nonstationary Poisson process. Thus, the two components separately are Poisson, and the rate of the component II process approaches that of component I as t → ∞. The ground process has H-intensity λ + (n − m)µ

7.3.


251

and HII -intensity λ(2 − e−µt ); its Hg -intensity is that of a Gauss–Poisson process; see Exercise 7.3.1. Similar distinctions need to be borne in mind with respect to the various compensators and martingales that can be formed with the two component processes. Thus, NI (t) − λt is both an H- and an HI -martingale, the process NII (t) − µ(NII (t) − NI (t)) is an H-martingale, and NII (t) − λt(1 − e−µt ) is an HII -martingale. We now turn to an MPP extension of Proposition 7.2.III, expressing the likelihood of a simple point process in terms of its conditional intensity. As there, reversing the construction that leads from the point process distributions to the H-intensity in (7.3.2) yields an explicit expression for the Janossy density of the MPP in terms of its conditional intensity (see below). Details of the proof are left to Exercise 7.3.2. Proposition 7.3.III. Let N be a regular MPP on [0, T ] × K for some finite positive T , and let (t1 , κ1 ), . . . , (tNg (T ) , κNg (T ) ) be a realization of N over the interval [0, T ]. Then, the likelihood L of such a realization is expressible in the form * + T Ng (T ) ∗ ∗ λ (ti , κi ) exp − λ (u, κ) du K (dκ) L= i=1

=

* Ng (T )

0

K

+ * Ng (T ) + ∗ ∗ λg (ti ) f (κi | ti ) exp −

i=1

T

λ∗g (u) du ,

(7.3.5)

0

i=1

where K is the reference measure on K. Its log likelihood ratio on [0, T ] relative to the compound Poisson process N0 with constant intensity λ and i.i.d. mark distribution with density π(·) is expressible as T Ng (T )

L λ∗ (ti , κi ) log = log [λ∗ (u, κ) − λπ(κ)] du K (dκ) − L0 λπ(κ) 0 K i=1

Ng (T )

=

i=1

log

λ∗g (ti ) − λ

0

T

Ng (T )

[λ∗g (u) − λ] du +

i=1

log

f ∗ (κi | ti ) . (7.3.6) π(κi )

The second form in equations (7.3.5) and (7.3.6) follows from the assumption that the densities over the mark space are proper (i.e. integrate to unity). The reversibility of the arguments leading to the representation of the conditional intensity function in (7.3.2) (see Exercise 7.3.2) implies the following MPP analogue of Proposition 7.2.IV. Proposition 7.3.IV. Let N be a regular MPP as in Proposition 7.3.III. Then, the conditional intensity function with respect to the internal history H determines the probability structure of N uniquely. The next proposition gives specific examples of such characterizations, makeing more explicit the distinction between point processes with independent and unpredictable marks introduced already in Section 6.4.

252


Proposition 7.3.V. Let N be a regular MPP on R+ × K with H-intensity expressible as λ∗ (t, κ) = λ∗g (t)f ∗ (κ | t), (7.3.7) where λ∗g (t) is the H-intensity of the ground process. Then N is (i) a compound Poisson process if λ∗g (t) = λ(t) and f ∗ (κ | t) = f (κ | t) for deterministic functions λ(t) and f (κ | t); (ii) a process with independent marks if λ∗g (t) equals the Hg -intensity for the ground process and f ∗ (κ | t) = f (κ | t) as in (i); and (iii) a process with unpredictable marks if f ∗ (κ | t) = f (κ | t) as in (i). Proof. In a process with independent marks, the ground process and the marks are completely decoupled (i.e. they are independent processes), whereas for a process with unpredictable marks, the marks can influence the subsequent evolution of the process, though the ground process does not influence the distribution of the marks. The compound Poisson process is the special case of a Poisson process with independent marks. The forms of the conditional intensities follow readily from these comments, which merely reflect the definitions of these three types of MPP given in Definition 6.4.III and preceding Lemma 6.4.VI. The lemma is then a consequence of the uniqueness assertion in Proposition 7.3.IV. Some details and examples are given in Exercise 7.3.5. The following nonlinear generalization of the Hawkes process is important for its range of applications. It has been used as a model for neuron firing in Brémaud and Massoulié (1994, 1996), and it also embraces a range of other examples, including both ordinary and space–time versions of the ETAS model [Examples 6.4(d) and 7.2(f)]. Example 7.3(b) Nonlinear, marked Hawkes processes [see Example 7.2(b)]. We start by extending the basic Hawkes process N to a nonlinear version with conditional intensity [see (7.2.6)] t ∗ λ (t) = Φ λ + µ(t − u) N (du) , (7.3.8) 0

where the nonnegative function Φ is in general nonlinear but satisfies certain boundedness and continuity conditions; in particular, it is required to be Lipschitz with Lipschitz constant α ≤ 1. Such a nonlinear Hawkes process can immediately be extended to a nonlinear marked Hawkes process by giving the points independent marks with density f (κ) so that the conditional intensity function for the marked version is t µ(t − u) Ng (du) f (κ) = λ∗g (t)f (κ). (7.3.9) λ∗ (t, κ) = Φ λ + 0

The marks here make no contribution to the current risk, nor to the evolution of the ground process, which therefore has the same structure as the process N of (7.3.8). Consequently, in (7.3.9) we have Ng = N .

7.3.


253

By contrast, generalizing the ETAS model of Example 6.4(d) and using its notation, we may equally well consider extensions in which the conditional intensity has the form ∗ λ (t, κ) = Φ λ + ψ(χ)µ(t − u) N (du × dχ) f (κ), (7.3.10) (0,t)×K

where ψ(χ) modifies the strength of the infectivity density µ(·) according to the mark χ. In this case, the process has unpredictable marks that, depending on the form of ψ(·), can influence substantially the evolution of the ground process. In both cases, the likelihood for a finite observation period [0, T ] decouples and, following the second form in (7.3.6), can be written as * + T

∗ ∗ log L = log λg (ti ) − λg (u) du + log f (κi ) 0

i:0≤ti ≤T

i:0≤ti ≤T

≡ log L1 + log L2 , where λ∗g (t)

=Φ λ+

ψ(κ)µ(t − u) N (du × dκ) .

(0,t)×K

In many parametric models, no parameter appears in both L1 and L2 , so each term can be maximized separately. It is not necessary here to limit the mark to a measure of the size of the accompanying event. As suggested in Example 6.4(d), elements in the mark space may comprise both size and spatial components, κ ∈ K and y ∈ Y, say. Then we can write, for example, λ∗ (t, κ, x) = Φ λ + ψ(χ)µ(t − u)g(x − y) N (du × dχ × dy) f (κ), (0,t)×K×Y

where the spatial density g(·), like f (·), has been normalized to have unit integral and determines the positions of the offspring about the ancestor. Because of the independent sizes κi here, the log likelihood again separates into two terms, the first of which is analogous to log L1 above but includes an integration over both space and time. From a model-building point of view, it is of critical importance to establish conditions for the existence of stationary versions of the process and for convergence to equilibrium. General conditions are given by Brémaud and Massoulié (1996) and discussed further in Chapters 13 and 14. In the special case corresponding to the space–time ETAS model, where the function Φ is linear (and can be taken to be the identity function), the process retains the basic branching structure, and a sufficient condition for the existence of a stationary version is the subcriticality of the underlying branching component, as outlined already in Example 6.4(d). It is, of course, quite possible to devise models where the mark distributions are dependent on the evolution of the process. A simple example is given below.

254


Example 7.3(c) Processes governed by a Markovian rate process. Several models for both simple and marked processes are governed by an underlying Markov process, X(t) say, which both influences and is influenced by the evolving point process. Typically, in the marked case, both the ground process intensity and the mark distribution depend on the current value of X(t). Two simple models of this type are the simple stress-release model in Example 7.2(g) and the Cox process with Markovian rate function considered in Exercise 7.2.7. To illustrate possible ramifications of such models, consider first a Hawkes process with exponential infectivity density µ(x) = µe−µx . In this case, the Markovian process X(t) is given by the sum X(t) = µ i: 0 0),

where A, c and p are nonnegative parameters and p is commonly close to zero. It is a delicate question to determine the time at which the aftershocks merge indistinguishably into the general background activity for the region. Leaving aside the problem of defining precisely what is meant by this statement, the visual pattern can be much enhanced by first transforming the time scale by the compensator Λ∗ (t) = (A/p)[c−p − (t + c)−p ], (t ≥ 0) of the model above. When the rate of aftershock activity has decayed to about the level of the background activity, the dominant factor in the observed rate changes from the aftershock decay term to the steady background rate, increasing the observed rate above what would be expected from modelling the aftershock sequence. The change point is hard to pinpoint visually on the original time scale, but on the transformed time scale, it shows up relatively clearly as a deviation above the diagonal y = x near the end of the observation sequence. See e.g. Ogata (1988) and Utsu et al. (1995) for illustrations and further details. Residual analysis can also be adapted to more specific problems as below. Example 7.4(d) Using the ETAS model to test for relative quiescence in seismic data. At shallow depths (0–20 km or so), the ETAS model of Example 6.4(d) usually provides a reasonable first approximation to the time– magnitude history of moderate or small-size earthquake events in an observation region. For this reason, departures from the ETAS model, or changes in its apparent parameter values, can be used as an indicator of anomalous seismic activity that may be associated with the genesis of a forthcoming large event. In particular, a reduction in activity below that anticipated by the ETAS model may signify the onset of a period of seismic quiescence, a much debated indicator of a larger event. The task of searching for changes in rate is here complicated by the high level of clustering characteristic of earthquake activity, which makes the evaluation of appropriate confidence levels particularly difficult. Again, the task can be much facilitated by first transforming the occurrence times according to the best-fitting ETAS model and then carrying out the change-point test on the transformed data. The problem is then reduced to that of testing for a change point in a constant-rate Poisson process, a relatively straightforward and well-studied problem. Ogata (1988, 1992, 2001) has developed detailed procedures, including a modification to the usual AIC criterion, to take into account the nonstandard character of the change-point problem (the additional parameters are absent in the null hypothesis rather than being fitted to a special numerical value; Davies’ (1987) work on the problem of hypothesis testing when parameters vanish under H0 is pertinent). Some further details are given in the exercises. Exercise 7.4.2 indicates extensions to the marked point process case.

264


As with the other procedures we have illustrated in this chapter, the results on random time changes can be generalized relatively straightforwardly to other types of evolutionary point processes (notably multivariate and marked point processes) but only with more difficulty to spatial point patterns (see Chapter 14). We indicate below the extensions to multivariate and marked point processes; for more discussion, see e.g. Brown and Nair (1988). These extensions hinge on the uniqueness of the compensator with respect to the internal history H; see Proposition 7.3.IV for regular MPPs. Consider first a multivariate point process. Here each component could be transformed by its own compensator, as a result of which we would obtain a multivariate Poisson process in which each component has unit rate. But would these components then be independent? The answer to this question depends crucially on the histories used to define the compensators. If the full internal history is used for each component, then any dependence between the original components is taken into account and a Poisson process with independent, equally likely components is obtained. On the other hand, if each component is transformed according to its own internal history, the components of the resulting multivariate Poisson process will have equal (unit) rates but in general will not be independent. The next example provides a simple illustration. Example 7.4(e) Bivariate Poisson process [see Example 7.3(a)]. The model consists of an initial stream of input points from a Poisson process at constant rate λ and an associated stream of output points formed by delaying the initial points by random times exponentially distributed with mean 1/µ independently for each initial point. Integrating the full H-conditional intensities at (7.3.2), the corresponding compensators are for component I a line of constant slope λ and for component II a broken straight line, with segments whose slopes are nonnegative multiples of µ, the breaks in the line occurring at the points of both processes, the slope increasing by µ whenever a component I point occurs and decreasing by µ whenever a component II point occurs. The transformed points from component I are identical with the original points apart from an overall linear change of scale. The time transformation for component II is more complex: the distances between points are stretched just after a component I point and shrunk after a component II point. Further, if for any t all points of component I have been cleared (i.e. their associated component II points have already occurred), the transformed time remains fixed until the next component I point arrives. In this way, the dependence between the two components is broken, and both component processes are transformed into unit-rate Poisson processes. A similar conclusion holds even if either or both components is augmented by the addition of the points from an independent Poisson process or processes: the relative scales of the time changes compensate for any differences in the original component rates, producing always a unit rate in the trans-

7.4.

Random Time Change and a Goodness-of-Fit Test

265

formed process, while any dependence between the two components is still broken as explained above. Consider now the case of a regular MPP. If the support of the mark distribution is no longer finite, then effectively we have an infinite family of different components; clearly it is not possible to turn them all into unit-rate Poisson processes and hope to retain an MPP as output. To achieve such a result, at least the rates of the components should be adjusted to produce a transformed process with finite ground rate. Here is one way of proceeding. Suppose that the H-conditional intensity of the original process can be represented in the form λ∗ (t, κ) = λ∗g (t)f ∗ (κ | t), where f ∗ (κ | t) is a probability density with respect to the reference measure K (·), which we take here to be itself a probability measure so that t (dκ) = K f (κ | t) K (dκ) = 1. Let A(t, U ) = U 0 λ∗ (s, κ) ds K (dκ) be K K t the full H-compensator for the process, and write Aκ (t) = 0 λ∗ (s, κ) ds. To avoid complications in defining the inverse functions, we suppose both λ∗g (t) and f ∗ (κ | t) are strictly positive for all t and κ. Now consider the transformation that takes the pair (t, κ) into the pair (Aκ (t), κ). We claim that the transformed process is a stationary compound Poisson process with unit ground rate and mark distribution K (·). To establish this result, we appeal to the uniqueness theorem for compensators (Proposition 7.3.IV). The crucial computation, corresponding to equation (7.4.2), is (dτ × dκ)] = E[N (dy × dκ)] ≈ λ∗ (y, κ) dy (dκ) = dτ (dκ), E[N K K ∗ where y = A−1 κ (τ ), so that dy = dτ /λ (y, κ). The last form can be identified with the compensator for a stationary compound Poisson process with ground ˜ g = 1 and mark distribution (·). The uniqueness theorem completes rate λ K the proof. The results for both multivariate and marked point processes are summarized in the following proposition (a more careful discussion of the arguments above is given in Chapter 14).

Proposition 7.4.VI. (a) Let {Nj (t): j = 1, . . . , J} be a multivariate point process defined on [0, ∞) with a finite set of components, full internal history H, and left-continuous H-intensities λ∗j (t). Suppose that for j = 1, . . . , J, the t conditional intensities are strictly positive and that Λ∗j (t) = 0 λ∗j (s) ds → ∞ as t → ∞. Then, under the simultaneous random time transformations t → Λ∗j (t),

(j = 1, . . . , J)

the process {(N1 (t), . . . , NJ (t)): t ≥ 0} is transformed into a multivariate Poisson process with independent components each having unit rate.

266


(b) Let N (t, κ) be an MPP defined on [0, ∞) × K, where K is a c.s.m.s. with Borel sets BK and reference probability measure K (·), and let H denote the full internal history. Suppose that the H-conditional intensity λ∗ (t, κ) = λ∗g (κ)f ∗ (κ | t) exists, is K -a.e. left-continuous in t and strictly positive on t [0, ∞) × K, and that Λ∗κ (t) = 0 λ∗ (s, κ) ds → ∞ as t → ∞ K -a.e. Then, under the random time transformations (t, κ) → (Λ∗κ (t), κ), with unit the MPP N is transformed into a compound Poisson process N ground rate and stationary mark distribution K (·). Example 7.4(f) ETAS model [see Example 6.4(d)]. This can serve as a typical example of a process with unpredictable marks. The conditional intensity factorizes into the form [see equation (7.3.10)] λ∗ (t, κ) = λ0 + ν eα(χ−κ0 ) g(t − s) N (ds × dχ) f (κ) ≡ λ∗g (t)f (κ), (−∞,t)×K

where f (·), the density of the magnitude distribution, is commonly assumed to have ∞an exponential form on K = [0, ∞). For stationarity, we require ρ = ν 0 eακ f (κ) dκ < 1. Under these conditions, it is natural to take the reference measure on K to be f itself, in which case all the densities relative to the reference measure are equal to unity. Consequently, the multiple time changes here all reduce to the same form: t (t, κ) → (Λ∗g (t), κ), where Λ∗g (t) = λ∗g (s) ds. 0

In other words, under the random time change associated with the ground process, the original ETAS process is transformed into a compound Poisson process with unit ground rate and stationary mark density f . Such transformations open the way to corresponding extensions of the procedures described earlier for testing the process. In particular, checking the constancy of the mark distribution simplifies the detection of changes in the relative rates of events of different magnitudes. Similar remarks apply to other examples with unpredictable marks, such as the stress-release models of Examples 7.2(g) and 7.3(d). Schoenberg (1999) gives a random-time change for transforming spatial point processes to Poisson.

Exercises and Complements to Section 7.4 7.4.1 Consider a two-point process t1 , t2 , (t1 < t2 ) on [0, T ], where (t1 , t2 − t1 ) has continuous bivariate d.f. F (t, u). Find the compensator and define the random time change explicitly in terms of F . The Poisson process here has to be conditioned on the occurrence of two points within the interval [0, T ]. [Hint: Example 7.4(b) treats the one-point case.]

7.5.

Simulation and Prediction Algorithms

267

7.4.2 Marked point process extension of Algorithm 7.4.III. Following the discussion around equation (7.3.2), suppose there is given a family of conditional hazard functions hn (u | (t1 , κ1 ), . . . , (tn−1 , κn−1 )) and corresponding conditional mark distributions fn (κ | (t1 , κ1 ), . . . , (tn−1 , κn−1 ); u). Formulate in detail a sequence of simulation steps to solve successively the pairs of equations

tn

hn (u | (t1 , κ1 ), . . . , (tn−1 , κn−1 )) du = Yn ,

tn−1 κn

fn (κ | (t1 , κ1 ), . . . , (tn−1 , κn−1 ); u) dκ = Un .

0

7.4.3 (Continuation). Using steps analogous to the simulation argument above, provide an alternative, constructive proof of Proposition 7.4.VI. 7.4.4 Extension of Ogata’s residual analysis to multivariate and marked point processes. Develop algorithms, analogous to those in Algorithm 7.4.V, for testing multivariate and marked point processes. [Hint: In the multivariate case, test both (a) that the ground process for the transformed process is a unit-rate Poisson process and (b) that the marks are i.i.d. with equal probabilities. In the marked case, take the reference measure to be, say, a unit exponential distribution, and replace (b) with a test for a set of i.i.d. unit exponential variates.]

7.5. Simulation and Prediction Algorithms In the next two sections, we broach the topics of simulation, prediction, and prediction assessment. In modelling, the existence of a logically consistent simulation algorithm for some process is tantamount to a constructive proof that the process exists. Furthermore, simulation methods have become a key component in evaluating the numerical characteristics of a model, in checking both qualitative and quantitative features of the model, and in the centrally important task of model-based prediction. A brief survey of the principal approaches to point process simulation and of the theoretical principles on which these approaches are based therefore seemed to us an important complement to the rest of the text. This section provides a brief introduction to simulation methods for evolutionary models; that is, for models retaining a time-like dimension that then dictates the probability structure through the conditional intensity function. Simulation methods can be developed also for spatial point patterns (see Chapter 15), but considerable conceptual simplicity results from the ability to order the evolution of the process in ‘time’. The growth in importance of Markov chain Monte Carlo methods for simulating spatial processes is a tacit acknowledgement of the fact that such methods introduce an artificial time dimension even into problems where no such dimension is originally present. Two general approaches are commonly used for simulating point processes in time. The first we have already considered in Algorithm 7.4.III; it involves simulating the successive intervals, making use of the description of the

268


conditional intensity function as a family of hazard functions as in equation (7.2.3). Its main disadvantage as a general method is that it requires repeated numerical solution of the equation defining the inverse. The thinning methods outlined in the present section, by contrast, require only evaluations of the conditional intensity function. Although the difference in computational time between these two methods is not huge, it is the main reason why the thinning method is given greater prominence in this section. In addition, the theoretical basis behind thinning methods is of interest in its own right. The most important theoretical result is a construction, originating in Kerstan (1964) and refined and extended in Brémaud and Massoulié (1996), that has something of the character of a converse to Proposition 7.4.I. There we transformed a point process with general conditional intensity to a Poisson process; here we convert a Poisson process back into a process with general conditional intensity. For this purpose, we use an auxiliary coordinate in the say, on the state space, so we consider a unit-intensity Poisson process, N consist of pairs (xj , yj ). product space X = R × R+ . The realizations of N Also, let Ht denote the σ-algebra of events defined on a simple point process over the interval [0, t) and H the history {Ht }. The critical assumption below is that λ∗ is H-adapted. , H be defined as above, let λ∗ (t) be a nonnegative, Proposition 7.5.I. Let N left-continuous, H-adapted process, and define the point process N on R by dt × (0, λ∗ (t)] . N (dt) = N

(7.5.1)

Then N has H-conditional intensity λ∗ (t). Proof. Arguing heuristically, it is enough to note that & ˜ dt × (0, λ∗ (t−)] & Ht− = λ∗ (t) dt. E[N (dt) | Ht− ] = E N There is no requirement in this proposition that the conditional intensity be a.s. uniformly bounded as was required in the original Shedler–Lewis algorithm. When such a bound exists, it leads to straightforward versions of the thinning algorithm, as in Algorithm 7.5.II below. The result can be further extended in various ways, for example to situations where more general histories are permitted or where the initial process is not Poisson but has a conditional intensity function that almost surely bounds that of the process to be simulated; see Exercises 7.5.1–2. Example 7.5(a) Standard renewal process on [0, ∞). We suppose the process starts with an event at t = 0. Let h(u) denote the hazard function for the lifetime distribution of intervals between successive points, so that [see Exercise 7.2.3(a)] the conditional intensity function has the form λ∗ (t) = h(t − tN (t) )

(t ≥ 0),

7.5.


269

where tN (t) is the time of occurrence of the last event before time t. However, rather than on N . To this end, λ∗ (t) should be defined on the history of N . With t0 = 0, define we first define the sequence of points ti in terms of N sequentially tn+1 = min{xi : xi > tn and yi < h(xi − tn )}

(n = 0, 1, . . .)

and then define λ∗ (t) as above. Notice that the right-hand side of this expression is Ft -measurable and the whole process is F -adapted. Thinning algorithms generally follow much the same lines as in Proposition 7.5.I and the example above. The main difficulty arises from the range of yi being unbounded, which provides a flexibility that is difficult to match in practice. The original Shedler–Lewis algorithm (Lewis and Shedler, 1976; see also Exercise 2.1.6) was for an inhomogeneous Poisson process in a time interval where the intensity is bounded above by some constant, M say. Then, the auxiliary dimension can be taken as the bounded interval (0, M ) rather than the whole of R+ , or equivalently the yi could be considered i.i.d. uniformly distributed random variables on the interval (0, M ). Equivalently again, the time intensity could be increased from unity to M and the yi taken as i.i.d. uniform on (0, 1), which leads to the basic form of the thinning algorithm outlined in the algorithm below. In discussing the simulation algorithms below, it is convenient to introduce the term list-history to stand for the actual record of times, or times and marks, of events observed or simulated up until the current time t. We shall denote such a list-history by H, or Ht if it is important to record the current time in the notation. Thus, a list-history H is just a vector of times {t1 , . . . , tN (t) } or a matrix of times and marks {(t1 , κ1 ), . . . , (tN (t) , κN (t) )}. We shall denote the operation of adding a newly observed or generated term to the list-history by H → H ∪ tj or H → H ∪ (tj , κj ). In the discussion of conditioning relations such as occur in the conditional intensity, the listhistory Ht bears to the σ-algebra Ht a relationship similar to that between an observed value x of a random variable X and the random variable X itself. The algorithms require an extension of Proposition 7.5.I to the situation where the process may depend on an initial history H0 ; we omit detail but note the following. Such a history will be reflected in the list-history by a set of times or times and marks of events observed prior to the beginning of the simulation. This is an important feature when we come to prediction algorithms and wish to start the simulation at the ‘present’, taking into account the real observations that have been observed up until that time. It is also important in the simulation of stationary processes, for which the simulation may be allowed to run for some initial period (−B, 0) before simulation proper begins. The purpose is to allow the effects of any transients from the initial conditions to become negligible. Finding the optimal length of such a preliminary ‘burn-in’ period is an important question in its own right. Its solution depends on the rate at which the given process converges toward equilibrium

270


from the initial state, but in general this is a delicate question that is affected by the choice of initial state as well as decay parameters characteristic of the process as a whole. Suppose, then, that the process to be simulated is specified through its conditional intensity λ∗ (t), that there exists a finite bound M such that λ∗ (t) ≤ M for all possible past histories, and that the process is to be simulated over a finite interval [0, A) given some initial list-history H0 . Algorithm 7.5.II. Shedler–Lewis Thinning Algorithm for processes with bounded conditional intensity. 1. Simulate x1 , . . . , xi according to a Poisson process with rate M (for example, by simulating successive interval lengths as i.i.d. exponential variables with mean 1/M ), stopping as soon as xi > A. 2. Simulate y1 , . . . , yi as a set of i.i.d. uniform (0, 1) random variables. 3. Set k = 1, j = 1. 4. If xk > A, terminate. Otherwise, evaluate λ∗ (xk ) = λ(xk | Hxk ). 5. If yk ≤ λ∗ (xk )/M , set tj = xk , update H to H ∪ tj , and advance j to j + 1. 6. Advance k to k + 1 and return to step 4. 7. The output consists of the list {j; t1 , . . . , tj }. This algorithm is relatively simple to describe. In the more elaborate versions that appear shortly, it is convenient to include a termination condition (or conditions), of which steps 1 and 4 above are simple. In general, we may need some limit on the number of points to be generated that lies outside the raison d’être of the algorithm. While this algorithm works well enough in its original context of fixed intensity functions, its main drawback in applications to processes with random conditional intensities is the need for a bound on the intensity that holds not only over (0, A) but also over all histories of the process up to time A. To meet this difficulty, Ogata (1981) suggested a sequential variant of the algorithm that overcomes this difficulty, requiring only a local boundedness condition on the conditional intensity. A minor variant of his approach is outlined in Algorithm 7.5.IV. For the sake of clarity, we return to the representation of the conditional intensity function in terms of successive hazard functions, much as in Definition 7.2.II, but allowing all such functions to depend on an initial history H0 , namely hn (s | H0 , t1 , . . . , tn−1 ), for 0 < t1 < · · · < tn−1 < s ≤ A. For every t in (0, A) and associated σ-algebra Ht , we suppose there are given two quantities, a local bound M (t | Ht ) and a time interval of length L(t | Ht ), satisfying the following conditions.

7.5.


271

Condition 7.5.III. There exist functions M (t | Ht ), L(t | Ht ) such that, for all initial histories H0 , all t ∈ [0, ∞), for every n = 1, 2, . . . , and all sequences t1 , . . . , tn−1 with 0 < t1 < · · · < tn−1 < t, the hazard functions satisfy hn (t + u | H0 , t1 , . . . , tn−1 ) ≤ M (t | Ht )

(0 ≤ u < L(t | Ht ) ).

Placing the bound on the hazard function is equivalent to placing the bound on the conditional intensity function under the constraint that no additional points of the process occur in the interval (t, t + u) under scrutiny. As soon as a new point does occur, in general the hazard function will change and a new bound will be required. Thus, the bound holds until either the time step L(·) has elapsed or a new point of the process occurs. For the algorithm below, the list-history Ht consists of {H0 , t1 , . . . , tN (t) }, where N (t) is the number of points ti satisfying 0 ≤ ti < t. For brevity, we mostly write M (t) and L(t) for M (t | Ht ) and L(t | Ht ). Ogata (1981) gives extended discussion and variants of the procedure. Algorithm 7.5.IV. Ogata’s modified thinning algorithm. 1. Set t = 0, i = 0. 2. Stop if the termination condition is met; otherwise, compute M (t | Ht ) and L(t | Ht ). 3. Generate an exponential r.v. T with mean 1/M (t) and an r.v. U uniformly distributed on (0, 1). 4. If T > L(t), set t = t + L(t) and return to step 2. 5. If T ≤ L(t) and λ∗ (t + T )/M (t) > U , replace t by t + T and return to step 2. 6. Otherwise, advance i by 1, set ti = t + T , replace t by ti , update H to H ∪ ti , and return to step 2. 7. The output is the list {i; t1 , . . . , ti }. The technical difficulties of calculating suitable values for M (t) and L(t) vary greatly according to the character of the process being simulated. In an example such as a Hawkes process, at least when the hazard functions decrease monotonically after an event, it would be enough in principle to consider only t = ti (i.e. points of the process) and set M (ti ) = λ∗ (ti +). This leads to a very inefficient algorithm, however, since the hazard decreases rapidly and a large number of rejected trial points could be generated. A simple modification is to set M (t) = λ∗ (t) and L(t) = 12 λ∗ (t+), irrespective of whether or not t is a point of the process. Such a choice gives a reasonable compromise between setting the bound too high, and so generating excessive trial points, and setting it too low, thus requiring too many iterations of step 3. The next example is a process with an increasing hazard, where the intervention of step 3 is virtually mandatory. Example 7.5(b) Self-correcting or stress-release model. We discuss the simulation of the model of Example 7.2(g). As described there, points {ti } occur

272


at a rate governed by the conditional intensity function λ(t) = Ψ[X(t)], where X(t) is an unobserved Markov jump process that increases linearly between jump times ti at which it decreases by an amount κi , so that X(t) = X(0) + νt −

i:ti L(t), set t = t + L(t), update the list-history H, and return to step 2. 5. If T ≤ L(t) and λ∗0 (t + T )/M (t) < U , replace t by t + T , update the list-history H, and return to step 2. 6. Advance i by 1, set ti = t + T , replace t by ti , and generate a mark κi from the distribution with density f (κ | ti ). 7. Update the list-history H to H ∪ (ti , κi ), and return to step 2. 8. The output is the list {i; (t1 , κ1 ), . . . , (ti , κi )}. In Example 7.5(b) above, for example, simulation proceeds as if the process has nonanticipating marks until step 6 is reached, at which point the appropriate value φ[X(t)] must be read into the simulation routine for producing values according to the tapered Pareto distribution. By way of illustrating Algorithm 7.5.V, we consider the extension of Example 7.5(b) to the linked stress-release model. Example 7.5(c) Simulating the linked stress-release model [see Example 7.3(d)]. In this model, there are two types of marks: the region in which the event occurs (as a surrogate for spatial location) and the size of the event. The basic form of the conditional intensity is given in equation (7.3.14). A key step in the simulation is updating the list-history. This will consist of a matrix or list type object with one column for each coordinate of the events being described: here the times ti , their regions Ki , and their magnitudes Mi . When the simulation is started, the list-history may contain information from real or simulated data from the past in order to allow the simulation to join ‘seamlessly’ onto the past. Each time a new event is simulated, its coordinates are added to the list-history. Since the simulation of the next event depends only on the form of the conditional intensity, as determined by the current list-history, and additional random numbers, it can proceed on an event-by-event basis. First, the time of the next event in the ground process is simulated, then the region is selected with probabilities proportional to the relative values of the conditional intensities for the different regions at that time, and then a magnitude is selected from the standard magnitude distribution (this distribution is fixed in the standard model, but it can also be made stress- or region-dependent).

274


The prediction of point processes, in all but a few very special cases where explicit algorithms are available, goes hand-in-hand with simulation. The quantities that one would like to predict, such as the time to the next event, the probability of an event occurring within a given interval in the future, or the costs caused by events in the future, are commonly nonlinear functionals of the future of the process. They rarely fall into any general category for which analytic expressions are available. Since, on the other hand, simulation of a point process is relatively straightforward once its conditional intensity function is known, and moreover can be extended to situations where an arbitrary initial history can be incorporated into the conditional intensity, it is indeed natural to see prediction as an application and extension of the preceding procedures. Suppose there is given a realization of the point process on some finite interval (a, b). To link up with the preceding algorithms, we identify the origin t = 0 with the end point b of the interval so that, in our earlier notation, the realization on (a, b) forms part of the initial history H0 . Suppose for the sake of definiteness that our aim is to predict a particular quantity V that can be represented as a functional of a finite segment of the future of the process. To fulfil our aim, we estimate the distribution of V . An outline of a prediction procedure is as follows. 1. Choose a time horizon (0, A) sufficient to encompass the predicted quantity of interest (we need not insist here that A be a fixed number, provided the stopping rule is clearly defined and can be incorporated into the simulation algorithm). 2. Simulate the process forward over (0, A) using the known structure of the conditional intensity function and initial history H0 . 3. Extract from the simulation the value V of the functional that it is required to predict. 4. Repeat steps 2 and 3 sufficiently often to obtain the required precision for the prediction. 5. The output consists of the empirical distribution of the values of V obtained from the successive simulations. In step 5 above, it is often convenient to summarize the empirical distribution by key characteristics, such as its mean, standard deviation, and selected quantiles. Not all prediction exercises fit exactly into this schema, but many are variations on it. Example 7.5(d) Prediction of a Wold process with exponential intervals [see Exercise 4.5.8 and Example 4.6(b)]. In the notation used previously, let an interval preceded by an interval of length x have parameter λ(x) [and hence mean 1/λ(x)]. Suppose that we wish to predict the time X0 to the next event and the length X1 of the ensuing complete interval, given the current list-history consisting of the times t0 , t−1 , . . . of the preceding events, where 0 denotes the present time so 0 > t0 > t−1 > · · · .

7.5.


275

The quantity V of the preceding discussion is the pair (X0 , X1 ). The particular specification of the model here implies that the joint density function of (X0 , X1 ) equals λ |t0 − t−1 | e−λ(|t0 −t−1 |)X0 λ |t0 | + X0 e−λ(|t0 |+X0 )X1 ; then simulation via the model should lead to a joint histogram that in principle is an approximation to this function. For pragmatic purposes, we may be satisfied with the first moments 1 E(X0 | H0 ) = λ |t0 − t−1 |

and E(X1 | H0 ) =

0

∞

λ(|t0 − t−1 |) −λ(|t0 −t−1 |)u du. e λ(|t0 | + u)

Exercises and Complements to Section 7.5 7.5.1 Extended form of Proposition 7.5.I. Let F be a history on [0, ∞), λ1 (t), λ2 (t) be two nonnegative, left-continuous (or more generally predictable), historydependent candidates for conditional intensity functions, and let N ∗ (dt × ds) be an F -adapted unit-rate Poisson process on R+ × R that is unpredictable in the sense that its evolution for s > t is independent of the history up to t. Let N (t) on R+ consist of the time coordinates ti from those points of N ∗ lying in the region min{λ1 (t), λ2 (t)} < s < max{λ1 (t), λ2 (t)}. Then N is F -adapted and has conditional intensity |λ1 (t) − λ2 (t)|. [In most cases, as in Proposition 7.5.I, the history will be that generated by the Poisson process itself, but the generalization opens the way to conditioning on external variables. See Brémaud and Massoulié (1996) and Massoulié (1998).] 7.5.2 Extension of thinning Algorithm 7.5.II. In the setup for Algorithm 7.5.II, suppose that the xi are simulated from a process with conditional intensity λ+ (t) that satisfies a.s. λ+ (t) ≥ λ∗ (t) (0 < t < T ) and that the thinning probability at time t is equal to the ratio λ∗ (t)/λ+ (t). Show that the thinned process is again the point process with intensity λ∗ (t). [See Ogata (1981).] 7.5.3 Simulation algorithms for Boolean models. Devise a simulation procedure for the Boolean model of Example 6.4(d) with a view to describing distributions of functionals such as the intensity function or a joint intensity (‘correlation’). 7.5.4 Show how Algorithm 7.5.V can be applied to a pure linear birth process. 7.5.5 Simulation of cluster processes. Brix and Kendall (2002) describe a technique for the perfect simulation of a cluster point process in a given region A (hence, the simulations have no edge effects—this is an analogue of having no ‘burnin’ period). The crucial step is to replace the parent process Nc , say, by a process which has at least one offspring point in the observation region.

276


7.6. Information Gain and Probability Forecasts We come now to the problem of assessing probability forecasts of the type described in the previous section. A distinction needs to be made here between assessing the probability forecast as such and assessing a decision procedure based on the probability forecast. Commonly, when probability forecasts for weather and other phenomena are being assessed, a threshold probability level is established, and the forecast is counted as a ‘success’ if either the forecast probability rises above the threshold level and a target event occurs within the forecasting period or region or the forecast probability falls below the threshold level and no event occurs. The assessment is then based on the 2 ×2 table of observed and forecast successes and failures, and a variety of scores for this purpose have been developed and studied (see e.g. Shi et al., 2001). In effect, such a procedure converts the probability forecast into a decision rule, and it is the decision rule rather than the forecast that is assessed. In fact, many decision rules can be based on the same probability forecast, depending on the application in view. For example, in earthquake forecasts, one relevant decision for a government might be whether or not to issue a public earthquake warning; but other potential users, such as insurance companies, emergency service coordinators, and managers of gas, power, or transport companies, might prefer to initiate actions at quite different probability levels and would therefore score the forecasts quite differently. Our concern is with assessing the probability forecasts as such. The basic criterion we shall use for this purpose is the binomial or entropy score, in which the forecast is scored by the negative logarithm − log pˆk of the forecast probability pˆk of the outcome k that actually occurs. If outcome k has true probability pk of occurring, then a ‘good’ set of forecasts should have pˆk ≈ pk for outcome k, and therefore the expected score is approximately − k pk log pk , which is just the entropy of the distribution {pk } (up to a multiplicative factor in not using logarithms to base 2). This leads us to a preliminary discussion of the entropy of point process models, a study taken further in Chapter 14. The entropy score itself, summed over a sequence of forecasts based on a specific parametric model, is nothing other than the log likelihood of the model. In this sense, the discussion highlights an alternative interpretation of the likelihood principle. Maximizing the likelihood from within a family of models amounts to finding the model with the best forecast performance in the sense of the entropy score. Equally, testing the model on the basis of its forecasting performance amounts to testing the model on the basis of its likelihood. Other criteria, such as the goodness-of-fit of first- and second-moment properties, may be less relevant to selecting a model for its forecasting ability. In any case, the analysis and assessment of probability forecasts is a topic of importance in its own right, and it is this point of view that motivates the present discussion. To bring some of the underlying issues into focus, consider first the simpler problem of producing and assessing probability forecasts for a sequence of

7.6.

Information Gain and Probability Forecasts

277

i.i.d. multinomial trials in which observation Yi , for i = 1, . . . , N , may have one of K different outcomes 1, . . . , K, say, with respective true probabilities K pk = Pr{outcome is k} = Pr{Yi = k} (the trials are i.i.d.) and k=1 pk = 1. Suppose that there is a record available of observations {Y1 , . . . , YN } on N N independent trials, and write pˆk = N −1 i=1 δYi k ≡ Nk /N for the sample proportion of outcomes equal to k (k = 1, . . . , K). What should be our forecast for trial N + 1? In accordance with our general prescription, the forecast should be in the form of a set of probabilities based on an assumed model (i.e. a model for which the underlying probabilities are assumed known). In this simple situation, it is intuitively obvious that the {pk } are also the probabilities that we would use to forecast the different possible outcomes of the next event. However, it is also possible to base this choice on somewhat more objective grounds, namely that our choice should maximize some expected score, suitably chosen. Denote the candidate probabilities for the forecast by ak . In accordance with the discussion above, we consider here the likelihood ratio score SLR =

N

i=1

ak aYi = N pˆk log , πYi πk K

log

(7.6.1)

k=1

where {πk } is a set of reference probabilities. The use of the logarithm of the ratio ak /πk rather than the simple logarithm log ak has two benefits: it introduces a natural standard against which the forecasts using the given model can be compared; and it overcomes dimensionality problems in the passage from discrete to continuous contexts (Exercise 7.6.1 gives some further discussion). This score function has the character of a skill score, for which higher values show greater skills. Taking expected values has the effect of replacing the empirical frequencies pˆk by pk in the second form of (7.6.1). Elementary computations then show that the score SLR is optimized by the choice ak = pk ; i.e. the procedure that optimizes the expected score is to use the model probabilities as the forecasting probabilities. Specifically, the optimum values achieved by following the procedure above are given by E(SLR ) = N H(P ; Π),

(7.6.2)

where P , Π denote the distributions with elements pk , πk , respectively, and H(·) is the relative or generalized entropy or Kullback–Leibler distance between the two distributions. The appearance of the entropy here should not come as a surprise, as it is nothing other than the expected value of (minus) a log probability, or more generally a log likelihood. In terms of SLR , the distribution that is hardest to predict is the discrete uniform distribution, which has maximum entropy amongst distributions on K points. If we use the uniform as the reference distribution {πk }, the change in the expected score as the model distribution moves away from the maximum

278


entropy distribution will be referred to as the expected information gain. It represents the improvement in the predictability of the model used relative to the reference model. The greatest expected gains, corresponding to the most effective predictions, will be achieved when the model distribution is largely concentrated on one or a small number of distinguished values. The ratio pk /πk of the model probability pk to the reference probability πk for any particular distinguished value k is sometimes called the probability gain for k. Now let us examine how these ideas carry over to the point process context. We start with a discrete-time framework, such as would arise if the forecasts were being made regularly, after the elapse of a fixed time interval (weekly, monthly, etc.). We also assume that the process is marked, with the marks taking one of the finite set of values {1, . . . , K}. In effect, this merely extends the discussion from the case of independent to dependent trials, with the assumption that the trials are indexed by a time parameter so that the evolutionary character is maintained. Alternatively, and more conveniently for our purposes, we may consider the model as a multivariate point process in discrete time. Rather than using the sequence of marks Yn (n = 1, 2, . . .) as before, introduce Xkn = δYn k , and let the K component simple point processes Nk (n) count the number ofpoints with mark k up to ‘time’ n, with Nk (0) = 0 for n each k, so Nk (n) = i=1 Xki . An argument similar to that given previously shows that the forecasting probability that optimizes the expected value of the score at step n, given the history Hn−1 up to time n − 1, is p∗kn = E(Xkn | Hn−1 ), where H is the full history of the process, recording information on the marks as well as the occurrence times. If, as a reference process, we take the process of i.i.d. trials having fixed probabilities πkn = fk , then the total entropy score over a period of T time units can be written log

T K

p∗ L = Xkn log kn , L0 πkn n=1

(7.6.3)

k=1

which is just the likelihood ratio for the given process relative to the reference process. This formulation shows clearly that the total entropy score for the multivariate process is the sum of the entropy scores of the component processes. There is no implication here that the component processes are independent; dependence comes through the joint dependence of the components on the full past history. In the case of a univariate process, for which the only possible outcomes are 0 and 1, the formula in (7.6.3) simplifies to the binomial score T

p∗n 1 − p∗n L Xn log . (7.6.4) = + (1 − Xn ) log log L0 πn 1 − πn n=1 Equation (7.6.3) assumes a form closer to that used previously for the likelihood of a multivariate point process if we reserve one mark, 0 say, for

7.6.


279

the null event; that is, the event that no event of any other type occurs. Let us assume in addition that the ground process is simple, so that at most one nonnull notations event can occur in any one time instant, and introduce the ∗ p∗n = k p∗kn for the conditional intensity of the ground process, fk|n = p∗kn /p∗n for the conditional distribution of the mark, Kgiven the past history and the occurrence of an event at n, and Xn = k=1 Xkn for the ground process itself. Let us also choose the reference probabilities in the form πkn = fk πn for k = 0, π0n = 1 − πn , corresponding to a discrete-time analogue of a continuous-time compound Poisson process. Then we can rewrite (7.6.3) as * K + T ∗

p∗n fk|n L 1 − p∗n log = Xkn log + (1 − Xn ) log L0 fk πn 1 − πn n=1 k=1 * + T K ∗

fk|n p∗n 1 − p∗n = Xn log . (7.6.5) + (1 − Xn ) log + Xkn log πn 1 − πn fk n=1 k=1

Taking expectations of the nth term, given the past up to time n − 1, gives the conditional relative entropy or conditional information gain In =

K

p∗kn log

k=1

= p∗n log

p∗kn 1 − p∗n + (1 − p∗n ) log pkn 1 − pn

K ∗

fk|n p∗n 1 − p∗n ∗ + (1 − p∗n ) log + p∗n fk|n log . pn 1 − pn fk

(7.6.6)

k=1

It is the conditional relative entropy of the nth observation, given the information available prior to the nth step. Note that this quantity is still a random variable since it depends on the random past through the conditioning σ-algebra Hn−1 . It reduces to the zero random variable when p∗kn = πkn but is otherwise positive, as follows from Jensen’s inequality. In the special case of a univariate process, it reduces to In∗ = p∗n log The relation

p∗n 1 − p∗n + (1 − p∗n ) log . πn 1 − πn

(7.6.7)

∗ ∗ ) | Hn−1 ] = E(In∗ | Hn−1 ) + In−1 E[(In∗ + In−1

yields the joint conditional entropy of Xn and Xn+1 , given the information available at the (n − 1)th step. Continuing in this way, we obtain * N + & & N

&

L && ∗ & E (7.6.8) H In & H0 = E In∗ | H0 = E log 0 , L0 & n=1 n=1 the joint entropy of the full set of observations, conditional on the information available at the beginning of the observation period. Dividing this quantity

280


by N , we obtain the average expected information gain per time step. This quantity is of particular interest when the whole setup is stationary and the expectations in (7.6.8) have the same value, namely the expected information gain per unit time. We shall denote this quantity by G. In this situation, we expect the log likelihood to increase roughly linearly with the number of observations, with the expected increment being equal to G. To avoid difficulties with transient effects near n = 0, the histories in the stationary case should cover the infinite past rather than the past since some fixed starting time. Following the notation in later chapters, write p†n = E[Xn+1 | H(−∞,n] ] and set πn = E(p†n ) = E(Xn ) = p, say. Then, G can be expressed as K

f† 1 − p†n p† G = E p†n log n + (1 − p†n ) log log k . + p†n p 1−p fk

(7.6.9)

k=1

The first term represents the information gain from the ground process and the second the additional information gain that comes from predicting the values of the marks, given the ground process. Overall, G represents the expected improvement in forecasting skill, as measured by the entropy score, if we move from using the background probabilities as the forecast to using the time-varying model probabilities. G ranges from 0, when the trials are i.i.d. and the model probabilities coincide with those of the reference model, to a maximum when the model trials are completely predictable, related to the absolute entropy of the independent trials model. To see this last point, suppose, to take a specific case, that the background model is for i.i.d. trials with equal probabilities 1/K for each outcome. Now write G in the form * G = E p†n log p†n + (1 − p†n ) log(1 − p†n ) − p†n log p + (1 − p†n ) log(1 − p) + p∗n

K

k=1

† fk|n

log

† fk|n

fk

+ (7.6.10)

and suppose that with high probability, p†n is close to either one or zero † is also close to one, so that the process is highly and that one of the fk|n predictable. Then, both the first two terms in the first sum above are very small, while in the second sum either p†n itself is very small or it is close to one and the remaining sum is close to the value − log(1/K). After taking expectations, recalling E(p†n ) = p, G reduces to approximately −[p log p + (1 − p) log(1 − p) + p log(1/K)], the absolute entropy of the independent trials model with equal probabilities for each outcome. In general, the final term will be of the form p E[log fk† ], where fk† is the background probability of the outcome k † that is successfully predicted. In summary, we have the following statement.

7.6.


281

Proposition 7.6.I. For a stationary, multivariate, discrete-time process, with full internal history F , overall occurrence rate p, and background model as defined above, G, the expected information gain per time step, is given by (7.6.9) above. It is a characteristic of the model and lies in the range 0 ≤ G ≤ −[p log p + (1 − p) log(1 − p) + p E(log fk† )], where fk† is the background probability of the outcome k † that is successfully predicted. G takes the lower end point of the range when the increments Xnk are independent and the upper end point when perfect prediction is possible. Example 7.6(a) Discrete Hawkes process: logistic autoregression. This model defines a univariate process in which p∗n has the general form log

K K

p∗n = a + a X = a + I{Xn−i =1} ai , 0 i n−i 0 1 − p∗n i=1 i=1

(7.6.11)

where the ai are parameters and, to accommodate the stationarity requirement, F is taken to be the complete history H† , so that Hn† is generated by the Xi with −∞ < i ≤ n. For simplicity, we examine just the case of a first-order autoregression; there are then just two parameters, a0 and a1 , in 1 : 1 correspondence with the probabilities π1|0 = Pr{Xn = 1 | Xn−1 = 0} and π1|1 = Pr{Xn = 1 | Xn−1 = 1}, respectively. Three extreme cases arise. If π1|0 is close to 0 and π1|1 is close to 1, then a realization will consist of long sequences of 0s followed by long sequences of 1s, and any prediction should approximate the weatherman’s rule: tomorrow’s weather will be the same as today’s. If π1|1 is close to 0 and π1|0 is close to 1, then the realization will be an almost perfect alternation of 0s and 1s, and any prediction rule should approximate the antiweatherman’s rule: tomorrow’s weather will be the opposite of today’s. In the third case, π1|0 and π1|1 are both close to 12 , and the sequence will consist of more or less random occurrences of 0s and 1s, and no good prediction rule will be possible. To examine such effects quantitatively, let us choose the parameters a0 , a1 so that π1|0 and π1|1 can be written π1|0 = ,

π1|1 = 1 − ρ .

The stationary probability p solves the equation p = pπ1|1 + (1 − p)π1|0 so p = 1/(1 + ρ). Thus, the parameter controls the mean length of runs of the same digit, and the parameter ρ controls the relative probabilities of 0s and 1s. We examine the behaviour of the predictions for small . When Xn−1 = 0, we take as our prediction p∗n = π1|0 = , and when Xn−1 = 1 we take p∗n = π1|1 = 1 − ρ. The information gain when Xn−1 = r for r = 0, 1 is then Jr = π1|r log

π1|r 1 − π1|r + (1 − π1|r ) log . p 1−p

282


The expected information gain per forecast is G = pJ1 +(1−p)J0 . Substituting for π1|0 , π1|1 and p, we find that, for small , G = Hp + 2ρ log + O(), where Hp is as in Proposition 7.6.I. As decreases, the expected information gain approaches Hp , whereas if = 1/(1 + ρ), then π1|0 = π1|1 = 1/(1 + ρ) and G = 0. We have stressed that the expected information gain is a function of the model: it is an indicator of its inherent predictability. In practice, other factors may intervene to produce an observed mean information gain that is well below that predicted by the model. This may happen, in particular, if the data are being fitted by a poor model. There would then be substantial long-run discrepancies between the actual data and the data that would be produced by simulation from the model. In such a case, the average information gain over a long sequence of trials could be well below the expected model value. In this sense, the mean information gain, representing the average likelihood per observation, forms the basis for a kind of goodness-of-fit test for the model. We turn now to the problem of transferring these ideas to the continuoustime, point process context. In practice, forecasts cannot be issued continuously but only after intervals of greater or smaller length. We therefore adopt the following framework. Suppose there is given a finite interval (0, T ) and a partition T into subintervals {0 < tT ,1 < · · · < tT ,N = T }. Forecasts are to be made at the end of each subinterval (i.e. at the time points {tT ,k }) for the probability of an event occurring in the next subinterval. Suppose further that the given partition is a member of a dissecting family of partitions Tn in the sense of Appendix A1.6: as n → ∞, the norm T = max |tT ,k − tT ,k−1 | → 0 so that the partitions ultimately distinguish points of (0, T ), and the intervals appearing in the partitions are rich enough in total to generate the Borel sets of (0, T ). Our aim is to relate the performance of the forecasts on the finite partition to the underlying properties of the point process. For this purpose, Lemmas A1.6.IV, on convergence to a Radon–Nikodym derivative, and A1.6.V, on the relative entropy of probability measures on nested partitions, play a key role. To apply these lemmas, we must relate the partitions of the interval (0, T ) to the partitions of the measurable space (Ω, E) on which the probabilities are defined. Here it is enough to note that a partition of the interval into N subintervals induces a partition of (Ω, E) into the (K + 1)N events corresponding to all possible sequences obtained by noting whether or not the subinterval contains a point of the process and, if so, noting the mark of the first point occurring within the subinterval. From Lemma A1.6.IV, it follows that, as the partitions are refined, the probability gains p∗nk /πnk converge (P × )-a.e. to the corresponding ratio of intensities λ∗ (t, k)/λ0 (t, k). Lemma A1.6.V then implies that the corresponding relative entropies increase to a limit bounded above by the point process

7.6.


283

relative entropy. The latter can be obtained directly by taking expectations of the point process likelihood ratio. Specifically, starting from the MPP log likelihood ratio at (7.6.5), taking expectations when the reference measure corresponds to a compound Poisson process with constant rate λ0 and mark distribution fk , the relative entropy H(PT ; P0,T ) equals * K + T

T λ∗ (t, k) ∗ ∗ E dt − [λg (t) − λ0 ] dt (7.6.12) λ (t, k) log λ0 fk 0 k=1 0 T T λ∗g (t) ∗ =E λg (t) log dt − [λ∗g (t) − λ0 ] dt λ0 0 0 T K

fk∗ (t) ∗ ∗ + λg (t) fk (t) log dt , (7.6.13) fk 0 k=1

where λ∗g (t) is the conditional intensity for the ground process. A proof of this result for the univariate case when H is the internal history and the likelihood reduces to the Janossy density is outlined in Exercise 7.6.3. The general case, as well as a more complete discussion of the convergence of the p∗nk to λ∗ (t, k), is taken up in Chapter 14. When the process is stationary and λ∗ is replaced by λ† (i.e. the conditioning is taken with respect to the infinite past), the relative entropy in (7.6.12) reduces to a multiple of T . If further we assume that λ0 = E[λ†g (0)] ≡ mg , then (7.6.12) can be written ! * K † +"

fk|0 λ†g (0) † + mg E . (7.6.14) log H(PT ; P0,T ) = T E λg (0) log fk λ0 k=1

Again, we can write G for the coefficient of T and refer to it as the mean entropy or expected information gain per unit time. It is worth noting that here G can be written in the two alternative forms * K † + K

fk|0 λ†g (0) λ† (0) † + mg E , = log E λ†k (0) log k G = E λg (0) log λ0 fk λk k=1

λ†k (0)

† λ†g (0)fk|0

k=1

where = and λk = mg fk . The first form represents a division of the information gain into components due to forecasting the occurrence times of the points and their marks, while the second represents a division of the information gain into components corresponding to the individual marks. This equality does not hold in general for the approximating discrete-time processes because the two forms then correspond to different ways of scoring situations where more than one point of the process falls into a single time step. As in the discrete case, the quantity G is a characteristic of the model. It represents an upper bound to the expected information gains per unit time that could be obtained from any approximating discrete model. The results are summarized in the proposition below.

284


Proposition 7.6.II. Let N (t, κ) be a stationary regular MPP, let † † λ† (t, κ) dt = λ†g (t)fκ|t dt = E[dt N (t, κ) | Ht− ]

denote its complete H† -conditional intensity, and suppose that λ†g (0) † G = E λg (0) log < ∞, mg where mg = E[λ†g (0)]. If T is any finite partition of the interval (0, T ) and GT the associated average expected information gain per unit time, then GT ≤ G and, as Tn increases through any nested sequence of partitions generating the Borel sets in (0, T ), GTn ↑ G† ≡ limn→∞ GTn ≤ G. Proof. The result follows from further applications of Lemmas A1.6.IV and A1.6.V, but a formal proof requires a more careful discussion of conditioning and predictability than given here and is deferred to Chapter 14. Since G here is a property of the model, it can be evaluated analytically or numerically (by simulation). The model value of G can then be compared with the mean likelihood T −1 log L obtained by applying the model to a set of data, this latter being just the mean entropy score per unit time for the given model with the given data. If the model is close to the true model for the data, the estimate of G obtained in this way should be close to the model G. When the data do not match the model well, the predictive power of the model should be below that obtained when the model is applied to matching data and hence below the theoretical G of the model. In such a situation, the estimated G from the likelihood will generally come out well below the true G of the model (as well as below the unknown G of the true model). The difference between the model and estimated values of G can therefore serve as a basis for model testing and is in fact so used in contingency table contexts, corresponding roughly to the discrete time-models considered earlier in this section. Some of these points are illustrated in the following two examples. Example 7.6(b) Renewal process. Consider a stationary renewal process with interval distribution having density f (x), assumed at least left-continuous. Then λ† (t) = f (Bt )/S(Bt ), where Bt has the distribution of a stationary backward recurrence time. For the mean rate and the expected information gain per unit time, we obtain, respectively, f (Bt ) , m = E[λ† (t)] = E S(Bt ) f (Bt ) λ† (t) f (Bt ) G = E λ† (t) log =E , (7.6.15) log m S(Bt ) mS(Bt )

7.6.


285

the two expectations on the extreme right-hand sides

being with respect to ∞ the distribution of Bt , which has density y f (u) du µ, where µ is the mean interval length [see (4.2.5) or Exercise 3.4.1]. Substituting and simplifying, we find m = 1/µ and ∞ f (y) dy . (7.6.16) f (y) log G=m 1+ m 0 The same result can be obtained from the general result that, for a stationary process, the expected information gain per unit time is just m times the expected information gain per interval, where the latter is defined to be GI = E 0

∞

f † (x) log

f † (x) dx , f0 (x)

with f † (x) the density of the distribution of an interval given the history up to its start, and f0 (x) is the density of an interval under the reference measure. Here, given m, the exponential distribution with mean 1/m has maximum entropy so we take f0 (x) = me−mx in the expression above, corresponding precisely to the choice of the Poisson process with rate m used in the counting process description. Now suppose that probability forecasts are made for a forecasting period of length ∆ ahead. The probability of an event occurring in the interval (t, t + ∆), given the past history Ft† , is given by

p∗ (∆ | X) = S(X) − S(X + ∆) S(X), say, where S(x) is the survivor function for the interval distribution, and X is the backward recurrence time. In the stationary case, writing p0 = 1 − e−m∆ and taking expectations with respect to the stationary form of the backward recurrence time distribution, we consider the quantity G∆ = E[I∆ | Ht† ] 1 p∗ (∆ | X) 1 − p∗ (∆ | X) = E p∗ (∆ | X) log . + [1 − p∗ (∆ | X)] log ∆ p0 1 − p0 (7.6.17) It represents the average expected information gain for forecasts of length ∆, is independent of t and can be shown to satisfy G∆ ≤ G = lim∆→0 G∆ . See Exercise 7.6.4 for details and some numerical illustrations. The next model both illustrates the ideas of Proposition 7.6.II in a relatively simple context and adds a cautionary note to the discussion of probability forecasts for point processes. Example 7.6(c) Marked Hawkes process with exponential infectivity function [see Example 7.3(b)]. Consider an MPP with complete conditional intensity of the form

286


λ† (t, κ) = µ0 + ψ(κi )βe−β(t−ti ) f (κ). {i:ti 0, every point in X can be approximated by points in D; i.e. given x ∈ X , there exists d ∈ D such that ρ(x, d) < . The space X is separable if there exists a countable dense set, also called a separability set. If X is a separable metric space, the spheres with rational radii and centres on a countable dense set form a countable base for the topology. Given two topological spaces (X1 , U1 ) and (X2 , U2 ), a mapping f (·) from (X1 , U1 ) to (X2 , U2 ) is continuous if the inverse image f −1 (U ) of every open set U ∈ U2 is an open set in U1 . If both spaces are metric spaces, the mapping is continuous if and only if for every x ∈ X1 and every > 0, there exists δ > 0 such that ρ2 (f (x ), f (x)) < whenever ρ1 (x , x) < δ, where ρi is the metric in Xi for i = 1, 2; we can express this more loosely as f (x ) → f (x) whenever x → x. A homeomorphism is a one-to-one continuous-bothways mapping between two topological spaces. A famous theorem of Urysohn asserts that any complete separable metric space (c.s.m.s.) can be mapped homeomorphically into a countable product of unit intervals. A Polish space is a space that can be mapped homeomorphically into an open subset of a c.s.m.s. The theory developed in Appendix 2 can be carried through for an arbitrary Polish space with only minor changes, but we do not seek this greater generality. A set K in a topological space (X , U) is compact if every covering of K by a family of open sets contains a finite subcovering; i.e. K ⊆ α Uα , Uα ∈ U, N implies the existence of N < ∞ and α1 , . . . , αN such that K ⊆ i=1 Uαi . It is relatively compact if its closure K is compact. In a separable space, every open covering contains a countable subcovering, and consequently it is sufficient to check the compactness property for sequences of open sets rather than general families. More generally, for a c.s.m.s., the following important characterizations of compact sets are equivalent. Proposition A1.2.II (Metric Compactness Theorem). Let X be a c.s.m.s. Then, the following properties of a subset K of X are equivalent and each is equivalent to the compactness of K. (i) (Heine–Borel property) Every countable open covering of K contains a finite subcovering. (ii) (Bolzano–Weierstrass property) Every infinite sequence of points in K contains a convergent subsequence with its limit in K. (iii) (Total boundedness and closure) K is closed, and for every > 0, K can be covered by a finite number of spheres of radius . (iv) Every sequence {Fn } of closed subsets of K with nonempty finite interN sections (i.e. n=1 Fn = ∅ for N < ∞, ∞the finite intersection property) has nonempty total intersection (i.e. n=1 Fn = ∅). The space X itself is compact if the compactness criterion applies with X in place of K. It is locally compact if every point of X has a neighbourhood with compact closure. A space with a locally compact second countable topology

372

APPENDIX 1. Some Basic Topology and Measure Theory Concepts

is always metrizable. In a c.s.m.s., local compactness implies σ-compactness: the whole space can be represented as a countable union of compact sets (take the compact closures of the neighbourhoods of any countable dense set). Any finite-dimensional Euclidean space is σ-compact, but the same does not apply to infinite-dimensional spaces such as C[0, 1] or the infinite-dimensional Hilbert space 2 . A useful corollary of Proposition A1.2.II is that any closed subset F of a compact set in a complete metric space is again compact, for by (ii) any infinite sequence of points of F has a limit point in K, and by closure the limit point is also in F ; hence, F is compact.

A1.3. Finitely and Countably Additive Set Functions Let A be a class of sets in X , and ξ(·) a real- or complex-valued function defined on A. ξ(·) is finitely additive on A if for finite families {A1 , . . . , AN } of disjoint sets from A, with their union also in A, there holds N N ξ Ai = ξ(Ai ). i=1

i=1

If a similar result holds for sequences of sets {Ai : i = 1, 2, . . .}, then ξ is countably additive (equivalently, σ-additive) on A. A countably additive set function on A is a measure if it is nonnegative; a signed measure if it is realvalued but not necessarily nonnegative; and a complex measure if it is not necessarily real-valued. A determining class for a particular type of set function is a class of sets with the property that if two set functions of the given type agree on the determining class, then they coincide. In this case, we can say that the set function is determined by its values on the determining class in question. The following proposition gives two simple results on determining classes. The first is a consequence of the representation of any element in a ring of sets as a disjoint union of the sets in any generating semiring; the second can be proved using a monotone class argument and the continuity lemma A1.3.II immediately following. Proposition A1.3.I. (a) A finitely additive, real- or complex-valued set function defined on a ring A is determined by its values on any semiring generating A. (b) A countably additive real- or complex-valued set function defined on a σ-ring S is determined by its values on any ring generating S. Proposition A1.3.II (Continuity Lemma). Let µ(·) be a finite real- or complex-valued, finitely additive set function defined on a ring A. Then, µ is countably additive on A if and only if for every decreasing sequence {An : n = 1, 2, . . .} of sets with An ↓ ∅, µ(An ) → 0.

A1.3.

Finitely and Countably Additive Set Functions

373

So far, we have assumed that the set functions take finite values on all the sets for which they are defined. It is frequently convenient to allow a nonnegative set function to take the value +∞; this leads to few ambiguities and simplifies many statements. We then say that a finitely additive set function ξ(·) defined on an algebra or σ-algebra A is totally finite if, for all unions of disjoint sets A1 , . . . , AN in A, there exists M < ∞ such that N

& & &ξ(Ai )& ≤ M. i=1

In particular, a nonnegative, additive set function µ is totally finite if and only if µ(X ) < ∞. A finitely additive set function is σ-finite ∞ if there exists a sequence of sets {An : n = 1, 2, . . .} ∈ A such that X ⊆ n=1 An and for each n the restriction of ξ to An , defined by the equation ˆ ξ(A) = ξ(A ∩ An )

(A ∈ A),

is totally finite, a situation we describe more briefly by saying that ξ is totally finite on each An . The continuity lemma extends to σ-finite set functions with the proviso that we consider only sequences for which |µ(An )| < ∞ for some n < ∞. (This simple condition, extending the validity of Proposition A1.3.II to σ-finite set functions, fails in the general case, however, and it is then better to refer to continuity from below.) We state next the basic extension theorem used to establish the existence of measures on σ-rings. Note that it follows from Proposition A1.3.I that when such an extension exists, it must be unique. Theorem A1.3.III (Extension Theorem). A finitely additive, nonnegative set function defined on a ring R can be extended to a measure on σ(R) if and only if it is countably additive on R. As an example of the use of the theorem, we cite the well-known result that a right-continuous monotonically increasing function F (·) on R can be used to define a measure on the Borel sets of R (the sets in the smallest σ-ring containing the intervals) through the following sequence of steps. (i) Define a nonnegative set function on the semiring of half-open intervals (a, b] by setting µF (a, b] = F (b) − F (a). (ii) Extend µF by additivity to all sets in the ring generated by such intervals (this ring consists, in fact, of all finite disjoint unions of such half-open intervals). (iii) Establish countable additivity on this ring by appealing to compactness properties of finite closed intervals. (iv) Use the extension theorem to assert the existence of a measure extending the definition of µF to the σ-ring generated by the half-open intervals— that is, the Borel sets. The intrusion of the topological notion of compactness into this otherwise measure-theoretic sequence is a reminder that in most applications there is a

374


close link between open and measurable sets. Generalizing the corresponding concept for the real line, the Borel sets in a topological space are the sets in the smallest σ-ring (necessarily a σ-algebra) BX containing the open sets. A Borel measure is any measure defined on the Borel sets. The properties of such measures when X is a c.s.m.s. are explored in Appendix 2. Returning to the general discussion, we note that no simple generalization of the extension theorem is known for signed measures. However, there is an important result, that shows that in some respects the study of signed measures can always be reduced to the study of measures. Theorem A1.3.IV (Jordan–Hahn Decomposition). Let ξ be a signed measure defined on a σ-algebra S. Then, ξ can be written as the difference ξ = ξ+ − ξ− of two measures ξ + , ξ − on S, and X can be written as the union of two disjoint sets U + , U − in S such that, for all E ∈ S, ξ + (E) = ξ(E ∩ U + )

ξ − (E) = −ξ(E ∩ U − ),

and

and hence in particular, ξ + (U − ) = ξ − (U + ) = 0. The measures ξ + and ξ − appearing in this theorem are called upper and lower variations of ξ, respectively. The total variation of ξ is their sum Vξ (A) = ξ + (A) + ξ − (A). It is clear from Theorem A1.3.IV that

n(IP)

Vξ (A) = sup

|ξ(Ai )|,

IP(A) i=1

where the supremum is taken over all finite partitions IP of A into disjoint measurable sets. Thus, ξ is totally bounded if and only if Vξ (X ) < ∞. In this case, Vξ (A) acts as a norm on the space of totally bounded signed measures ξ on S; it is referred to as the variation norm and sometimes written Vξ (X ) = ξ.

A1.4. Measurable Functions and Integrals A measurable space is a pair (X , F), where X is the space and F a σ-ring of sets defined on it. A mapping f from a measurable space (X , F) into a measurable space (Y, G) is G-measurable (or measurable for short) if, for all A ∈ G, f −1 (A) ∈ F. Note that the inverse images in X of sets in G form a σ-ring H = f −1 (G), say, and the requirement for measurability is that H ⊆ F. By specializing to the case where Y is the real line R with G the σ-algebra of Borel sets generated by the intervals, BR , the criterion for measurability simplifies as follows. Proposition A1.4.I. A real-valued function f : (X , F) → (R, BR ) is Borel measurable if and only if the set {x: f (x) ≤ c} is a set in F for every real c.

A1.4.

Measurable Functions and Integrals

375

The family of real-valued (Borel) measurable functions on a measurable space (X , F) has many striking properties. It is closed under the operations of addition, subtraction, multiplication, and (with due attention to zeros) division. Moreover, any monotone limit of measurable functions is measurable. If X is a topological space and F the Borel σ-field on X , then every continuous function on X is measurable. The next proposition provides an important approximation result for measurable functions. Here a simple function is a finite linear combination of indicator functions of measurable sets; that is, a function of the form N s(x) = k=1 ck IAk (x), where c1 , . . . , cN are real and A1 , . . . , AN are measurable sets. Proposition A1.4.II. A nonnegative function f : (X , F) → (R+ , BR+ ) is measurable if and only if it can be represented as the limit of a monotonically increasing sequence of simple functions. Now let µ be a measure on F. We call the triple (X , F, µ) a finite or σfinite measure space according to whether µ has the corresponding property; in the special case of a probability space, when µ has total mass unity, the triple is more usually written (Ω, E, P), where the sets of the σ-algebra E are interpreted as events, a measurable function on (Ω, E) is a random variable, and P is a probability measure. We turn to the problem of defining an integral (or in the probability case N an expectation) with respect to the measure µ. If s = k=1 ck IAk is a nonnegative simple function, set N

s(x) µ(dx) = s dµ = ck µ(Ak ), X

X

k=1

where we allow +∞ as a possible value of the integral. Next, for any nonnegative measurable function f and any sequence of simple functions {sn } approximating f from below, set f dµ = lim sn dµ n→∞

X

X

and prove that the limit is independent of the particular sequence of simple functions used. Finally, for any measurable function f , write + f+ (x) = f (x) = max f (x), 0 ,

f− (x) = f+ (x) − f (x),

and if f+ dµ and f− dµ are both finite (equivalently, X |f | dµ is finite), say that f is integrable and then define, for any integrable function f , f dµ = f+ dµ − f− dµ. X

X

X

The resulting abstract Lebesgue integral is well defined, additive, linear, order-preserving, and enjoys strikingly elegant continuity properties. These

376


last are set out in the theorem below, where we say fn → f µ-almost everywhere (µ-a.e., or a.e. µ) if the (necessarily measurable) set {x: fn (x) → f (x)} has µ-measure zero. In the probability case, we refer to almost sure (a.s.) rather than a.e. convergence. Theorem A1.4.III (Lebesgue Convergence Theorems). The following results hold for a sequence of measurable functions {fn : n = 1, 2, . . .} defined on the measure space (X , F, µ) : (a) (Fatou’s Lemma) If fn ≥ 0, lim inf fn (x) µ(dx) ≤ lim inf fn (x) µ(dx). X n→∞

n→∞

X

(b) (Monotone Convergence Theorem) If fn ≥ 0 and fn ↑ f µ-a.e., then f is measurable and lim fn dµ = f dµ n→∞

X

X

in the sense that either both sides are finite, and then equal, or both are infinite. (c) (Dominated Convergence Theorem) If |fn (x)| ≤ g(x) where g(·) is integrable, and fn → f µ-a.e., then lim fn dµ = f dµ. n→∞

X

X

If f is an integrable function, the indefinite integral of f over any measurable subset can be defined by def def f dµ = IA f dµ, ξf (A) = A

X

where IA is the indicator function of A. It is clear that ξf is totally finite and finitely additive on S. Moreover, it follows from the dominated convergence theorem that if An ∈ S and An ↓ ∅, then IAn f → 0 and hence ξf (An ) → 0. Thus, ξf is also countably additive; that is, a signed measure on S. This raises the question of which signed measures can be represented as indefinite integrals with respect to a given µ. The essential feature is that the ξ-measure of a set should tend to zero with the µ-measure. More specifically, ξ is absolutely continuous with respect to µ whenever µ(A) = 0 implies ξ(A) = 0; we then have the following theorem. Theorem A1.4.IV (Radon–Nikodyn Theorem). Let (X , F, µ) be a σ-finite measure space and ξ a totally finite measure or signed measure on F. Then, there exists a measurable integrable function f such that ξ(A) = f (x) µ(dx) (all A ∈ F) (A1.4.1) A

if and only if ξ is absolutely continuous with respect to µ; moreover, f is a.e. uniquely determined by (A1.4.1), in the sense that any two functions satisfying (A1.4.1), for all A ∈ F must be equal µ-a.e.

A1.5.

Product Spaces

377

The function f appearing in (A1.4.1) is usually referred to as a Radon– Nikodym derivative of ξ with respect to µ, written dξ/dµ. Lemma A1.6.III below shows one way in which the Radon–Nikodym derivative can be expressed as a limiting ratio. There is an obvious extension of Theorem A1.4.IV to the case where ξ is σ-finite; in this extension, (A1.4.1) holds for subsets A of any member of the denumerable family of measurable sets on which ξ is totally finite. Finally, we consider the relation between a fixed σ-finite measure µ and an arbitrary σ-finite signed measure ξ. ξ is said to be singular with respect to µ if there is a set E in F such that µ(E) = 0 and for all A ∈ F, ξ(A) = ξ(E ∩ A) so that also µ(E c ) = 0 and µ(A) = µ(A ∩ E c ). We then have the following theorem. Theorem A1.4.V (Lebesgue Decomposition Theorem). Let (X , F, µ) be a σ-finite measure space and ξ(·) a finite or σ-finite signed measure on F. Then, there exists a unique decomposition of ξ, ξ = ξs + ξac , into components that are, respectively, singular and absolutely continuous with respect to µ.

A1.5. Product Spaces If X , Y are two spaces, the Cartesian product X × Y is the set of ordered pairs {(x, y): x ∈ X , y ∈ Y}. If X and Y are either topological or measure spaces, there is a natural way of combining the original structures to produce a structure in the product space. Consider first the topological case. If U , V are neighbourhoods of the points x ∈ X , y ∈ Y with respect to topologies U, V, define a neighbourhood of the pair (x, y) as the product set U × V . The class of product sets of this kind is closed under finite intersections because (U × V ) ∩ (A × B) = (U ∩ A) × (V ∩ B). It can therefore be taken as the basis of a topology in X × Y; it is called the product topology and denoted X ⊗ Y [we follow e.g. Brémaud (1981) in using a distinctive product sign as a reminder that the product entity here is generated by the elements of the factors]. Most properties enjoyed by the component (or coordinate) topologies are passed on to the product topology. In particular, if X , Y are both c.s.m.s.s, then X × Y is also a c.s.m.s. with respect to any one of a number of equivalent metrics, of which perhaps the simplest is ρ((x, y), (u, v)) = max(ρX (x, u), ρY (y, v)). More generally, if {Xt : t ∈ T } is a family of spaces, the Cartesian product X =

× (Xt)

t∈T

may be defined as the set of all functions x: T →

t

Xt such that x(t) ∈ Xt .

378


A cylinder set in this space is a set in which restrictions are placed on a finite subset of the coordinates, on x(t1 ), . . . , x(tN ), say, the values of the other coordinates being unrestricted in their appropriate spaces. A family of basic open sets in X can be defined by choosing open sets {Ut ⊆ Xti , i = 1, . . . , N } and requiring x(ti ) ∈ Ui , i = 1, . . . , N . The topology generated by the class of cylinder sets of this form is called the product topology in X . A remarkable property of this topology is that if the coordinate spaces Xt are individually compact in their respective topologies, then X is compact in the product topology. On the other hand, if the individual Xt are metric spaces, there are again many ways in which X can be made into a metric space [e.g. by using the supremum of the distances ρt (x(t), y(t)) ], but the topologies they generate are not in general equivalent among themselves nor to the product topology defined earlier. Turning now to the measure context, let (X , F, µ) and (Y, G, ν) be two measure spaces. The product σ-ring F ⊗ G is the σ-ring generated by the semiring of measurable rectangles A × B with A ∈ F, B ∈ G. The product measure µ × ν is the extension to the σ-ring of the countably additive set function defined on such rectangles by (µ × ν)(A × B) = µ(A) ν(B) and extended by additivity to the ring of all finite disjoint unions of such rectangles. If µ, ν are both finite, then so is µ × ν; similarly, if µ, ν are σfinite, so is µ × ν. The product measurable space is the space (X × Y, F ⊗ G), and the product measure space is the space (X × Y, F ⊗ G, µ × ν). All the definitions extend easily to the products of finite families of measure spaces. In the probability context, they form the natural framework for the discussion of independence. In the context of integration theory, the most important results pertain to the evaluation of double integrals, the question we take up next. Let H = F ⊗ G and π = µ × ν. If C is H-measurable, its sections Cx = {y: (x, y) ∈ C},

C y = {x: (x, y) ∈ C}

are, respectively, G-measurable for each fixed x and F-measurable for each fixed y. (The converse to this result, that a set whose sections are measurable is H-measurable, is false, however.) Similarly, if f (x, y) is H-measurable, then regarded as a function of y, it is G-measurable for each fixed x, and regarded as a function of x, it is F-measurable for each fixed y. Introducing integrals with respect to µ, ν, write f (x, y) ν(dy) if the integrand is ν-integrable, Y s(x) = +∞ otherwise; f (x, y) µ(dx) if the integrand is µ-integrable, X t(y) = +∞ otherwise. We then have the following theorem.

A1.5.

Product Spaces

379

A1.5.I (Fubini’s Theorem). Let (X , F, µ) and (Y, G, ν) be σ-finite measure spaces, and let (Z, H, π) denote the product measure space. (a) If f is H-measurable and π-integrable, then s(x) is F-measurable and µ-integrable, t(y) is G-measurable and ν-integrable, and

f dπ = Z

s dµ = X

t dν. Y

(b) If f is H-measurable and f ≥ 0, it is necessary and sufficient for f to be π-integrable that either s be µ-integrable or t be ν-integrable. Not all the important measures on a product space are product measures; in the probability context, in particular, it is necessary to study general bivariate probability measures and their relations to the marginal and conditional measures they induce. Thus, if π is a probability measure on (X × Y, F ⊗ G), we define the marginal probability measures πX and πY to be the projections of π onto (X , F) and (Y, G), respectively; i.e. the measures defined by πX (A) = π(A × Y)

πY (B) = π(X × B).

and

We next investigate the possibility of writing a measure on the product space as an integral (or a mixture of conditional probabilities), say π(A × B) = A

Q(B | x) πX (dx),

(A1.5.1)

where Q(B | x) may be regarded as the conditional probability of observing the event B given the occurrence of x. Such a family is also known as a disintegration of π. Proposition A1.5.II. Given a family {Q(· | x): x ∈ X } of probability measures on (Y, G) and a probability measure πX on (X , F), the necessary and sufficient condition that (A1.5.1) should define a probability measure on the product space (Z, H) is that, as a function of x, Q(B | x) be F-measurable for each fixed B ∈ G. When this condition is satisfied, for every H-measurable, nonnegative function f (·, ·),

f dπ =

Z

X

πX (dx)

Y

f (x, y) Q(dy | x).

(A1.5.2)

Indeed, the integral in (A1.5.1) is not defined unless Q(B | ·) is F-measurable. When it is, the right-hand side of (A1.5.2) can be extended to a finitely additive set function on the ring of finite unions of disjoint rectangle sets. Countable additivity and the extension to a measure for which (A1.5.2) holds then follow along standard lines using monotone approximation arguments.

380


The projection of π onto the space (Y, G), i.e. the measure defined by πY (B) =

X

Q(B | x) πX (dx),

is known as the mixture of Q(· | x) with respect to πX . The converse problem, of establishing the existence of a family of measures satisfying (A1.5.1) from a given measure and its marginal, is a special case of the problem of regular conditional probabilities (see e.g. Ash, 1972, Section 6.6). For any fixed B ∈ G, π(· × B) may be regarded as a measure on (X , F), that is clearly absolutely continuous with respect to the marginal πX . Hence, there exists a Radon–Nikodym derivative, QR (B | x) say, that is Fmeasurable, satisfies (A1.5.1), and should therefore be a candidate for the disintegration of π. The difficulty is that we can guarantee the behaviour of QR only for fixed sets B, and it is not clear whether, for x fixed and B varying, the family QR (B | x) will have the additivity and continuity properties of a measure. If {A1 , . . . , AN } is a fixed family of disjoint sets in G or if {Bn : n ≥ 1} is a fixed sequence in G with Bn ↓ ∅, then it is not difficult to show that ! QR

N i=1

" N &

& Ai & x = QR (Ai | x)

πX -a.e.,

i=1

QR (Bn | x) → 0

(n → ∞)

πX -a.e.,

respectively, but because there are uncountably many such relations to be checked, it is not obvious that the exceptional sets of measure zero can be combined into a single such set. The problem, in fact, is formally identical to establishing the existence of random measures and is developed further in Chapter 9. The following result is a partial converse to Proposition A1.5.II. Proposition A1.5.III (Existence of Regular Conditional Probabilities). Let (Y, G) be a c.s.m.s. with its associated σ-algebra of Borel sets, (X , F) an arbitrary measurable space, and π a probability measure on the product space (Z, H). Then, with πX (A) = π(A × Y) for all A ∈ F, there exists a family of kernels Q(B | x) such that (i) Q(· | x) is a probability measure on G for each fixed x ∈ X ; (ii) Q(B | ·) is an F-measurable function on X for each fixed B ∈ G; and (iii) π(A × B) = A Q(B | x) πX (dx) for all A ∈ F and B ∈ B. We consider finally the product of a general family of measurable spaces, {(XT , Ft ): t ∈ T }, where T is an arbitrary (finite, countable, or uncountable) indexing set. Once again, the cylinder sets play a basic role. A measurable cylinder set in X = ×t∈T (Xt ) is a set of the form C(t1 , . . . , tN ; B1 , . . . , BN ) = {x(t): x(ti ) ∈ Bi , i = 1, . . . , N },

A1.5.

Product Spaces

381

where Bi ∈ Fti is measurable for each i = 1, . . . , N . Such sets form a semiring, their finite disjoint unions form a ring, and the generated σ-ring we denote by 0 F∞ = Ft . t∈T

This construction can be used to define a product measure on F∞ , but greater interest centres on the extension problem: given a system of measures π(σ) defined on finite subfamilies F(σ) = Ft1 ⊗ Ft2 ⊗ · · · ⊗ FtN , where (σ) = {t1 , . . . , tN } is a finite selection of indices from T , when can they be extended to a measure on F∞ ? It follows from the extension theorem A1.3.III that the necessary and sufficient condition for this to be possible is that the given measures must give rise to a countably additive set function on the ring generated by the measurable cylinder sets. As with the previous result, countable additivity cannot be established without some additional assumptions; again it is convenient to put these in topological form by requiring each of the Xt to be a c.s.m.s. Countable additivity then follows by a variant of the usual compactness argument, and the only remaining requirement is that the given measures should satisfy the obviously necessary consistency conditions stated in the theorem below. Theorem A1.5.IV (Kolmogorov Extension Theorem). Let T be an arbitrary index set, and for t ∈ T suppose (Xt , Ft ) is a c.s.m.s. with its associated Borel σ-algebra. Suppose further that for each finite subfamily (σ) = {t1 , . . . , tN } of indices from T , there is given a probability measure π(σ) on F(σ) = Ft1 ⊗ · · · ⊗ FtN . In order that there exist a measure π on F∞ such that for all (σ), π(σ) is the projection of π onto F(σ) , it is necessary and sufficient that for all (σ), (σ1 ), (σ2 ), (i) π(σ) depends only on the choice of indices in (σ), not on the order in which they are written down; and (ii) if (σ1 ) ⊆ (σ2 ), then π(σ1 ) is the projection of π(σ2 ) onto F(σ1 ) . Written out more explicitly in terms of distribution functions, condition (i) becomes (in an obvious notation) the condition of invariance under simultaneous permutations: if p1 , . . . , pN is a permutation of the integers 1, . . . , N , then (N ) (N ) Ft1 ,...,tN (x1 , . . . , xN ) = Ftp1 ,...,tp (xp1 , . . . , xpN ). N

Similarly, condition (ii) becomes the condition of consistency of marginal distributions, namely that (N +k)

(N )

Ft1 ,...,tN ,s1 ,...,sk (x1 , . . . , xN , ∞, . . . , ∞) = Ft1 ,...,tN (x1 , . . . , xN ). The measure π induced on F∞ by the fidi distributions is called their projective limit. Clearly, if stochastic processes have the same fidi distributions, they must also have the same projective limit. Such processes may be described as being equivalent or versions of one another. See Parthasarathy (1967, Sections 5.1–5) for discussion of Theorem A1.5.IV in a slightly more general form and for proof and further details.

382


A1.6. Dissecting Systems and Atomic Measures The notion of a dissecting system in Definition A1.6.I depends only on topological ideas of separation and distinguishing one point from another by means of distinct sets, though we use it mainly in the context of a metric space where its development is simpler. If (X , U) is a topological space, the smallest σ-algebra containing the open sets is called the Borel σ-algebra. If f : X → R is any real-valued continuous function, then the set {x: f (x) < c} is open in U and hence measurable. It follows that f is measurable. Thus, every continuous function is measurable with respect to the Borel σ-algebra. Definition A1.6.I (Dissecting System). The sequence T = {Tn } of finite partitions Tn = {Ani : i = 1, . . . , kn } (n = 1, 2, . . .) consisting of Borel sets in the space X is a dissecting system for X when (i) (partition properties) Ani ∩ Anj = ∅ for i = j and An1 ∪ · · · ∪ Ankn = X ; (ii) (nesting property) An−1,i ∩ Anj = Anj or ∅; and (iii) (point-separating property) given distinct x, y ∈ X , there exists an integer n = n(x, y) such that x ∈ Ani implies y ∈ / Ani . Given a dissecting system T for X , properties (i) and (ii) of Definition A1.6.I imply that there is a well-defined nested sequence {Tn (x)} ⊂ T such that ∞ Tn (x) = {x}, so µ(Tn (x)) → µ{x} (n → ∞) n=1

because µ is a measure and {Tn (x)} is a monotone sequence. Call x ∈ X an atom of µ if µ({x}) ≡ µ{x} > 0. It follows that x is an atom of µ if and only if µ(Tn (x)) > (all n) for some > 0; indeed, any in 0 < ≤ µ{x} will do. We use δx (·) to denote Dirac measure at x, being defined on Borel sets A by 1 if x ∈ A, δx (A) = 0 otherwise. More generally, an atom of a measure µ on a measurable space (X , F) is any nonempty set F ∈ F such that if G ∈ F and G ⊆ F , then either G = ∅ or G = F . However, when X is a separable metric space, it is a consequence of Proposition A2.1.IV below that the only possible atoms of a measure µ on (X , F) are singleton sets. A measure with only atoms is purely atomic; a diffuse measure has no atoms. Given > 0, we can identify all atoms of µ of mass µ{x} ≥ , and then using a sequence {j } with j ↓ 0 as j → ∞, all atoms of µ can be identified. Because µ is σ-finite, it can have at most countably many atoms, so identifying them as {xj : j = 1, 2, . . .}, say, and writing bj = µ{xj }, the measure µa (·) ≡

∞

j=1

bj δxj (·),

A1.6.

Dissecting Systems and Atomic Measures

383

which clearly consists only of atoms, is the atomic component of the measure µ. The measure ∞

µd (·) ≡ µ(·) − µa (·) = µ(·) − bj δxj (·) j=1

has no atoms and is the diffuse component of µ. Thus, any measure µ as above has a unique decomposition into atomic and diffuse components. Lemma A1.6.II. Let µ be a nonatomic measure and {Tn } a dissecting system for a set A with µ(A) < ∞. Then n ≡ supi µ(Ani ) → 0 as n → ∞. Proof. Suppose not. Then there exists δ > 0 and, for each n, some set An,in , say, with An,in ∈ Tn and µ(An,in ) > δ. Because Tn is a dissecting system, the nesting implies that there exists An−1,in−1 ∈ Tn−1 and contains An,in , so µ(An−1,in−1 ) > δ. Consequently, we can assume there exists a nested sequence of sets An,in for which µ(An,in ) > δ, and hence δ ≤ lim µ(An,in ) = µ(lim An,in ), n

n

equality holding here because µ is a measure and {An,in } is monotone. But, because Tn is a dissecting system, limn An,in is either empty or a singleton set, {x } say. Thus, the right-hand side is either µ(∅) = 0 or µ({x}) = 0 because µ is nonatomic (i.e. δ ≤ 0), which is a contradiction. Dissecting systems can be used to construct approximations to Radon– Nikodym derivatives as follows (e.g. Chung, 1974, Chapter 9.5, Example VIII). Lemma#A1.6.III (Approximation of Radon–Nikodym Derivative). Let T = $ {Tn } = {Ani : i = 1, . . . , kn } be a nested family of measurable partitions of the measure space (Ω, E, µ), generating E and let ν be a measure absolutely continuous with respect to µ, with Radon–Nikodym derivative dν/dµ. Define λn (ω) =

kn

i=1

Then, as n → ∞, λn →

IAni (ω)

ν(Ani ) µ(Ani )

(ω ∈ Ω).

dν , µ-a.e. and in L1 (µ) norm. dµ

As a final result involving dissecting systems, given two probability measures P and P0 on (Ω, E), define the relative entropy of the restriction of P and P0 to a partition T = {Ai } of (Ω, E) by

P (Ai ) . P (Ai ) log H(P ; P0 ) = P 0 (Ai ) i Additivity of measures, convexity of x log x on x > 0, and the inequality (a1 + a2 )/(b1 + b2 ) ≤ a1 /b1 + a2 /b2 , valid for nonnegative ar and positive br , r = 1, 2, establishes the result below. Lemma A1.6.IV. Let T1 , T2 be measurable partitions of (Ω, E) with T1 ⊆ T2 and P, P0 two probability measures on (Ω, E). Then, the relative entropies of the restrictions of P, P0 to Tr satisfy H1 (P ; P0 ) ≤ H2 (P ; P0 ).

APPENDIX 2

Measures on Metric Spaces

A2.1. Borel Sets and the Support of Measures If (X , U) is a topological space, the smallest σ-algebra containing the open sets is called the Borel σ-algebra. If f : X → R is any real-valued continuous function, then the set {x: f (x) < c} is open in U and hence measurable. It follows that f is measurable. Thus, every continuous function is measurable with respect to the Borel σ-algebra. It is necessary to clarify the relation between the Borel sets and various other candidates for useful σ-algebras that suggest themselves, such as (a) the Baire sets, belonging to the smallest σ-field with respect to which the continuous functions are measurable; (b) the Borelian sets, generated by the compact sets in X ; and (c) if X is a metric space, the σ-algebra generated by the open spheres. We show that, with a minor reservation concerning (b), all three concepts coincide when X is a c.s.m.s. More precisely, we have the following result. Proposition A2.1.I. Let X be a metric space and U the topology induced by the metric. Then (i) the Baire sets and the Borel sets coincide; (ii) if X is separable, then the Borel σ-algebra is the smallest σ-algebra containing the open spheres; (iii) a Borel set is Borelian if and only if it is σ-compact; that is, if it can be covered by a countable union of compact sets. In particular, the Borel sets and the Borelian sets coincide if and only if the whole space is σ-compact. 384

A2.1.

Borel Sets and the Support of Measures

385

Proof. Part (i) depends on Lemma A2.1.II below, of interest in its own right; (ii) depends on the fact that when X is separable, every open set can be represented as a countable union of open spheres; (iii) follows from the fact that all closed subsets of a compact set are compact and hence Borelian. Lemma A2.1.II. Let F be a closed set in the metric space X , U an open set containing F , and IF (·) the indicator function of F . Then, there exists a sequence of continuous functions {fn (x)} such that (i) 0 ≤ fn (x) ≤ 1 (x ∈ X ); (ii) fn (x) = 0 outside U ; (iii) fn (x) ↓ IF (x) as n → ∞. Proof. Let fn (x) = ρ(x, U c )/[ρ(x, U c ) + 2n ρ(x, F )], where for any set C ρ(x, C) = inf ρ(x, y). y∈C

Then, the sequence {fn (x)} has the required properties. It is clear that in a separable metric space the Borel sets are countably generated. Lemma A2.1.III exhibits a simple example of a countable semiring of open sets generating the Borel sets. Lemma A2.1.III. Let X be a c.s.m.s., D a countable dense set in X , and S0 the class of all finite intersections of open spheres Sr (d) with centres d ∈ D and rational radii. Then (i) S0 and the ring A0 generated by S0 are countable; and (ii) S0 generates the Borel σ-algebra in X . It is also a property of the Borel sets in a separable metric space, and of considerable importance in the analysis of sample-path properties of point processes and random measures, that they include a dissecting system defined in Definition A1.6.I. Proposition A2.1.IV. Every separable metric space X contains a dissecting system. Proof. Let {d1 , d2 , . . .} = D be a separability set for X (i.e. D is a countable dense set in X ). Take any pair of distinct points x, y ∈ X ; their distance apart equals 2δ ≡ ρ(x, y) > 0. We can then find dm , dn in D such that ρ(dm , x) < δ, ρ(dn , y) < δ, so the spheres Sδ (dm ), Sδ (dn ), which are Borel sets, certainly separate x and y. We have essentially to embed such separating spheres into a sequence of sets covering the whole space. For the next part of the proof, it is convenient to identify one particular element in each Tn (or it may possibly be a null set for all n sufficiently large) as An0 ; this entails no loss of generality. Define the initial partition {A1i } by A11 = S1 (d1 ), A10 = X \ A11 . Observe that X is covered by the countably infinite sequence {S1 (dn )}, so the sequence

386

APPENDIX 2. Measures on Metric Spaces

n of sets {An0 } defined by An0 = X \ r=1 S1 (dr ) converges to the null set. For n = 2, 3, . . . and i = 1, . . . , n, define "c ! n Bni = S1/2n−i (di ), Bn0 = Bni , i=1

so that {Bni :i = 0, . . . , n} coversX . By setting Cn0 = Bn0 , Cn1 = Bn1 , and Cni = Bni \ Bn1 ∪ · · · ∪ Bn,i−1 , it is clear that {Cni : i = 0, 1, . . . , n} is a partition of X . Let the family {Ani } consist of all nonempty intersections of the $ , setting in particular An0 = An−1,0 ∩ Cn0 = An0 . Then # form An−1,j ∩ Cnk {Ani }: n = 1, 2, . . . clearly consists of nested partitions of X by Borel sets, and only the separation property has to be established. Take distinct points x, y ∈ X , and write δ = ρ(x, y) as before. Fix the integer r ≥ 0 by 2−r ≤ min(1, δ) < 2−r+1 , and locate a separability point dm such that ρ(dm , x) < 2−r . Then x ∈ S1/2r (dm ) = Bm+r,m , and consequently x ∈ Cm+r,j for some j = 1, . . . , m. But by the triangle inequality, for any z ∈ Cm+r,j , ρ(x, z) < 2

and

2−(m+r−j) < 2δ = ρ(x, y),

so the partition {Cm+r,i }, and hence also {Am+r,j }, separates x and y. Trivially, if T is a dissecting system for X , the nonempty sets of T ∩ A (in an obvious notation) constitute a dissecting system for any A ∈ BX . If A is also compact, the construction of a dissecting system for A is simplified by applying the Heine–Borel theorem to extract a finite covering of A from the countable covering {S2−n (dr ): r = 1, 2, . . .}. Definition A2.1.V. The ring of sets generated by finitely many intersections and unions of elements of a dissecting system is a dissecting ring.

A2.2. Regular and Tight Measures In this section, we examine the extent to which values of a finitely or countably generated set function defined on some class of sets can be approximated by their values on either closed or compact sets. Definition A2.2.I. (i) A finite or countably additive, nonnegative set function µ defined on the Borel sets is regular if, given any Borel set A and > 0, there exist open and closed sets G and F , respectively, such that F ⊆ A ⊆ G and µ(G − A) < and µ(AF ) < . (ii) It is compact regular if, given any Borel set A and > 0, there exists a compact set C such that C ⊆ A and µ(A − C) < . We first establish the following.

A2.2.

Regular and Tight Measures

387

Proposition A2.2.II. If X is a metric space, then all totally finite measures on BX are regular. Proof. Let µ be a totally finite, additive, nonnegative set function defined on BX . Call any A ∈ BX µ-regular if µ(A) can be approximated by the values of µ on open and closed sets in the manner of Definition A2.2.I. The class of µ-regular sets is obviously closed under complementation. It then follows from the inclusion relations Gα − Fα ⊆ (Gα − Fα ) (A2.2.1a) α

and

α

Gα −

α

Fα ⊆

α

α

α

α

G α − Fα

⊆

(Gα − Fα )

(A2.2.1b)

α

that the class is an algebra if µ is finitely additive and aσ-algebra if µ is countably additive. In the latter case, the countable union α Fα in (A2.2.1a) may N not be closed, but we can approximate µ α Fα by µ i=1 Fαi to obtain a set that is closed and has the required properties; similarly, in (A2.2.1b) we N can approximate µ α Gα by µ i=1 Aαi . Moreover, if µ is σ-additive, the class also contains all closed sets, for if F is closed, the halo sets F = S (x) = {x: ρ(x, F ) < } (A2.2.2) x∈F

form, for a sequence of values of tending to zero, a family of open sets with the property F ↓ F ; hence, it follows from the continuity lemma A1.3.II that µ(F ) → µ(F ). In summary, if µ is countably additive, the µ-regular sets form a σ-algebra containing the closed sets, and therefore the class must coincide with the Borel sets themselves. Note that this proof does not require either completeness or separability. Compact regularity is a corollary of this result and the notion of a tight measure. Definition A2.2.III (Tightness). A finitely or countably additive set function µ is tight if, given > 0, there exists a compact set K such that µ(X −K) is defined and µ(X − K) < . Lemma A2.2.IV. If X is a complete metric space, a Borel measure is compact regular if and only if it is tight. Proof. Given any Borel set A, it follows from Proposition A2.2.II that there exists a closed set C ⊆ A with µ(A − C) < 12 . If µ is tight, choose K so that µ(X − K) < 12 . Then, the set C ∩ K is a closed subset of the compact set K and hence is itself compact; it also satisfies µ(A − C ∩ K) ≤ µ(A − C) + µ(A − K) < ,

388


which establishes the compact regularity of µ. If, conversely, µ is compact regular, tightness follows on taking X = K. Proposition A2.2.V. If X is a c.s.m.s., every Borel measure µ is tight and hence compact regular. Proof. Let D be a separability set for X ; then for fixed n, d∈D S1/n (d) = X , and so by the continuity lemma A1.3.II, there is a finite set d1 , . . . , dk(n) such that k(n) S1/n (di ) < n . µ X− 2 i=1 k(n) Now consider K = n i=1 S1/n (di ) . It is not difficult to see that K is closed and totally bounded, and hence compact, by Proposition A1.2.II and that µ(X − K) < . Hence, µ is tight. The results above establish compact regularity as a necessary condition for a finitely additive set function to be countably additive. The next proposition asserts its sufficiency. The method of proof provides a pattern that is used with minor variations at several important points in the further development of the theory. Proposition A2.2.VI. Let A be a ring of sets from the c.s.m.s. X and µ a finitely additive, nonnegative set function defined and finite on A. A sufficient condition for µ to be countably additive on A is that, for every A ∈ A and > 0, there exists a compact set C ⊆ A such that µ(A − C) < . Proof. Let {An } be a decreasing sequence of sets in A with An ↓ ∅; to establish countable additivity for µ, it is enough to show that µ(An ) → 0 for every such sequence. Suppose to the contrary that µ(An ) ≥ α > 0. By assumption, there exists for each n a compact set Cn for which Cn ⊆ An and µ(An − Cn ) < α/2n+1 . By (A2.2.1), An −

Ck ⊆

k

Since A is a ring, every finite union from the finite additivity of µ,

(Ak − Ck ).

k

n

k=1 (Ak

− Ck ) is an element of A, so

n n

α µ An − Ck ≤ < 2n+1 k=1

1 2 α.

k=1

n Thus, the intersection k=1 Ck is nonempty for each n, nand it follows from the finite intersection part of Proposition A1.2.II that k=1 Ck is nonempty. This gives us the required contradiction to the assumption An ↓ ∅.

A2.2.

Regular and Tight Measures

389

Corollary A2.2.VII. A finite, finitely additive, nonnegative set function defined on the Borel sets of X is countably additive if and only if it is compact regular. We can now prove an extension of Proposition A2.2.VI that plays an important role in developing the existence theorems of Chapter 9. It is based on the notion of a self-approximating ring and is a generalization of the concept of a covering ring given in Kallenberg (1975). Definition A2.2.VIII (Self-Approximating Ring). A ring A of sets of the c.s.m.s. X is a self-approximating ring if, for every A ∈ A and > 0, there exists a sequence of closed sets {Fk (A; )} such that (i) Fk (A; ) ∈ A (k = 1, 2, . . .); (ii) each set Fk (A; ) is contained within a sphere of radius ; and ∞ (iii) k=1 Fk (A; ) = A. Kallenberg uses the context where X is locally compact, in which case it is possible to require the covering to be finite so that the lemma below effectively reduces to Proposition A2.2.VI. The general version is based on an argument in Harris (1968). The point is that it allows checking for countable additivity to be reduced to a denumerable set of conditions. Lemma A2.2.IX. Let A be a self-approximating ring of subsets of the c.s.m.s. X and µ a finitely additive, nonnegative set function defined on A. In order that µ have an extension as a measure on σ(A), it is necessary and sufficient that for each A ∈ A, using the notation of Definition A2.2.VIII, ! lim µ

m→∞

m

" Fi (A; )

= µ(A).

(A2.2.3)

i=1

Proof. Necessity follows from the continuity lemma. We establish sufficiency by contradiction: suppose that µ is finitely additive and satisfies (A2.2.3) but that µ cannot be extended to a measure on σ(A). From the continuity lemma, it again follows that there exists α > 0 and a sequence of sets An ∈ A, with An ↓ ∅, such that µ(An ) ≥ α. (A2.2.4) mk For each k, use (A2.2.3) to choose a set Fk = i=1 Fi (A; k −1 ) that is closed, can be covered by a finite number of k −1 spheres, and satisfies µ(Ak − Fk ) ≤ α/2n+1 . From (A2.2.1), we have A − additivity of µ, implies that ! µ

k j=1 k j=1

Fj ⊆

k

j=1 (Aj

" Fj

≥ 12 α > 0.

− Fj ), which, with the

390


Thus, the sets Fj have the finite intersection property. kTo show that their complete intersection is nonempty, choose any xk ∈ j=1 Fj . Since F1 can be covered by a finite number of 1-spheres, there exists a subsequence {xk } that is wholly contained within a sphere of radius 1. Turning to F2 , we can select a further subsequence xk , which for k ≥ 2 lies wholly within a sphere of radius 12 . Proceeding in this way by induction, we finally obtain by a diagonal selection argument a subsequence {xkj } such that for j ≥ j0 all terms are contained within a sphere of radius 1/j0 . This is enough to show that {xkj } is a Cauchy sequence that, since X is complete, k has a limit point x ¯, say. For each k, the xj are in n=1 Fn for all sufficiently large j. Since the sets are closed, this implies that x ¯ ∈ Fk for every k. But ∞ this implies also that x ¯ ∈ Ak and hence x ¯ ∈ k=1 Ak , which contradicts the assumption that An ↓ ∅. The contradiction shows that (A2.2.4) cannot hold and so completes the proof of the lemma. Let us observe finally that self-approximating rings do exist. A standard example, which is denumerable and generating as well as self-approximating, is the ring C generated by the closed spheres with rational radii and centres on a countable dense set. To see this, consider the class D of all sets that can be approximated by finite unions of closed sets in C in the sense required by condition (iii) of Definition A2.2.VIII. This class contains all open sets because any open set G can be written as a denumerable union of closed spheres, with their centres at points of the countable dense set lying within G, and rational radii bounded by the nonzero distance from the given point of the countable dense set to the boundary of G. D also contains all closed spheres in C; for example, suppose is given, choose any positive rational δ < , and take the closed spheres with centres at points of the countable dense set lying within the given sphere and having radii δ. These are all elements of C, and therefore so are their intersections with the given closed sphere. These intersections form a countable family of closed sets satisfying (iii) of Definition A2.2.VIII for the given closed sphere. It is obvious that D is closed under finite unions and that, from the relation ! ∞ " ! ∞ " ∞ ∞ Fj ∩ Fk = (Fj ∩ Fk ), j=1

k=1

j=1 k=1

D is also closed under finite intersections. Since D contains all closed spheres and their complements that are open, D contains C. Thus, every set in C can be approximated by closed spheres in C, so C is self-approximating as required.

A2.3. Weak Convergence of Measures We make reference to the following notions of convergence of a sequence of measures on a metric space (see Section A1.3 for the definition of · ).

A2.3.

Weak Convergence of Measures

391

Definition A2.3.I. Let {µn : n ≥ 1} and µ be totally finite measures in the metric space X . (i) µn → µ weakly if f dµn → f dµ for all bounded continuous functions f on X . (ii) µn → µ vaguely if f dµn → f dµ for all bounded continuous functions f on X vanishing outside a compact set. (iii) µn → µ strongly (or in variation norm) if µn − µ → 0. The last definition corresponds to strong convergence in the Banach space of all totally finite signed measures on X , for which the total variation metric constitutes a genuine norm. The first definition does not correspond exactly to weak convergence in the Banach-space sense, but it reduces to weak star (weak*) convergence when X is compact (say, the unit interval) and the space of signed measures on X can be identified with the adjoint space to the space of all bounded continuous functions on X . Vague convergence is particularly useful in the discussion of locally compact spaces; in our discussion, a somewhat analogous role is played by the notion of weak hash convergence (w# -convergence; see around Proposition A2.6.II below); it is equivalent to vague convergence when the space is locally compact. Undoubtedly, the central concept for our purposes is the concept of weak convergence. Not only does it lead to a convenient and internally consistent topologization of the space of realizations of a random measure, but it also provides an appropriate framework for discussing the convergence of random measures conceived as probability distributions on this space of realizations. In this section, we give a brief treatment of some basic properties of weak convergence, following closely the discussion in Billingsley (1968) to which we refer for further details. Theorem A2.3.II. Let X be a metric space and {µn : n ≥ 1} and µ measures on BX . Then, the following statements are equivalent. (i) µn → µ weakly. (ii) µn (X ) → µ(X ) and lim supn→∞ µn (F ) ≤ µ(F ) for all closed F ∈ BX . (iii) µn (X ) → µ(X ) and lim inf n→∞ µn (G) ≥ µ(G) for all open G ∈ BX . (iv) µn (A) → µ(A) for all Borel sets A with µ(∂A) = 0 (i.e. all µ-continuity sets). Proof. We show that (i) ⇒ (ii) ⇔ (iii) ⇒ (iv) ⇒ (i). Given a closed set F , choose any fixed ν > 0 and construct a [0, 1]-valued continuous function f that equals 1 on F and vanishes outside F ν [see (A2.2.2) and Lemma A2.1.II]. We have for each n ≥ 1 µn (F ) ≤ f dµn ≤ µn (F ν ), so if (i) holds,

lim sup µn (F ) ≤ n→∞

f dµ ≤ µ(F ν ).

392


But F ν ↓ F as ν ↓ 0, and by the continuity Lemma A1.3.II we can choose ν so that, given any > 0, µ(F ν ) ≤ µ(F ) + . Since is arbitrary, the second statement in (ii) follows, while the first is trivial if we take f = 1. Taking complements shows that (ii) and (iii) are equivalent. ¯ so supposing that (iii) holds When A is a µ-continuity set, µ(A◦ ) = µ(A), and hence (ii) also, we have on applying (ii) to A¯ and (iii) to A◦ that ¯ ≤ µ(A) ¯ = µ(A◦ ) lim sup µn (A) ≤ lim sup µn (A) ≤ lim inf µn (A◦ ) ≤ lim inf µn (A). Thus, equality holds throughout and µn (A) → µ(A) so (iv) holds. Finally, suppose that (iv) holds. Let f be any bounded continuous function on X , and let the bounded interval [α , α ] be such that α < f (x) < α for all x ∈ X . Call α ∈ [α , α ] a regular value of f if µ{x: f (x) = α} = 0. At most a countable number of values can be irregular, while for any α, β that are regular values, {x: α < f (x) ≤ β} is a µ-continuity set. From the boundedness of f on X , given any > 0, we can partition [α , α ] by a finite set of points α0 = α , . . . , αN = α with αi−1 < αi ≤ αi−1 + for i = 1, . . . , N , and from the countability of the set of irregular points (if any), we can moreover assume that these αi are all regular points of f . Defining Ai = {x: αi−1 < f (x) ≤ αi } for i = 1, . . . , N and then fL (x) =

N

αi−1 IAi (x),

fU (x) =

i=1

N

αi IAi (x),

i=1

each Ai is a µ-continuity set, fL (x) ≤ f (x) ≤ fU (x), and by (iv), fL dµ =

N

αi−1 µ(Ai ) = lim

n→∞

i=1

≤ lim

n→∞

fU dµn =

N

αi−1 µn (Ai ) = lim

n→∞

i=1

fL dµn

fU dµ,

by at most µ(X ). Since is arbitrary and the extreme terms here differing f dµ ≤ f dµ ≤ f dµ , it follows that we must have f dµ → L n n U n n f dµ for all bounded continuous f ; that is, µn → µ weakly. Since the functions used in the proof that (i) implies (ii) are uniformly continuous, we can extract from the proof the following useful condition for weak convergence. Corollary A2.3.III. µn → µ weakly if and only if f dµn → f dµ for all bounded and uniformly continuous functions f : X → R. Billingsley calls a class C of sets with the property that µn (C) → µ(C)

(all C ∈ C)

implies

µn → µ

weakly

(A2.3.1)

A2.3.

Weak Convergence of Measures

393

a convergence-determining class. In this terminology, (iv) of Theorem A2.3.II asserts that the µ-continuity sets form a convergence-determining class. Any convergence-determining class is necessarily a determining class, but the converse need not be true. In particular circumstances, it may be of considerable importance to find a convergence-determining class that is smaller than the classes in Theorem A2.3.II. While such classes often have to be constructed to take advantage of particular features of the metric space in question, the general result below is also of value. In it, a covering semiring is a semiring with the property that every open set can be represented as a finite or countable union of sets from the semiring. If X is separable, an important example of such a semiring is obtained by first taking the open spheres Srj (dk ) with centres at the points {dk } of a countable dense set and radii {rj } forming a countable dense set in (0, 1), then forming finite intersections, and finally taking proper differences. Proposition A2.3.IV. Any covering semiring, together with the whole space X , forms a convergence-determining class. Proof. Let G be an open set so that by assumption we have G=

∞

Ci

for some Ci ∈ S,

i=1

where S is a generating semiring. Since the limit µ in (A2.3.1) is a measure, given > 0, we can choose a finite integer K such that " " ! K ! K 1 Ci ≤ 2 , i.e. µ(G) ≤ µ Ci + 12 . µ G− i=1

i=1

K

Further, since C is a semiring, i=1 Ci can be represented as a finite union of disjoint sets in C. From (A2.3.1), it therefore follows that there exists N such that, for n ≥ N , " ! K " ! K µ Ci ≤ µn Ci + 12 . i=1

Hence,

i=1

! µ(G) ≤ lim inf µn n→∞

K i=1

" Ci

+ ≤ lim inf µn (G) + . n→∞

Since is arbitrary, (iii) of Theorem A2.3.II is satisfied, and therefore µn → µ weakly. We investigate next the preservation of weak convergence under mappings from one metric space into another. Let X , Y be two metric spaces with associated Borel σ-algebras BX , BY , and f a measurable mapping from (X , BX ) into (Y, BY ) [recall that f is continuous at x if ρY f (x ), f (x) → 0 whenever ρX (x , x) → 0].

394


Proposition A2.3.V. Let (X , BX ), (Y, BY ) be metric spaces and f a measurable mapping of (X , BX ) into (Y, BY ). Suppose that µn → µ weakly on X and µ(Df ) = 0; then µn f −1 → µf −1 weakly. Proof. Let B be any Borel set in BY and x any point in the closure of f −1 (B). For any sequence of points xn ∈ f −1 (B) such that xn → x, either ¯ Arguing similarly for x ∈ Df or f (xn ) → f (x), in which case x ∈ f −1 (B). the complement, ∂{f −1 (B)} ⊆ f −1 (∂B) ∪ Df . (A2.3.2) Now suppose that µn → µ weakly on BX , and consider the image measures µn f −1 , µf −1 on BY . Let B be any continuity set for µf −1 . It follows from (A2.3.2) and the assumption of the proposition that f −1 (B) is a continuity set for µ. Hence, for all such B, (µn f −1 )(B) = µn (f −1 (B)) → µ(f −1 (B)) = (µf −1 )(B); that is, µn f −1 → µf −1 weakly.

A2.4. Compactness Criteria for Weak Convergence In this section, we call a set M of totally finite Borel measures on X relatively compact for weak convergence if every sequence of measures in M contains a weakly convergent subsequence. It is shown in Section A2.5 that weak convergence is equivalent to convergence with respect to a certain metric and that if X is a c.s.m.s., the space of all totally finite Borel measures on X is itself a c.s.m.s. with respect to this metric. We can then appeal to Proposition A1.2.II and conclude that a set of measures is compact (or relatively compact) if and only if it satisfies any of the criteria (i)–(iv) of that proposition. This section establishes the following criterion for compactness. Theorem A2.4.I (Prohorov’s Theorem). Let X be a c.s.m.s. Necessary and sufficient conditions for a set M of totally finite Borel measures on X to be relatively compact for weak convergence are (i) the total masses µ(X ) are uniformly bounded for µ ∈ M; and (ii) M is uniformly tight—namely, given > 0, there exists a compact K such that, for all µ ∈ M, µ(X − K) < .

(A2.4.1)

Proof. We first establish that the uniform tightness condition is necessary, putting it in the following alternative form. Lemma A2.4.II. A set M of measures is uniformly tight if and only if, for all > 0 and δ > 0, there exists a finite family of δ-spheres (i.e. of radius δ) S1 , . . . , SN such that N µ X − k=1 Sk ≤ (all µ ∈ M). (A2.4.2)

A2.4.

Compactness Criteria for Weak Convergence

395

Proof of Lemma. If the condition holds, we can find, for every k = 1, 2, . . . , a finite union Ak of spheres of radius 1/k such that µ(X − Ak ) ≤ /2k for all ∞ µ ∈ M. Then, the set K = k=1 Ak is totally bounded and hence compact, and for every µ ∈ M, µ(X − K) ≤

∞

µ(X − Ak ) < .

k=1

Thus, M is uniformly tight. Conversely, if M is uniformly tight and, given , we choose a compact K to satisfy (A2.4.1), then for any δ > 0, K can be covered by a finite set of δ-spheres, so (A2.4.2) holds. Returning now to the main theorem, suppose if possible that M is relatively compact but (A2.4.2) fails forsome > 0 and δ > 0. Since we assume X is ∞ separable, we can write X = k=1 Sk , where each Sk is a δ-sphere. On the other hand, for every finite n, we can find a measure µn ∈ M such that ∞ (A2.4.3a) µn X − k=1 Sk ≥ . If in fact M is relatively compact, there exists a subsequence {µnj } that converges weakly to some limit µ∗ . From (A2.4.3a), we obtain via (ii) of Theorem A2.3.II that, for all N > 0, N N µ∗ X − k=1 Sk ≥ lim supnj →∞ µnj X − k=1 Sk ≥ . N This contradicts the requirement that, because X − k=1 Sk ↓ ∅, we must have N µ∗ X − k=1 Sk → 0. Thus, the uniform tightness condition is necessary. As it is clear that no sequence {µn } with µn (X ) → ∞ can have a weakly convergent subsequence, condition (i) is necessary also. Turning to the converse, we again give a proof based on separability, although in fact the result is true without this restriction. We start by constructing a countable ring R from the open spheres with rational radii and centres in a countable dense set by taking first finite intersections and then proper differences, thus forming a semiring, and finally taking all finite disjoint unions of such differences. Now suppose that {µn : n ≥ 1} is any sequence of measures from M. We have to show that {µn } contains a weakly convergent subsequence. For any A ∈ R, condition (i) implies that {µn (A)} is a bounded sequence of real numbers and therefore contains a convergent subsequence. Using a diagonal selection argument, we can proceed to ext ract subsequences {µnj } for which the µn (A) approach a finite limit for each of the countable number of sets A ∈ R. Let us write µ∗ (A) for the limit and for brevity of notation set µnj = µj . Thus, we have µj (A) → µ∗ (A)

(all A ∈ R).

(A2.4.3b)

396


This might seem enough to set up a proof, for it is easy to see that µ∗ inherits finite additivity from the µj , and one might anticipate that the uniform tightness condition could be used to establish countable additivity. The difficulty is that we have no guarantee that the sets A ∈ R are continuity sets for µ∗ , so (A2.4.3b) cannot be relied on to give the correct value to the limit measure. To get over this difficulty, we have to develop a more elaborate argument incorporating the notion of a continuity set. For this purpose, we introduce the class C of Borel sets, which are µ∗ -regular in the following sense: given C ∈ C, we can find a sequence {An } of sets in R and an associated sequence of open sets Gn such that An ⊇ Gn ⊇ C and similarly a sequence of sets Bn ∈ R and closed sets Fn with C ⊇ Fn ⊇ Bn , the two sequences {An }, {Bn } having the property lim inf µ∗ (An ) = lim sup µ∗ (Bn ) = µ(C),

say.

(A2.4.4)

We establish the following properties of the class C. (1◦ ) C is a ring: Let C, C be any two sets in C, and consider, for example, the difference C − C . If {An }, {Gn }, {Bn }, {Fn } and {An }, {Gn }, {Bn }, {Fn } are the sequences for C and C , respectively, then An − Bn ⊇ Gn − Fn ⊇ C − C ⊇ Fn − Gn ⊇ Bn − An , with Gn − Fn open, Fn − Gn closed, and the outer sets elements of R since R is a ring. From the inclusion (An − Bn ) − (Bn − An ) ⊆ (An − Bn ) ∪ (An − Bn ), we find that µ∗ (An − Bn ) and µ∗ (Bn − An ) have common limit values, which we take to be the value of µ(C − C ). Thus, C is closed under differences, and similar arguments show that C is closed also under finite unions and intersections. (2◦ ) C is a covering ring: Let d be any element in the countable dense set used to construct R, and for rational values of r define h(r) = µ∗ Sr (d) . Then h(r) is monotonically increasing, bounded above, and can be uniquely extended to a monotonically increasing function defined for all positive values of r and continuous at all except a countable set of values of r. It is clear that if r is any continuity point of h(r), the corresponding sphere Sr (d) belongs to C. Hence, for each d, we can find a sequence of spheres S n (d) ∈ C with radii n → 0. Since any open set in X can be represented as a countable union of these spheres, C must be a covering class. (3◦ ) For every C ∈ C, µj (C) → µ(C): Indeed, with the usual notation, we have µ∗ (An ) = lim µj (An ) ≥ lim sup µj (C) ≥ lim inf µj (C) j→∞

j→∞

j→∞

≥ lim µj (Bn ) = µ∗ (Bn ). j→∞

A2.4.

Compactness Criteria for Weak Convergence

397

Since the two extreme members can be made as close as we please to µ(C), the two inner members must coincide and equal µ(C). (4◦ ) µ is finitely additive on C: This follows from (3◦ ) and the finite additivity of µj . (5◦ ) If M is uniformly tight, then µ is countably additive on C: Suppose that {Ck } is a sequence of sets from C, with Ck ↓ ∅ but µ(Ck ) ≥ α > 0. From the definition of C, we can find for each Ck a set Bk ∈ R and a closed set Fk such that Ck ⊇ Fk ⊇ Bk and µ∗ (Bk ) > µ(Ck ) − α/2k+1 . Then lim inf µj (Fk ) ≥ lim µj (Bk ) = µ∗ (Bk ) ≥ α − α/2k+1 , j→∞

j→∞

! and µ(Ck ) − lim inf j→∞

µj

" equals

Fn

n=1

! lim sup µj j→∞

k

Ck −

"

k

≤

Fn

n=1

≤

k

lim sup µj (Cn − Fn )

n=1

j→∞

k

µ(Cn ) − lim inf µj (Fn ) ≤ j→∞

n=1

hence,

! lim inf j→∞

k

µj

1 2 α;

" Fn

≥ 12 α

(all k).

n=1

If now M is uniformly tight, there exists a compact set K such that µ(X − K) < 14 α for all µ ∈ M. In particular, therefore, ! µj

k n=1

" Fn −µj

!

k n=1

" (Fn ∩K)

α < , 4

! so

lim inf µj j→∞

k n=1

" (Fn ∩K)

≥

α . 4

k But this is enough to show that, for each k, the sets n=1 Fn ∩ K are nonempty, and since (if X is complete) each is a closed subset of the compact set K, it follows from Theorem A1.2.II that their total intersection is ∞ nonempty. Since their total intersection is contained in n=1 Cn , this set is also nonempty, contradicting the assumption that Cn ↓ ∅. We can now complete the proof of the theorem without difficulty. From the countable additivity of µ on C, it follows that there is a unique extension of µ to a measure on BX . Since C is a covering class and µj (C) → µ(C) for C ∈ C, it follows from Proposition A2.3.III that µj → µ weakly or, in other words, that the original sequence µn contains a weakly convergent subsequence, as required.

398


A2.5. Metric Properties of the Space MX Denote by MX the space of all totally finite measures on BX , and consider the following candidate (the Prohorov distance) for a metric on MX , where F is a halo set as in (A2.2.2): d(µ, ν) = inf{: ≥ 0, and for all closed F ⊆ X , µ(F ) ≤ ν(F ) + and ν(F ) ≤ µ(F ) + }.

(A2.5.1)

If d(µ, ν) = 0, then µ(F ) = ν(F ) for all closed F , so µ(·) and ν(·) coincide. If d(λ, µ) = δ and d(µ, ν) = , then λ(F ) ≤ µ(F δ ) + δ ≤ µ(F δ ) + δ ≤ ν (F δ ) + δ + ≤ ν(F δ+ ) + δ + , with similar inequalities holding when λ and ν are interchanged. Thus, the triangle inequality holds for d, showing that d is indeed a metric. The main objects of this section are to show that the topology generated by this metric coincides with the topology of weak convergence and to establish various properties of MX as a metric space in its own right. We start with an extension of Theorem A2.3.II. Proposition A2.5.I. Let X be a c.s.m.s. and MX the space of all totally finite measures on BX . Then, each of the following families of sets in MX is a basis, and the topologies generated by these three bases coincide: (i) the sets {ν: d(ν, µ) < } for all > 0 and µ ∈ MX ; (ii) the sets {ν: ν(Fi ) < µ(Fi ) + for i = 1, . . . , k, |ν(X ) − µ(X )| < } for all > 0, finite families of closed sets F1 , . . . , Fk , and µ ∈ MX ; (iii) the sets {ν: ν(Gi ) > µ(Gi ) − for i = 1, . . . , k, |ν(X ) − µ(X )| < } for all > 0, finite families of open sets G1 , . . . , Gk , and µ ∈ MX . Proof. Each of the three families represents a family of neighbourhoods of a measure µ ∈ MX . To show that each family forms a basis, we need to verify that, if G, H are neighbourhoods of µ, ν in the given family, and η ∈ G ∩ H, then we can find a member J of the family such that η ∈ J ⊆ G ∩ H. Suppose, for example, that G, H are neighbourhoods of µ, ν in the family (ii) [(ii)-neighbourhoods for short], corresponding to closed sets F1 , . . . , Fn , F1 , . . . , Fm , respectively, and with respective bounds , , and that η is any measure in the intersection G ∩ H. Then we must find closed sets Ci and a bound δ, defining a (ii)-neighbourhood J of η such that, for any ρ ∈ J, ρ(Fi ) < µ(Fi ) + ρ(Fj ) < µ(Fj ) +

(i = 1, . . . , n), (j = 1, . . . , m),

and |ρ(X − µ(X )| < . For this purpose, we may take Ci = Fi , i = 1, . . . , n; Ci+j = Fj , j = 1, . . . , m, and δ = min{δ1 , . . . , δn ; δ1 , . . . , δm ; 12 , 12 ], where

A2.5.

Metric Properties of the Space MX

δi = µ(Fi ) + − η(Fi ) δj

=

µ(Fj )

+ −

η(Fj )

399

(i = 1, . . . , n), (j = 1, . . . , m).

For ρ ∈ J thus defined, we have, for i = 1, . . . , n, ρ(Fi ) < η(Fi ) + δ = η(Fi ) + µ(Fi ) + 1 − η(Fi ) = µ(Fi ) + 1 , while |ρ(X ) − µ(X )| < 1 . Thus J ⊆ G, and similarly J ⊆ H. The proof for family (iii) follows similar lines, while that for family (i) is standard. To check that the three topologies are equivalent, we show that for any µ ∈ MX , any (iii)-neighbourhood of µ contains a (ii)-neighbourhood, which in turn contains a (i)-neighbourhood, and that this in turn contains a (iii)neighbourhood. Suppose there is given then a (iii)-neighbourhood of µ, as defined in (iii) of the proposition, and construct a (ii)-neighbourhood by setting Fi = Gci , i = 1, . . . , n, and taking 12 in place of . Then, for any ν in this neighbourhood, ν(Gi ) = ν(X ) − ν(Gci ) > µ(X ) − 12 − µ(Gci ) − 12 = µ(Gi ) − . Since the condition on |µ(X ) − νX )| carries across directly, this is enough to show that ν lies within the given (iii)-neighbourhood of µ. Given next a (ii)-neighbourhood, defined as in the proposition, we can find a δ with 0 < δ < 12 for which, for i = 1, . . . , n, µ(Fiδ ) < µ(Fi ) + 12 . Consider the sphere in MX with centre µ and radius δ, using the weak-convergence metric d. For any ν in this sphere, ν(Fi ) < µ(Fiδ ) + δ < µ(Fi ) + 21 + 21 = µ(Fi ) + , while taking F = X in the defining relation for d gives ν(X ) − 12 < µ(X ) < ν(X + 21 ; thus ν also lies within the given (ii)-neighbourhood. Finally, suppose there is given a (i)-neighbourhood of µ, Sµ say, defined by the relations, holding for all closed F and given > 0, {ν: ν(F ) < µ(F ) + ; µ(F ) < ν(F ) + }. We have to construct a (iii)-neighbourhood of µ that lies within Sµ . To this end, we first use the separability of X to cover X with a countable union of spheres S1 , S2 , . . . , each of radius 13 or less, and each a continuity set for µ. Then, choose N large enough so that RN = X − ∪N 1 Si , which is also a continuity set for µ, satisfies µ(RN ) < 13 . We now define a (iii)-neighbourhood of µ by taking the finite family of sets A consisting of all finite unions of the Si , i = 1, . . . , N , all finite unions of the closures of their complements Sic , and RN , and setting Gµ = {ν : ν(A) < µ(A) + 13 , A ∈ A, |ν(X ) − µ(X )| < 13 }. Given an arbitrary closed F in X , denote by F ∗ the union of all elements of A that intersect F , so that F ∗ ∈ A and F ∗ ⊆ F ∗ ⊆ F . Then, for ν ∈ Gµ ,

400


ν(F ) ≤ ν(F ∗ ) + ν(RN ) < µ(F ∗ ) + 13 + ν(RN ) < µ(F ∗ ) + 13 + µ(RN ) + 13 < µ(F ) + . Further, µ(F ) ≤ µ(F ∗ ) + µ(RN ) < µ(F ∗ ) + 13 = µ(X ) − µ[(F ∗ )c ] + 13 . But µ(X ) < ν(X ) + 13 , and µ[(F ∗ )c ] ≥ ν[(F ∗ )c ] − 13 , so that on substituting µ(F ) < ν(X ) − ν[(F ∗ )c ] + = ν(F ∗ ) + < ν(F ) + . These inequalities show that ν ∈ Sµ and hence Gµ ⊆ Sµ . The weak convergence of µn to µ is equivalent by Theorem A2.3.II to µn → µ in each of the topologies (ii) and (iii) and hence by the proposition to d(µn , µ) → 0. The converse holds, so we have the following. Corollary A2.5.II. For µn and µ ∈ MX , µn → µ weakly if and only if d(µn , µ) → 0. If A is a continuity set for µ, then we have also µn (A) → µ(A). However, it does not appear that there is a basis, analogous to (ii) and (iii) of Proposition A2.5.I, corresponding to this form of the convergence. Having established the fact that the weak topology is a metric topology, it makes sense to ask whether MX is separable or complete with this topology. Proposition A2.5.III. If X is a c.s.m.s. and MX is given the topology of weak convergence, then MX is also a c.s.m.s. Proof. We first establish completeness by using the compactness criteria of the preceding section. Let {µn } be a Cauchy sequence in MX ; we show that it is uniformly tight. Let positive and δ be given, and choose positive η < min( 13 , 12 δ). From the Cauchy property, there is an N for which d(µn , µN ) < η for n ≥ N . Since µN itself is tight, X can be covered by a sequence of spheres S1 , S2 , . . . of radius η and there is a finite K for which K µN (X ) − µN i=1 Si < η. For n > N , since d(µn , µN ) < η, µn (X ) − µN (X ) < η so

and

µN

K i=1

η K + η, Si < µn i=1 Si

η

K < µn (X ) − µn i=1 Si & K & & + η ≤ 3η < . ≤ |µn (X ) − µN (X )| + &µN (X ) − µN i=1 Si

µn (X ) − µn

K i=1

Si

It follows that for every and δ we can find a finite family of δ spheres whose union has µn measure within of µn (X ), uniformly in n. Hence, the sequence {µn } is uniformly tight by Lemma A2.4.II and relatively compact by Theorem A2.4.I [since it is clear that the quantities µn (X ) are bounded when {µn } is

A2.5.

Metric Properties of the Space MX

401

a Cauchy sequence]. Thus, there exists a limit measure such that µn → µ weakly, which implies by Corollary A2.5.II that d(µn , µ) → 0. Separability is easier to establish, as a suitable dense set is already at hand in the form of the measures with finite support (i.e. those that are purely atomic with only a finite set of atoms). Restricting the atoms to the points of a separability set D for X and their masses to rational numbers, we obtain a countable family of measures, D say, which we now show to be dense in MX by proving that any sphere S (µ) ⊆ MX contains an element of D . To this end, first choose a compact set K such that µ(X \ K) < 12 , which is possible because µ is tight. Now cover K with a finite family of disjoint sets A1 , . . . , An , each with nonempty interior and of radius or less. [One way of constructing such a covering is as follows. First, cover K with a finite family of open spheres S1 , . . . , Sm , say, each of radius . Take A1 = S¯1 , A2 = S¯2 ∩Ac1 , A3 = S¯3 ∩ (A1 ∪ A2 )c , and so on, retaining only the nonempty sets in this construction. Then S2 ∩ Ac1 is open and either empty, in which case S2 ⊆ A1 so S¯2 ⊆ A¯1 and A2 is empty, or has nonempty interior. It is evident that each Ai has radius or less and that they are disjoint.] For each i, since Ai has nonempty interior, we can choose an element xi of the separability set for X with xi ∈ Ai , give xi rational mass µi such that µ(Ai ) ≥ µi ≥ µ(Ai ) − /(2N ),

with atoms and let µ denote a purely atomic measure at xi of mass µi . Then, denoting i:xi ∈F , for an arbitrary closed set F , with

µ (F ) = µi ≤ µ(Ai ) < µ(F ) + , where we have used the fact that i:xi ∈F Ai ⊆ F because Ai has radius at most . Furthermore,

µ(F ∩ Ai ) + 12 , µ(F ) < µ(K ∩ F ) + 12 ≤ where denotes i:Ai ∩F =∅ , so

µ(F ) ≤ µ (Ai ) + 12 + 12 < µ(F ) + . Consequently, d(µ, µ ) < , or equivalently, µ ∈ S (µ), as required. Denote the Borel σ-algebra on MX by B(MX ) so that from the results just established it is the smallest σ-algebra containing any of the three bases listed in Proposition A2.5.I. We use this fact to characterize B(MX ). Proposition A2.5.IV. Let S be a semiring generating the Borel sets BX of X . Then B(MX ) is the smallest σ-algebra of subsets of MX with respect to which the mappings ΦA : MX → R defined by ΦA (µ) = µ(A) are measurable for A ∈ S. In particular, B(MX ) is the smallest σ-algebra with respect to which the ΦA are measurable for all A ∈ BX .

402


Proof. Start by considering the class C of subsets A of X for which ΦA is B(MX )-measurable. Since ΦA∪B = ΦA + ΦB for disjoint A and B, and the sum of two measurable functions is measurable, C is closed under finite disjoint unions. Similarly, since ΦA\B = ΦA −ΦB for A ⊇ B, C is closed under proper differences and hence in particular under complementation. Finally, since a monotone sequence of measurable functions has a measurable limit, and ΦAn ↑ ΦA whenever An ↑ A, it follows that C is a monotone class. Let F be any closed set in X and y any positive number. Choose µ ∈ MX such that µ(F ) < y and set = y − µ(F ). We can then write {ν: ΦF (ν) < y} = {ν: ν(F ) < y} = {ν: ν(F ) < µ(F ) + }, showing that this set of measures is an element of the basis (ii) of Proposition A2.5.I and hence an open set in MX and therefore an element of B(MX ). Thus, C contains all closed sets, and therefore also C contains all open sets. From these properties of C, it now follows that C contains the ring of all finite disjoint unions of differences of open sets in X , and since C is a monotone class, it must contain all sets in BX . This shows that ΦA is B(MX )-measurable for all Borel sets A and hence a fortiori for all sets in any semiring S generating the Borel sets. It remains to show that B(MX ) is the smallest σ-algebra in MX with this property. Let S be given, and let R be any σ-ring with respect to which ΦA is measurable for all A ∈ S. By arguing as above, it follows that ΦA is also R-measurable for all A in the σ-ring generated by S, which by assumption is BX . Now suppose we are given > 0, a measure µ ∈ MX , and a finite family F1 , . . . , Fn of closed sets. Then, the set {ν: ν(Fi ) < µ(Fi ) + for i = 1, . . . , n and |ν(X ) − µ(X )| < } is an intersection of sets of R and hence is an element of R. But this shows that R contains a basis for the open sets of MX . Since MX is separable, every open set can be represented as a countable union of basic sets, and thus all open sets are in R. Thus, R contains B(MX ), completing the proof.

#

A2.6. Boundedly Finite Measures and the Space MX

For applications to random measures, we need to consider not only totally finite measures on BX but also σ-finite measures with the strong local finiteness condition contained in the following definition. Definition A2.6.I. A Borel measure µ on the c.s.m.s. X is boundedly finite if µ(A) < ∞ for every bounded Borel set A. We write M# X for the space of boundedly finite Borel measures on X and generally use the # notation for concepts taken over from finite to boundedly

A2.6.

Boundedly Finite Measures and the Space M# X

403

finite measures. The object of this section is to extend to M# X the results previously obtained for MX : while most of these extensions are routine, they are given here for the sake of completeness. Consider first the extension of the concept of weak convergence. Taking a fixed origin x0 ∈ X , let Sr = Sr (x0 ) for 0 < r < ∞ and introduce a distance function d# on M# X by setting d# (µ, ν) = 0

∞

e−r

dr (µ(r) , ν (r) ) dr, 1 + dr (µ(r) , ν (r) )

(A2.6.1)

where µ(r) , ν (r) are the totally finite restrictions of µ, ν to Sr and dr is the Prohorov distance between the restrictions. Examining (A2.5.1) where this distance is defined, we see that the infimum cannot decrease as r increases when the number of closed sets to be scrutinized increases, so as a function of r, dr is monotonic and thus a measurable function. Since the ratio dr /(1 + dr ) ≤ 1, the integral in (A2.6.1) is defined and finite for all µ, ν. The triangle inequality is preserved under the mapping x → x/(1 + x), while d# (µ, ν) = 0 if and only if µ and ν coincide on a sequence of spheres expanding to the whole of X , in which case they are identical. We call the metric topology generated by d# the w# -topology (‘weak hash’ topology) and write µk →w# µ for convergence with respect to this topology. Some equivalent conditions for w# -convergence are as in the next result. Proposition A2.6.II. Let {µk : k = 1, 2, . . .} and µ be measures in M# X; then the following conditions are equivalent. (i) µk →w# µ. (ii) X f (x) µk (dx) → X f (x) µ(dx) for all bounded continuous functions f (·) on X vanishing outside a bounded set. (n) (iii) There exists a sequence of spheres S (n) ↑ X such that if µk , µ(n) denote (n) the restrictions of the measures µk , µ to subsets of S (n) , then µk → µ(n) weakly as k → ∞ for n = 1, 2, . . . . (iv) µk (A) → µ(A) for all bounded A ∈ BX for which µ(∂A) = 0. Proof. We show that (i) ⇒ (iii) ⇒ (ii) ⇒ (iv) ⇒ (i). Write the integral in (A2.6.1) for the measures µn and µ as ∞ d# (µk , µ) = e−r gk (r) dr 0

so that for each k, gk (r) increases with r and is bounded above by 1. Thus, there exists a subsequence {kn } and a limit function g(·) such that gkn (r) → g(r) at all continuity points of g [this is just a version of the compactness criterion for vague convergence on R: r egard each gk (r) as the distribution function of a probability measure so that there exists a vaguely convergent subsequence; see Corollary A2.6.V or any standard proof of the Helly–Bray ∞ results]. By dominated convergence, 0 e−r g(r) dr = 0 and hence, since g(·)

404


is monotonic, g(r) = 0 for all finite r > 0. This being true for all convergent subsequences, it follows that gk (r) → 0 for such r and thus, for these r, (r)

dr (µk , µ(r) ) → 0

(k → ∞).

In particular, this is true for an increasing sequence of values rn , corresponding (r ) to spheres {Srn } ≡ {Sn }, say, on which therefore µk n → µ(rn ) weakly. Thus, (i) implies (iii). Suppose next that (iii) holds and that f is bounded, continuous, and vanishes outside some bounded set. Then, the support of f is contained in some (r) Sr , and hence f dµk → f dµ(r) , which is equivalent to (ii). When (ii) holds, the argument used to establish (iv) of Theorem A2.3.II shows that µk (C) → µ(C) whenever C is a bounded Borel set with µ(∂C) = 0. Finally, if (iv) holds and Sr is any sphere that is a continuity set for µ, (r) then by the same theorem µk → µ(r) weakly in Sr . But since µ(Sr ) increases monotonically in r, Sr is a continuity set for almost all r, so the convergence to zero of d# (µk , µ) follows from the dominated convergence theorem. Note that we cannot find a universal sequence of spheres, {S n } say, for which (i) and (ii) are equivalent because the requirement of weak convergence on S n that µk (S n ) → µ(S n ) cannot be guaranteed unless µ(∂S n ) = 0. While the distance fiunction d# of Definition A2.6.I depends on the centre x0 of the family {Sr } of spheres used there, the w# -topology does not depend on the choice of x0 . To see this, let {Sn } be any sequence of spheres expanding to X so that to any Sn we can first find n for which Sn ⊆ Srn and then find n for which Srn ⊆ Sn . Now weak convergence within a given sphere is subsumed by weak convergence in a larger sphere containing it, from which the asserted equivalence follows. It should also be noted that for locally compact X , w# -convergence coincides with vague convergence. The next theorem extends to w# -convergence the results in Propositions A2.5.III and A2.5.IV. # Theorem A2.6.III. (i) M# X with the w -topology is a c.s.m.s. # (ii) The Borel σ-algebra B(MX ) is the smallest σ-algebra with respect to which the mappings ΦA : M# X → R given by

ΦA (µ) = µ(A) are measurable for all sets A in a semiring S of bounded Borel sets generating BX and in particular for all bounded Borel sets A. Proof. To prove separability, recall first that the measures with rational masses on finite support in a separability set D for X form a separability set D for the totally finite measures on each Sn under the weak topology. Given ∞ > 0, choose R so that R e−r dr < 12 . For any µ ∈ M# X , choose an atomic measure µR from the separability set for SR such that µR has support in SR

A2.6.

Boundedly Finite Measures and the Space M# X

405

and dR (µR , µ(R) ) < 12 . Clearly, for r < R, we also have (r)

dr (µR , µ(r) ) < 12 . Substitution in the expression for d# shows that d# (µR , µ) < , establishing that the union of separability sets is a separability set for measures in M# X. To show completeness, let {µk } be a Cauchy sequence for d# . Then, each (r) sequence of restrictions {µk } forms a Cauchy sequence for dr and so has a limit νr by Proposition A2.5.III. The sequence {νr } of measures so obtained is clearly consistent in the sense that νr (A) = νs (A) for s ≤ r and Borel sets A of Sr . Then, the set function µ(A) = νr (A) is uniquely defined on Borel sets A of Sr and is nonnegative and countably additive on the restriction of MX to each Sr . We now extend the definition of µ to all Borel sets by setting µ(A) = lim νr (A ∩ Sr ), r→∞

the sequence on the right being monotonically increasing and hence having a limit (finite or infinite) for all A. It is then easily checked that µ(·) is finitely additive and continuous from below and therefore countably additive and so a boundedly finite Borel measure. Finally, it follows from (ii) of Proposition A2.6.II that µk →w# µ. To establish part (ii) of the theorem, examine the proof of Proposition A2.5.IV. Let C be the class of sets A for which ΦA is a B(M# X )-measurable mapping into [0, ∞). Again, C is a monotone class containing all bounded open and closed sets on X and hence BX as well as any ring or semiring generating BX . Also, if S is a semiring of bounded sets generating BX and ΦA is R-measurable for A ∈ S and some σ-ring R of sets on M# X , then ΦA is R-measurable for A ∈ BX . The proposition now implies that R(r) , the σ-algebra formed by projecting the measures in sets of R onto Sr , contains B(MSr ). Equivalently, R contains the inverse image of B(MSr ) under this projection. The definition of B(M# X ) implies that it is the smallest σ-algebra containing each of these inverse images. Hence, R contains B(M# X ). The final extension is of the compactness criterion of Theorem A2.4.I. Proposition A2.6.IV. A family of measures {µk } in M# X is relatively com(n) pact in the w# -topology on M# if and only if their restrictions {µα } to a X sequence of closed spheres S n ↑ X are relatively compact in the weak topology on MS n , in which case the restrictions {µF α } to any closed bounded F are relatively compact in the weak topology on MF . Proof. Suppose first that {µα } is relatively compact in the w# -topology on M# X and that F is a closed bounded subset of X . Given any sequence of the # µF α , there exists by assumption a w -convergent subsequence, µαk →w# µ say. From Proposition A2.6.II, arguing as in the proof of A2.3.II, it follows that for

406


all bounded closed sets C, lim supk→∞ µαk (C) ≤ µ(C). Hence, in particular, the values of µαk (F ) are bounded above. Moreover, the restrictions {µF αk } are uniformly tight, this property being inherited from their uniform tightness on a closed bounded sphere containing F . Therefore, the restrictions are relatively compact as measures on F , and there exists a further subsequence converging weakly on F to some limit measure, µ# say, on F . This is enough to show that the µF α themselves are relatively compact. Conversely, suppose that there exists a family of spheres Sn , closed or (n) otherwise, such that {µα } are relatively compact for each n. By diagonal (n) selection, we may choose a subsequence αk such that µαk → µ(n) weakly for every n and therefore that, if f is any bounded continuous function vanishing (n) outside a bounded set, then f dµαk → f dµ(n) . It is then easy to see (n) (n) (m) that the µα form a consistent family (i.e. µα coincides with µα on Sm for # n ≥ m) and so define a unique element µ of MX such that µαk →w# µ. The criterion for weak convergence on each Sn can be spelled out in detail from Prohorov’s Theorem A2.4.I. A particularly neat result holds in the case where X is locally (and hence countably) compact when the following terminology is standard. A Radon measure in a locally compact space is a measure taking finite values on compact sets. A sequence {µk } of such measures converges vaguely to µ if f dµk → f dµ for each continuous f vanishing outside a compact set. Now any locally compact space with a countable base is metrizable, but the space is not necessarily complete in the metric so obtained. If, however, the space is both locally compact and a c.s.m.s., it can be ◦ represented as the union of a sequence of compact sets Kn with Kn ⊆ Kn+1 , and then by changing to an equivalent metric if necessary, we can ensure that the spheres Sn are compact as well as closed (see e.g. Hocking and Young, 1961, Proposition 2.61); we assume this is so. Then, a Borel measure is a Radon measure if and only if it is boundedly finite, and vague convergence coincides with w# -convergence. The discussion around (A2.6.1) shows that the vague topology is metrizable and suggests one form for a suitable metric. Finally, Proposition A2.6.IV takes the following form. Corollary A2.6.V. If X is a locally compact c.s.m.s., then the family {µα } of Radon measures on BX is relatively compact in the vague topology if and only if the values {µα (A)} are bounded for each bounded Borel set A. Proof. Assume the metric is so chosen that closed bounded sets are compact. Then, if the µα (·) are relatively compact on each Sn , it follows from condition (i) of Theorem A2.4.I that the µα (Sn ) are bounded and hence that the µα (A) are bounded for any bounded Borel set A. Conversely, suppose the boundedness condition holds. Then, in particular, it holds for Sn , which is compact so the tightness condition (ii) of Theorem A2.4.I is satisfied trivially. Thus, the {µα } are relatively compact on each Sn and so by Proposition A2.6.IV are relatively compact in the w# - (i.e. vague) topology.

A2.7.

Measures on Topological Groups

407

A2.7. Measures on Topological Groups A group G is a set on which is defined a binary relation G × G → G with the following properties. (i) (Associative law) For all g1 , g2 , g3 ∈ G, (g1 g2 )g3 = g1 (g2 g3 ). (ii) There exists an identity element e (necessarily unique) such that for all g ∈ G, ge = eg = g. (iii) For every g ∈ G, there exists a unique inverse g −1 such that g −1 g = gg −1 = e. The group is Abelian if it also has the property (iv) (Commutative law) For all g1 , g2 ∈ G, g1 g2 = g2 g1 . A homomorphism between groups is a mapping T that preserves the group operations in the sense that (T g1 )(T g2 ) = T (g1 g2 ) and (T g1 )−1 = T g −1 . If the mapping is also one-to-one, it is an isomorphism. An automorphism is an isomorphism of the group onto itself. A subgroup H of G is a subset of G that is closed under the group operations and so forms a group in its own right. If H is nontrivial (i.e. neither {e} nor the whole of G), its action on G splits G into equivalence classes, where g1 ≡ g2 if there exists h ∈ H such that g2 = g1 h. These classes form the left cosets of G relative to H; they may also be described as the (left) quotient space G/H of G with respect to H. Similarly, H splits G into right cosets, which in general will not be the same as the left cosets. If G is Abelian, however, or more generally if H is a normal (or invariant) subgroup, which means that for every g ∈ G, h ∈ H, g −1 hg ∈ H, the right and left cosets coincide and the products of two elements, one from each of any two given cosets, fall into a uniquely defined third coset. With this definition of multiplication, the cosets then form a group in their own right, namely the quotient group. The natural map taking an element from G into the coset to which it belongs is then a homomorphism of G into G/H, of which H is the kernel; that is, the inverse image of the identity in the image space G/H. The direct product of two groups G and H, written G × H, consists of the Cartesian products of G and H with the group operation (g1 , h1 )(g2 , h2 ) = (g1 g2 , h1 h2 ), identity (eG , eH ), and inverse (g, h)−1 = (g −1 , h−1 ). In particular, if G is a group and H a normal subgroup, then G is isomorphic to the direct product H × G/H. G is a topological group if it has a topology U with respect to which the mapping (g1 , g2 ) → g1 g2−1 from G × G (with the product topology) into G is continuous. This condition makes the operations of left (and right) multiplication by a fixed element of G, and of inversion, continuous. A theory with wide applications results if the topology U is taken to be locally compact and second countable. It is then metrizable but not necessarily complete in the resulting metric. In keeping with our previous discussion, however, we frequently assume that G is a complete separable metric group (c.s.m.g.) as well

408


as being locally compact. If, as may always be done by a change of metric, the closed bounded sets of G are compact, we refer to G as a σ-group. Definition A2.7.I. A σ-group is a locally compact, complete separable metric group with the metric so chosen that closed bounded sets are compact. In this context, boundedly finite measures are Radon measures, and the concepts of weak and vague convergence coincide. A boundedly finite measure µ on the σ-group is left-invariant if (writing gA = {gx: x ∈ A}) µ(gA) = µ(A) or equivalently,

G

(g ∈ G, A ∈ BG ),

f (g −1 x) µ(dx) =

(A2.7.1)

f (x) µ(dx)

(A2.7.2)

G

for all f ∈ BC(G), the class of continuous functions vanishing outside a bounded (in this case compact) set. Right-invariance is defined similarly. A fundamental theorem for locally compact groups asserts that up to scale factors they admit unique left- and right-invariant measures, called Haar measures. If the group is Abelian, the left and right Haar measures coincide, as they do also when the group is compact, in which case the Haar measure is totally finite and is uniquely specified when normalized to have total mass unity. On the real line, or more generally on Rd , the Haar measure is just Lebesgue measure (·), and the uniqueness referred to above is effectively a restatement of results on the Cauchy functional equation. If G is a topological group and H a subgroup, the quotient topology on G/H is the largest topology on G/H making the natural map from G into G/H continuous. It is then also an open map (i.e. takes open sets into open sets). If it is closed, then the quotient topology for G/H inherits properties from the topology for G: it is Hausdorff, or compact, or locally compact if and only if G has the same property. These concepts extend to the more general context where X is a c.s.m.s. and H defines a group of one-to-one bounded continuous maps Th of X into itself such that Th1 (Th2 (x)) = Th1 h2 (x). Again we assume that H is a σ-group and that the {Th } act continuously on X , meaning that the mapping (h, x) → Th (x) is continuous from H × X into X . The action of H splits X into equivalence classes, where x1 ≡ x2 if there exists h ∈ H such that x2 = Th (x1 ). It acts transitively on X if for every x1 , x2 ∈ X there exists an h such that Th maps x1 into x2 . In this case, the equivalence relation is trivial: there exists only one equivalence class, the whole space X . In general, the equivalence classes define a quotient space Q, which may be given the quotient topology; with this topology, the natural map taking x into the equivalence class containing it is again both continuous and open. If the original topology on H is not adjusted to the group action, however, the quotient topology may not be adequate for a detailed discussion of invariant measures.

A2.7.

Measures on Topological Groups

409

Example A2.7(a). Consider R1 under the action of scale changes: x → αx (0 < α < ∞). Here H may be identified with the positive half-line (0, ∞) with multiplication as the group action. There are three equivalence classes, (−∞, 0), {0}, and (0, ∞), which we may identify with the three-point space Q = {−1, 0, 1}. The quotient topology is trivial (only ∅ and the whole of Q), whereas the natural topology for further discussion is the discrete topology on Q, making each of the three points both open and closed in Q. With this topology, the natural map is open but not continuous. It does have, however, a continuous (albeit trivial) restriction to each of the three equivalence classes and therefore defines a Borel mapping of X into Q. An important problem is to determine the structure of boundedly finite measures on X that are invariant under the group of mappings {Th }. In many cases, some or all of the equivalence classes of X under H can be identified with replicas of H so that we may expect the restriction of the invariant measures to such cosets to be proportional to Haar measure. When such an identification is possible, the following simple lemma can be used; it allows us to deal with most of the situations arising from concrete examples of invariant measures [see e.g. Bourbaki (1963) for further background]. Lemma A2.7.II (Factorization Lemma). Let X = H × Y, where H is a σ-group and Y is a c.s.m.s., and suppose that µ ∈ M# X is invariant under left multiplication by elements of H in the sense that, for A ∈ BX and B ∈ BY , µ(hA × B) = µ(A × B).

(A2.7.3)

Then µ = × κ, where is a multiple of left Haar measure on H and κ ∈ M# Y is uniquely determined up to a scalar multiple. Proof. Consider the set function µB (·) defined on BH for fixed B ∈ BY by µB (A) = µ(A × B). Then µB inherits from µ the properties of countable additivity and bounded finiteness and so defines an element of M# H . But then, from (A2.7.3), µB (hA) = µ(hA × B) = µ(A × B) = µB (A), implying that µB is invariant under left multiplication by elements of H. It therefore reduces to a multiple of left Haar measure on H, µB (A) = κ(B) = (A),

say.

Now the family of constants κ(B) may be regarded as a set function on BY , and, as for µB , this function is both countably additive and boundedly finite. Consequently, κ(·) ∈ M# Y , and it follows that µ(A × B) = µB (A) = (A)κ(B). In other words, µ reduces to the required product form on product sets, and since these generate BX , µ and the product measure × κ coincide. To apply this result to specific examples, it is often necessary to find a suitable product representation for the space on which the transformations act. The situation is formalized in the following statement.

410


Proposition A2.7.III. Let X be a c.s.m.s. acted on measurably by a group of transformations {Th : h ∈ H}, where H is a σ-group. Suppose, furthermore, that there exists a mapping ψ: H × Y → X , where Y is a c.s.m.s. and ψ is one-to-one, both ways measurable, takes bounded sets into bounded sets, and preserves the transformations {Th } in the sense that Th ψ(h, y) = ψ(h h, y)

(h ∈ H).

(A2.7.4)

Let µ be a measure on M# X that is invariant under the transformation Th . Then there exists a unique invariant measure κ ∈ M# Y such that, for BX measurable nonnegative functions f , f (x) µ(dx) = κ(dy) f ψ(h, y) (dh). (A2.7.5) X

Y

H

Proof. Let µ ˜ be the image of µ induced on H × Y by the mapping ψ; that is, µ ˜(A × B) = µ ψ(A × B) . Then, ˜(A × B) µ ˜(hA × B) = µ ψ(hA × B) = µ Th ψ(A × B) = µ ψ(A × B) = µ so that µ ˜ is invariant under the action of h ∈ H on the first argument. Moreover, if A and B are bounded sets in H and Y, respectively, then by assumption ψ(A × B) is bounded in X so that µ ˜ is boundedly finite whenever µ is boundedly finite. Lemma A2.7.II can now be applied and yields the result that µ ˜(A × B) = (A)κ(B) for some unique boundedly finite measure κ in M# Y . This relation establishes the truth of (A2.7.5) for indicator functions Iψ(A×B) (x) for A ∈ BH and B ∈ B(M# Y ). Using the usual approximation arguments, the result extends to simple functions f and thence to limits of these. It therefore holds for all nonnegative f such that f ◦ ψ is measurable on H × Y. But this is true for any f that is BX -measurable and so proves (A2.7.5). Example A2.7(b). Let µ be a measure on R2 that is invariant under rotations about the origin. These may be written Tθ for θ ∈ S, S denoting the circumference of the unit disk with addition modulo 2π. The equivalence classes consist of circles of varying radii centred on the origin, together with the isolated point {0}. The mapping (r, θ) → (r cos θ, r sin θ) takes the product space S × R+ into R2 \ {0} and is a representation of the required kind for R2 \ {0}. We therefore write µ as the sum of a point mass at the origin and a measure on R2 \ {0} that is invariant under rotations and can therefore be represented as the image of the uniform distribution around the circle and a measure κ on the positive half-line. Integration with respect to µ takes the form [see (A2.7.5)] ∞ 2π dθ f (x) µ(dx) = f (0)µ({0}) + κ(dr) f (r cos θ, r sin θ) . 2π R2 0+ 0

A2.8.

Fourier Transforms

411

A2.8. Fourier Transforms In this section, we collect together a few basic facts from classical Fourier transform theory. For brevity, most results are stated for Fourier transforms of functions on R ≡ R1 ; the corresponding results for Rd can be obtained by no more than changes in the domain of integration and appropriate bookkeeping with multiples of 2π. Both the Rd theory and the theory of Fourier series, which can be regarded as Fourier transforms of functions defined on the unit circle, are subsumed under the concluding comments concerned with Fourier transforms of functions defined on locally compact Abelian groups. We refer to texts such as Titchmarsh (1937) for more specific material on these topics. For any real- or complex-valued measurable (Lebesgue) integrable function f (·), its Fourier transform f˜(·) is defined by f˜(ω) =

∞

eiωx f (x) dx

−∞

(ω ∈ R).

(A2.8.1)

If f is real and symmetric, then so is f˜. In any case, f˜ is bounded and continuous, while the Riemann–Lebesgue lemma asserts that f (ω) → 0 as |ω| → ∞. Furthermore, if f˜ is integrable, then the inverse relation f (ω) =

1 2π

∞

eixω f˜(ω) dω

(A2.8.2)

−∞

holds. The theory is not symmetric with respect to f and f˜: for a more detailed account of the representation of a function by its inverse Fourier transform, see, for example, Titchmarsh (1937). A symmetrical theory results if we consider (real- or complex-valued) functions that are square integrable. We have the Plancherel identities for square integrable functions f and g,

∞

f (x)g(x) dx = −∞

1 2π

∞

g (ω) dω, f˜(ω)˜

(A2.8.3)

& & &f˜(ω)&2 dω.

(A2.8.4)

−∞

and, with g = f ,

& & &f (x)&2 dx = 1 2π −∞ ∞

∞

−∞

Here the Fourier transform cannot be obtained directly from (A2.8.1) but can be represented as a mean square limit

T

f˜(ω) = l.i.m. T →∞

eiωx f (x) dx, −T

(A2.8.5)

412


the existence of the finite integral following readily from the Schwarz inequality. Since the limit is defined only up to an equivalence, the theory is strictly between equivalence classes of functions—that is, elements of the Hilbert space L2 (R)—rather than a theory between individual functions. An important version for probability theory is concerned with the Fourier transforms of totally finite measures (or signed measures). If G is such a measure, its Fourier–Stieltjes transform g˜ is the bounded uniformly continuous function ∞ g˜(ω) = eiωx G(dx). (A2.8.6) −∞

If G is a probability measure, g˜(ω) is its characteristic function and g˜ is then a positive-definite function: for arbitrary finite families of real numbers ω1 , . . . , ωr and complex numbers α1 , . . . , αr , r r

αi α ¯ j g˜(ωi − ωj ) ≥ 0.

(A2.8.7)

i=1 j=1

Conversely, Bochner’s theorem asserts that any function continuous at ω = 0 and satisfying (A2.8.7) can be represented as the Fourier transform of a totally finite measure G on R with G(R) = g˜(0). If we take any real or complex integrable function f with any totally finite signed measure G and apply Fubini’s theorem to the double integral ∞ ∞ eiωx f (ω) G(dx) dω, −∞

−∞

which is certainly well defined, we obtain Parseval’s identity: ∞ ∞ ˜ f (ω)˜ g (ω) dω. f (x) G(dx) = −∞

(A2.8.8)

−∞

This identity is of basic importance because it shows that G is uniquely determined by g˜. Various more specific inversion theorems can be obtained by taking suitable choices of f followed by a passage to the limit: this approach is outlined in Feller (1966, Section XV.3), for example. In particular, the following two forms are traditional. (i) For continuity intervals (a, b) of G,

T

G((a, b)) = lim

T →∞

−T

e−iωa − e−iωb g˜(ω) dω. iω

(ii) For an atom a of G, 1 T →∞ 2T

T

G({a}) = lim

−T

e−iωa g˜(ω) dω.

A2.8.

Fourier Transforms

413

Much of the preceding theory can be extended without difficulty from R to the case of a locally compact Abelian topological group G. The characters of such a group are the continuous homomorphisms of the group into the complex numbers of modulus 1. If χ1 , χ2 are characters, then so are χ1 χ2 and χ−1 1 . Thus, the characters form a group in their own right, G say, the dual group for G. There is a natural topology on G, namely the smallest making the evaluation mapping eg (χ) ≡ χ(g) continuous for each g ∈ G, and with this topology G also is a locally compact Abelian topological group. If G = R, the characters are of the form eiωx (ω ∈ R), and G can be identified with another version of R. If G = Z, the group of integers, G is the circle group, and vice versa. In any case, the original group reappears as the dual of the dual group and if G is compact, G is discrete and conversely. G, denote Haar measure on G and G, respectively. If f : G → Now let H and H R is measurable and H-integrable, its Fourier transform f˜ is the function defined on G by ˜ f (χ) = χ(g)f (g) H(dg). (A2.8.9) G

If also f˜ is H-integrable, then the inverse relation χ(g)f˜(χ) H(dχ) f (g) = G

(A2.8.10)

is normed appropriately [otherwise, a normalizing holds, provided that H constant such as 1/(2π) in (A2.8.2) is needed]. Assuming that such a norming has been adopted, the appropriate analogues of (A2.8.4–8) remain true. In particular, we note the generalized Plancherel identity & & & & &f (g)&2 H(dg) = &f˜(χ)&2 H(dχ). (A2.8.11) G G

APPENDIX 3

Conditional Expectations, Stopping Times, and Martingales

This appendix contains mainly background material for Chapter 14. For further discussion and most proofs, we refer the reader to Ash (1972), Chung (1974), Brémaud (1981), and to various references cited in the text.

A3.1. Conditional Expectations Let (Ω, E, P) be a probability space (see Section A1.4), X a random variable (r.v.) with E|X| = Ω |X| P(dω) < ∞, and G a sub-σ-algebra of events from E. The conditional expectation of X with respect to G, written E(X | G) or EX|G (ω), is the G-measurable function (i.e. a random variable) defined up to values on a set of G of P-measure zero as the Radon–Nikodym derivative (G)

E(X | G) = EX|G (ω) = ξX (dω)/P (G) (dω),

where ξX (A) = A X(ω) P(dω) is the indefinite integral of X and the superscript (G) indicates that the set functions are to be restricted to G. The G-measurability of E(X | G) implies that X(ω) P(dω) = EX|G (ω) P(dω) (all U ∈ G), (A3.1.1) U

U

an equation, usually taken as the defining relation, that determines the conditional expectation uniquely. Extending (A3.1.1) from G-measurable indicator functions IU (ω) to more general G-measurable functions Y , we have, whenever E(|X|) and E(|XY |) exist, E(XY ) = Y (ω)X(ω) P(dω) = Y (ω)EX|G (ω) P(dω) = E[Y E(X | G)]. Ω

Ω

(A3.1.2) 414

A3.1.

Conditional Expectations

415

Now replacing Y by Y IU for U ∈ G and using (A3.1.1), there follows the factorization property of conditional expectations that for G-measurable r.v.s Y for which both E(|X|) and E(|XY |) exist, E(XY | G) = Y E(X | G)

a.s.

(A3.1.3)

Conditional expectations inherit many standard properties of ordinary expectations: k k & & αj Xj & G = αj E(Xj | G); (A3.1.4) Linearity: E j=1

j=1

Monotonicity: X ≤ Y a.s. implies E(X | G) ≤ E(Y | G) a.s.; (A3.1.5) Monotone convergence: Xn ≥ 0 and Xn ↑ Y a.s. imply that E(Xn | G) ↑ E(Y | G) a.s.; (A3.1.6) Jensen’s inequality: For convex measurable functions f : R → R for which E[|f (X)|] < ∞, f (E[X | G]) ≤ E[f (X) | G] a.s. (A3.1.7) in (A3.1.7), convexity means that f 12 (x + y) ≤ 12 [f (x) + f (y)] . If G1 and G2 are two sub-σ-algebras with G1 ⊆ G2 ⊆ E and E(|X|) < ∞ as before, the repeated conditioning theorem holds: E[E(X | G1 ) | G2 ] = E[E(X | G2 ) | G1 ] = E(X | G1 ),

(A3.1.8)

yielding as the special case when G = {∅, Ω} E[E(X | G)] = E(X).

(A3.1.9)

Two σ-algebras G and H are independent if, for all A ∈ G and B ∈ H, P(A ∩ B) = P(A)P(B). Given such G and H, if X is G-measurable and we seek E(X | H), we may expect it to reduce to yield E(X | H) = E(X).

(A3.1.10)

This is a special case of the principle of redundant conditioning: if the r.v. X is independent of H [i.e. σ(X) and H are independent σ-algebras] and G is independent of H, then E(X | G ∨ H) = E(X | G),

(A3.1.11)

reducing to (A3.1.10) for trivial G. Let X be a c.s.m.s. and X an X -valued r.v. on (Ω, E, P). Given a sub-σalgebra G of E, the conditional distribution of X given G is defined by analogy with (A3.1.1) by P(X ∈ A | G) = E(IA (X) | G)

(A ∈ BX ).

(A3.1.12)

416

APPENDIX 3. Conditional Expectations, Stopping Times, Martingales

As in Section A1.5, the question of the existence of regular conditional distributions arises. In our present context, we seek a kernel function Q(A, ω)

(A ∈ B(X ), ω ∈ Ω)

such that for fixed A, Q(A, ·) is a G-measurable function of ω [and we identify this with (A3.1.12)], while for fixed ω, we want Q(·, ω) to be a probability measure on B(X ). Introduce the set function π(·) defined initially for product sets A × U for A ∈ B(X ) and U ∈ G by IA (X(ω)) P(dω). (A3.1.13) π(A × U ) = U

Since π(·) is countably additive on such sets, it can be extended to a measure, clearly a probability, on (X × Ω, B(X ) ⊗ G). Then Proposition A1.5.III can be applied and yields the following formal statement in which we identify the kernel function Q(·, ·) sought above with P(X ∈ A | G). Proposition A3.1.I. Let X be a c.s.m.s., (Ω, E, P) a probability space, and X an X -valued r.v. defined on (Ω, E, P). If G is a sub-σ-algebra of E, then there exists a regular version of the conditional distribution PX∈·|G (ω) such that (i) PX∈·|G (ω) is a probability measure on B(X ) for each fixed ω; (ii) PX∈A|G (·) is a G-measurable function of ω for fixed A ∈ B(X ); and (iii) for each U ∈ G and A ∈ B(X ), PX∈A|G (ω) P(dω) = IA (X(ω)) P(dω). (A3.1.14) U

U

Observe that if G = E, then the conditional distribution PX∈·|G (ω) is the degenerate distribution concentrated on the point X(ω). In general, the conditional distribution represents a blurred image of this degenerate distribution, the blurring arising as a result of the incomplete information concerning X carried by the sub-σ-algebra G. The following question is of the nature of a converse to the proposition. Given (X , B(X )), (Ω, E, P) and a regular kernel Q(A, ω), can we find a refinement E ⊇ E and an E -measurable X -valued r.v. X such that Q(A, ω) coincides with PX∈A|G (ω)? If we confine ourselves to the original space, this may not necessarily be possible, but by extending Ω we can accomplish our aim. Take the probability space (Ω , E , P ) given by Ω = X × Ω, E = B(X ) ⊗ E and P = π as constructed via (A3.1.13) (identifying G there with E here), and consider the r.v. X: X × Ω → X for which X(ω ) = X(x, ω) = x. With the mapping T : Ω → Ω for which T (ω ) = T (x, ω) = ω, so that T −1 (E) is a sub-σ-algebra of E , we then have PX∈A|T −1 (E) (ω ) = Q(A, T (ω )) = Q(A, ω)

(A ∈ B(X )).

(A3.1.15)

A3.1.

Conditional Expectations

417

Often the conditioning σ-algebra G is itself generated by some real- or (more generally) c.s.m.s.-valued r.v. Y . Then E(X | G) is called the conditional expectation of X given Y and P(X ∈ A | G) the conditional distribution of X given Y , together with the suggestive notation E(X | Y ) or EX|Y (ω) and P(X ∈ A | G) or PX∈A|G (ω). Equation (A3.1.13) then implies, for any Borelmeasurable function h(·) such that the unconditional expectations exist, E[Xh(Y ) | Y ] = h(Y ) E(X | Y ).

(A3.1.16)

The terminology suggests that, although E(X | Y ) is defined as an r.v., its value should depend on ω only through Y (ω). Thus, if Y takes its values in a c.s.m.s. Y, we should look for a real-valued B(Y)-measurable function hX|Y (y) such that (A3.1.17) EX|Y (ω) = hX|Y Y (ω) a.s. That such a function exists is the assertion of the Doob representation theorem (e.g. Doob, 1953). It can be established by applying the argument around (A3.1.1) to the measures induced on B(Y) by the equations PY (B) = P(Y −1 (B)) (B ∈ B(Y)), ξX (B) = X(ω) P(dω), Y −1 (B)

and, noting that ξX PY on B(Y), by applying the Radon–Nikodym theorem. Since the product of a finite or denumerably infinite number of c.s.m.s.s can itself be regarded as a c.s.m.s., we state the theorem in the following general form. Proposition A3.1.II. Let (Ω, E, P) be a probability space, X an integrable real-valued r.v. on Ω, and G a sub-σ-algebra of E generated by a countable family of r.v.s Y = {Y1 , Y2 , . . .} taking their values in the c.s.m.s.s Y1 , Y2 , . . . respectively. Then, there exists a Borel-measurable function hX|Y (·): Y1 × Y2 × · · · → R such that EX|G (ω) = hX|Y (Y1 (ω), Y2 (ω), . . .) P-a.s.

(A3.1.18)

The proposition concerning regular conditional distributions can be transformed in a similar way, yielding a kernel PX∈A|Y (y1 , y2 , . . .), which is a probability distribution in A for each vector (y1 , y2 , . . .), a Borel-measurable function of the family (y1 , y2 , . . .) for each A, and satisfies PX∈A|G (ω) = PX∈A|Y (Y1 (ω), Y2 (ω), . . .) P-a.s. When densities exist with respect to some underlying measure µ such as Lebesgue measure on Rd , the conditional distributions have the form fX,Y (x, y1 , y2 , . . .) µ(dx) , PX∈A|Y (y1 , y2 , . . .) = A f (x, y1 , y2 , . . .) µ(dx) X X,Y where fX,Y (·) is the joint density for X, Y1 , Y2 , . . . in the product space X × Y1 × Y2 × · · ·, and a similar representation holds for the conditional expectation hX|Y (·).

418


A3.2. Convergence Concepts Most of the different notions of convergence and uniform integrability mentioned below are standard. Stable convergence is less familiar and is discussed in more detail. A sequence of r.v.s {Xn : n = 1, 2, . . .} on a common probability space (Ω, E, P) converges in probability to a limit r.v. X, also defined on (Ω, E, P), if for all > 0, (n → ∞). (A3.2.1) P{|Xn − X| > } → 0 The sequence converges almost surely to X if 1 = P{ω: Xn (ω) → X(ω) (n → ∞)} ∞ ∞ & & 1 ω: &Xm (ω) − X(ω)& < =P r r=1 n=1 m≥n ∞ ∞ & & 1 & & =P . ω: Xm (ω) − Xn (ω) < r r=1 n=1

(A3.2.2)

m≥n

Both these concepts readily generalize to the case where the r.v.s X and Xn are X -valued for some c.s.m.s. X by simply replacing the Euclidean distance |X − Y | by the metric ρ(X, Y ) for X, Y ∈ X . The a.s. convergence in (A3.2.2) implies convergence in probability; convergence in probability implies the existence of a subsequence {Xnk } that converges a.s. to the same limit. Returning to the real-valued case, for any given real p ≥ 1, {Xn } converges in the mean of order p (or in pth mean, or in Lp norm) if the pth moments exist and Xn − Xp ≡ [E(|Xn − X|p )]1/p → 0 (n → ∞), (A3.2.3) the norm here denoting the norm in the Banach space Lp (Ω, E, P) of equivalence classes of r.v.s with finite pth moments. Mean square convergence—i.e. convergence in L2 norm—has its own notation l.i.m. (Doob, 1953, p. 8) as in Section 8.4. For p = ∞, the space L∞ (Ω, E, P) consists of P-essentially bounded r.v.s X; that is, r.v.s X for which |X| ≤ M a.s. for some M < ∞; then X∞ = ess sup |X(ω)| = inf{M : |X(ω)| ≤ M a.s.}. (A3.2.4) If Xn → X in pth mean, then E(Xnp ) → E(X p ) (n → ∞). Chebyshev’s inequality, in the form for an Lp r.v. X, P{|X − a| > } ≤ −p E(|X − a|p )

( > 0, real a),

(A3.2.5)

shows that convergence in Lp norm implies convergence in probability. The converse requires the additional condition of uniform integrability. Definition A3.2.I. A family of real-valued r.v.s {Xt : t ∈ T } defined on the common probability space (Ω, E, P) is uniformly integrable if, given > 0, there exists M < ∞ such that

A3.2.

Convergence Concepts

419

|Xt |>M

|Xt (ω)| P(dω) <

(all t ∈ T ).

(A3.2.6)

Proposition A3.2.II. Let the r.v.s {Xn : n = 1, 2, . . .} and X be defined on a common probability space (Ω, E, P) and be such that Xn → X in probability. Then, a necessary and sufficient condition for the means to exist and for Xn → X in L1 norm is that the sequence {Xn } be uniformly integrable. Applied to the sequence {Xnp } and noting the inequality E(|Xn − X|p ) ≤ 2 [E(|Xn |p ) + E(|X|p )] (1 ≤ p < ∞), the proposition extends in an obvious way to convergence in Lp norm for 1 ≤ p < ∞. A weaker concept than convergence in Lp norm [i.e. strong convergence in the Banach space Lp (Ω, E, P)] is that of weak Lp convergence, namely, that if Xn and X ∈ Lp , then E(Xn Y ) → E(XY ) (n → ∞) for all Y ∈ Lq , where p−1 + q −1 = 1. Let Xn be X -valued for a c.s.m.s. X with metric ρ. Xn converges in distribution if P{Xn ∈ A} → P{X ∈ A} for all A ∈ B(X ) for which P{X ∈ ∂A} = 0. This type of convergence is not so much a constraint on the r.v.s as a constraint on the distributions they induce on B(X ): indeed, it is precisely weak convergence of their induced distributions. If Xn → X in probability (or, a fortiori, if Xn → X a.s. or in Lp norm), then from the inequalities P{Xn ∈ A} − P{X ∈ A} ≤ P {Xn ∈ A} ∩ {X ∈ A} ≤ P {Xn ∈ A} ∩ {X ∈ (A )c } + P{X ∈ A } − P{X ∈ A} p−1

≤ P{ρ(Xn , X) > } + P{X ∈ A } − P{X ∈ A}, it follows that Xn → X in distribution, also written Xn →d X. No general converse statement is possible except when X is degenerate; that is, X = a a.s. for some a ∈ X . For this exceptional case, Xn →d a means that for any positive , P{ρ(Xn , a) < } = P{Xn ∈ S (a)} → 1 (n → ∞), and this is the same as Xn → a in probability. A hybrid concept, in the sense that it depends partly on the r.v.s Xn themselves and partly on their distributions, is that of stable convergence. Definition A3.2.III. If {Xn : n = 1, 2, . . .} and X are X -valued r.v.s on (Ω, E, P) and F is a sub-σ-algebra of E, then Xn → X (F-stably) in distribution if for all U ∈ F and all A ∈ B(X ) with P{X ∈ ∂A} = 0, P({Xn ∈ A} ∩ U ) → P({X ∈ A} ∩ U )

(n → ∞).

(A3.2.7)

The hybrid nature of stable convergence is well illustrated by the facts that when F = {∅, Ω}, F-stable convergence is convergence in distribution, whereas when F ⊇ σ(X), we have a.s. convergence in probability because the regular version PX∈A|F (ω) of the conditional distribution appearing in P({X ∈ A} ∩ U ) = U PX∈A|F (ω) P(dω) can be taken as being {0, 1}-valued, and when such degenerate distributions for the limit r.v. occur, the concepts of convergence in distribution and in probability coincide, as already noted.

420


In general, stable convergence always implies weak convergence, and it may be regarded as a form of weak convergence of the conditional distributions P(Xn ∈ A | F). Just as weak convergence can be expressed in equivalent ways, so too can stable convergence as follows (see Aldous and Eagleson, 1978). Proposition A3.2.IV. Let {Xn }, X and F be as in Definition A3.2.III. Then, the conditions (i)–(iv) below are equivalent. (i) Xn → X (F-stably); that is, (A3.2.7) holds. (ii) For all F-measurable P-essentially bounded r.v.s Z and all bounded continuous h: X → R, E[Zh(Xn )] → E[Zh(X)]

(n → ∞).

(A3.2.8)

(iii) For all real-valued F-measurable r.v.s Y , the pairs (Xn , Y ) converge jointly in distribution to the pair (X, Y ). (iv) For all bounded continuous functions g: X × R → R and all real-valued F-measurable r.v.s Y , g(Xn , Y ) → g(X, Y )

(F-stably).

(A3.2.9)

If X = R , then any of (i)–(iv) is equivalent to condition (v). (v) For all real vectors t ∈ Rd and all P-essentially bounded F-measurable r.v.s Z, d

E[Z exp(it Xn )] → E[Z exp(it X)]

(n → ∞).

(A3.2.10)

Proof. Equation (A3.2.7) is the special case of (A3.2.8) with Z = IU (ω) and h(x) = IA (x) for U ∈ F and A ∈ B(X ), except that such h(·) is not in general continuous: as in the continuity theorem for weak convergence, (A3.2.8) can be extended to the case where h is bounded and Borel measurable and P{X ∈ ∂h} = 0, where ∂h is the set of discontinuities of h. When X = Rd , (A3.2.10) extends the well-known result that joint convergence of characteristic functions is equivalent to weak convergence of distributions. Note that all of (A3.2.7), (A3.2.8), and (A3.2.10) are contracted versions of the full statement of weak convergence in L1 of the conditional distributions; namely, that E(Z E[h(Xn ) | F]) → E(Z E[h(X) | F])

(n → ∞)

(A3.2.11)

for arbitrary (not necessarily F-measurable) r.v.s Z. However, (A3.2.11) can immediately be reduced to the simpler contracted forms by using the repeated conditioning theorem, which shows first that it is enough to consider the case that Z is F-measurable and second that when Z is F-measurable, the conditioning on F can be dropped. If Y is real-valued and F-measurable and in (A3.2.7) we set U = Y −1 (B) for B ∈ B(R), we obtain P{(Xn , Y ) ∈ A × B} → P{(X, Y ) ∈ A × B}, from which (iii) follows. Conversely, taking Y = IU in (iii) yields (A3.2.7).

A3.2.

Convergence Concepts

421

Finally, for any two real-valued F-measurable r.v.s Y, Z, repeated application of (iii) shows that (Xn , Y, Z) converges weakly in distribution to the triple (X, Y, Z). Applying the continuous mapping theorem (Proposition A2.2.VII) yields the result that the pairs (g(Xn , Y ), Z) converge weakly in distribution to (g(X, Y ), Z), which is equivalent to the stable convergence of g(Xn , Y ) to g(X, Y ) by (iii). Since stable convergence implies weak convergence, (iv) implies (iii). When the limit r.v. is independent of the conditioning σ-algebra F, we have a special case of some importance: (A3.2.7) and (A3.2.10) then reduce to the forms P(Xn ∈ A | U ) → P{X ∈ A} and

(P(U ) > 0)

E[Z exp(it Xn )] → E(Z) E[exp(it X)],

(A3.2.12) (A3.2.13)

respectively. In this case, the Xn are said to converge F-mixing to X. In applications, it is often the case that the left-hand sides of relations such as (A3.2.7) converge as n → ∞, but it is not immediately clear that the limit can be associated with the conditional distribution of a well-defined r.v. X. Indeed, in general there is no guarantee that such a limit r.v. will exist, but we can instead extend the probability space in such a way that on the extended space a new sequence of r.v.s can be defined with effectively the same conditional distributions as for the original r.v.s and for which there is F-stable convergence in the limit to a proper conditional distribution. Lemma A3.2.V. Suppose that for each # U ∈ F and for A $ in some covering ring generating B(X ), the sequences P({Xn ∈ A} ∩ U ) converge. Then, there exists a probability space (Ω , E , P ), a measurable mapping T : (Ω , E ) → (Ω, E), and an r.v. X defined on (Ω , E ) such that if F = T −1 F and Xn (ω ) = Xn (T ω ), then Xn → X (F -stably). Proof. Set Ω = X × Ω, and let E be the smallest σ-algebra of subsets of Ω containing both B(X ) ⊗ F and also X × E. Defining T by T (x, ω) = ω, we see that T is measurable. Also, for each A ∈ B(X ) and U ∈ F, the limit π(A × U ) = limn→∞ P({Xn ∈ A} ∩ U ) exists by assumption and defines a countably additive set function on such product sets. Similarly, we can set π(X × B) = limn→∞ P({Xn ∈ X } ∩ B) = P(B) for B ∈ E. Thus, π can be extended to a countably additive set function, P say, on E . Observe that F = T −1 F consists of all sets X × U for U ∈ F. Define also X (x, ω) = x. Then, for U = X × U ∈ F , P ({Xn ∈ A} ∩ U ) = P({Xn ∈ A} ∩ U ) → P (A × U ) = P ({X ∈ A} ∩ U ) so that Xn converges to X F-stably. Each of the conditions (i)–(v) of Proposition A3.2.IV consists of a family of sequences, involving r.v.s Xn converging in some sense, and the family of

422


the limits is identified with a family involving a limit r.v. X. It is left to the reader to verify via Lemma A3.2.V that if we are given only the convergence parts of any of these conditions, then the conditions are still equivalent, and it is possible to extend the probability space and construct a new sequence of r.v.s Xn with the same joint probability distributions as the original Xn together with a limit r.v. X such that Xn → X , F-stably, and so on. In a similar vein, there exists the following selection theorem for stable convergence. Proposition A3.2.VI. Let {Xn } be a sequence of X -valued r.v.s on (Ω, E, P) and F a sub-σ-algebra of E. If (i) either F is countably generated or F ⊇ σ(X1 , X2 , . . .), and (ii) the distributions of the {Xn } converge weakly on B(X ), then there exists an extended probability space (Ω , E , P ), elements T , F , Xn defined as in Lemma A3.2.V, a sequence {nk }, and a limit r.v. X such that {Xn k } converges to X , F-stably, as k → ∞. Proof. Suppose first that F is countably generated, and denote by R some countable ring generating F. For each U ∈ R, the measures on B(X ) defined by Qn (A; U ) = P({Xn ∈ A} ∩ U ) are uniformly tight because they are strictly dominated by the uniformly tight measures P({Xn ∈ A}). Thus, they contain a weakly convergent subsequence. Using a diagonal selection argument, the subsequence can be so chosen that convergence holds simultaneously for all U ∈ R. Therefore, we can assume that the sequence {Qnk (A; U )} converges as k → ∞ to some limit Q(A; U ) for all A that are continuity sets of this limit measure and for all U ∈ R. Given > 0 and B ∈ F, there exist U , V ∈ R such that U ⊆ B ⊆ V and P(U ) ≥ P(V ) − . Then, the two extreme terms in the chain of inequalities lim Qnk (A; U ) ≤ lim inf P({Xnj ∈ A} ∩ B)

k→∞

k→∞ j>k

≤ lim sup P({Xnj ∈ A} ∩ B) ≤ lim Qnk (A; V ) k→∞ j>k

k→∞

# $ differ by at most , so the sequence P({Xnk ∈ A} ∩ B) also converges. The construction of an extended probability space (Ω , E , P ) and a limit r.v. X now follows as in the lemma, establishing the proposition in the case where F is countably generated. To treat the case where F ⊇ σ(X1 , X2 , . . .), consider first the case where F = F0 ≡ σ(X1 , X2 , . . .). This is countably generated because X is separable and only a countable family of r.v.s is involved. Applying the selection argument and extension of the probability space, we can conclude from (A3.2.10) that E[Zh(Xn k )] → E[Zh(X )]

(any F0 -measurable Z).

(A3.2.14)

A3.3.

Processes and Stopping Times

423

Now let Z be any F -measurable r.v. (where F ⊃ F0 ). Because h(Xn k ) is F0 -measurable, we can write E[Z h(Xn k )] = E[ E(Z | F0 ) h(Xn k )], and the convergence follows from (A3.2.14) by the F0 -measurability of E(Z | F0 ). Thus, for any such Z , E[Z h(Xn k )] → E[Z h(X )], implying that Xn k → X (F0 -stably). A systematic account of the topology of stable convergence when F = E but no limit r.v. is assumed is given by Jacod and Memin (1984).

A3.3. Processes and Stopping Times This section is primarily intended as background material for Chapter 14, where the focus is on certain real-valued stochastic processes denoted {Xt (ω)} = {X(t, ω)} = {X(t)} on the positive time axis, t ∈ (0, ∞) ≡ R+ . Other time domains—finite intervals, or R, or (subsets of) the integers Z = {0, ± 1, . . .} —can be considered: it is left to the reader to supply appropriate modifications to the theory as needed. Our aim here is to give just so much of the measure-theoretic framework as we hope will make our text intelligible. For a detailed discussion of this framework, texts such as Dellacherie (1972), Dellacherie and Meyer (1978) or Elliott (1982) should be consulted. Condensed accounts of selected results such as given here are also given in Brémaud (1981), Kallianpur (1980), and Liptser and Shiryayev (1977). While a stochastic process X(t, ω) may be regarded as an indexed family of random variables on a common probability space (Ω, E, P), with index set here taken to be R+ , it is more appropriate for our purposes, as in the general theory, to regard it as a function on the product space R+ × Ω. The stochastic process X: R+ × Ω → B(R+ ) ⊗ E is measurable when this mapping is measurable; that is, for all A ∈ B(R), {(t, ω): X(t, ω) ∈ A} ∈ B(R+ ) ⊗ E,

(A3.3.1)

where the right-hand side denotes the product σ-algebra of the two σ-algebras there. As a consequence of this measurability and Fubini’s theorem, X(·, ω): R+ → R is a.s. measurable, while for measurable functions h: R → R, Y (ω) ≡

h(X(t, ω)) dt R+

is a random variable provided the integral exists. A stochastic process on R+ , if defined merely as an indexed family of r.v.s on a common probability space, is necessarily measurable if, for example, the trajectories are either a.s. continuous or a.s. monotonic and right-continuous.

424


The main topic we treat concerns the evolution of a stochastic process; that is, we observe {X(s, ω): 0 < s ≤ t} for some (unknown) ω and finite time interval (0, t]. It is then natural to consider the σ-algebra (X)

Ft

≡ σ{X(s, ω): 0 < s ≤ t}

generated by all possible such evolutions. Clearly, (X)

Fs(X) ⊆ Ft

for 0 < s < t < ∞. Of course, we may also have some foreknowledge of the process X, and this we represent by a σ-algebra F0 . Quite generally, an expanding family F = {Ft : 0 ≤ t < ∞} of sub-σ-algebras of E is called a filtration or history, and we concentrate on those histories that incorporate information on the process X. For this purpose, we want the r.v. X(t, ω) to be Ft -measurable (all t); we then say that X is F -adapted. We adopt the special notation (X)

H = {Ft

: 0 ≤ t ≤ ∞} ≡ {Ht : 0 ≤ t ≤ ∞},

(X) (X) (X) (X) where F0 = lim inf t>0 Ft = {∅, Ω} and F∞ = t>0 Ft , and call H the internal, minimal, or natural history of the process X, both of these last two names reflecting the fact that H is the smallest family of nested σ-algebras to which X is adapted. Any history of the form F = {F0 ∨ Ht : 0 ≤ t ≤ ∞} is called an intrinsic history. Suppose X is measurable and F -adapted. An apparently stronger condition to impose on X is that of progressive measurability with respect to F , meaning that for every t ∈ R+ and any A ∈ B(R), {(s, ω): 0 < s ≤ t, X(s, ω) ∈ A} ∈ B((0, t]) × Ft .

(A3.3.2)

Certainly, (A3.3.2) is more restrictive on X than (A3.3.1), and while (A3.3.2) implies (A3.3.1), the converse is not quite true. What can be shown, however, is that given any measurable F -adapted R-valued process X, we can find an F -progressively measurable process Y (that is therefore measurable and F adapted) that is a modification of X in the sense of being defined (like X) on (Ω, E, P) and satisfying P{ω: X(t, ω) = Y (t, ω)} = 1

(all t)

(A3.3.3)

(see e.g. Dellacherie and Meyer, 1978, Chapter IV, Theorems 29 and 30). The sets of the form [s, t] × U, 0 ≤ s < t, U ∈ Ft , t ≥ 0, generate a σ-algebra on R+ × Ω, which may be called the F -progressive σ-algebra. Then the requirement that the process X be F -progressively measurable may be rephrased as the requirement that X(t, ω) be measurable with respect to the F -progressive σ-algebra.

A3.3.


425

A more restrictive condition to impose on X is that it be F -predictable (the term F -previsible is also used). Call the sub-σ-algebra of B(R+ ) ⊗ E generated by product sets of the form (s, t] × U , where U ∈ Fs , t ≥ s, and 0 ≤ s < ∞, the predictable σ-algebra, denoted ΨF . (The terminology is well chosen because it reflects what can be predicted at some ‘future’ time t given the evolution of the process—as revealed by sets U ∈ Fs —up to the ‘present’ time s). Then X is F -predictable when it is ΨF -measurable; that is, for any A ∈ B(R), {(t, ω): X(t, ω) ∈ A} ∈ ΨF . The archetypal F -predictable process is left-continuous, and this is reflected in Lemma A3.3.I below, in which the left-continuous history F(−) ≡ {F 1 t− } associated with F appears: here, F0− = F0 and Ft− = lim sups 0). Then, by the same argument, {T (ω) ≤ t} = {ω: X(t, ω) − Y (ω) ≥ 0} ∈ Ft . Finally, when X is F -predictable, it is F(−) -adapted, and thus we can replace Ft by Ft− throughout. The next result shows that a process stopped at an F -stopping time T inherits some of the regularity properties of the original process. Here we use the notation X(t) (t ≤ T ), X(t ∧ T ) = X(T ) (t > T ). Proposition A3.3.IV. Let F be a history, T an F -stopping time, and X a process. Then X(t ∧ T ) is measurable, F -progressive, or F -predictable, according to whether X(t) itself is measurable, F -progressive, or F -predictable. In all these cases, if T < ∞ a.s., then X(T ) is an F∞ -measurable r.v. Proof. The product σ-algebra B(R+ ) ⊗ E is generated by sets of the form (a, ∞) × B for real finite a and B ∈ E. Since {(t, ω): (t ∧ T (ω), ω) ∈ (a, ∞) × B} = (a, ∞) × (B ∩ {T (ω) > a}) and B ∩ {T (ω) > a} ∈ E, if X is measurable, so is Y (t, ω) ≡ X(t ∧ T (ω), ω). The F -predictable σ-algebra ΨF is generated by sets of a similar product form but with B ∈ Fa . Since {T (ω) > a} ∈ Fa , (a, ∞) × (B ∩ {T (ω) > a}) is also a set generating ΨF , and thus if X is F -predictable, so is Y as before. Suppose now that X is F -progressive so that for given t in 0 < t < ∞, {X(s, ω): o < s ≤ t} is measurable as a process on (0, t] with probability space (Ω, Ft , P). Then, the first argument shows that Y (s) ≡ X(s ∧ T ) is a measurable process on this space; that is, X(t ∧ T ) is F -progressive. On the set {T < ∞}, X(t∧T ) → X(T ) as t → ∞, so when P{T < ∞} = 1, X(T ) is an r.v. as asserted. As an important corollary to this result, observe that if X is F -progressive and a.s. integrable on finite intervals, then

A3.3.


427

Y (t, ω) =

t

X(s, ω) ds 0

is F -progressive, Y (T ) is an r.v. if T < ∞ a.s., and Y (t ∧ T ) is again F progressive. We conclude this section with some remarks about the possibility of a converse to Lemma A3.3.I. In the case of a quite general history, no such result of this kind holds, as is shown by the discussion in Dellacherie and Meyer (1978), especially around Chapter IV, Section 97. On the other hand, it is shown in the same reference that when X is defined on the canonical # measure space (M# [0,∞) , B(M[0,∞) )), the two concepts of being F(−) -adapted and F -predictable can be identified, a fact exploited in the treatment by Jacobsen (1982). The situation can be illustrated further by the two indicator processes VT− (t, ω) ≡ I{T (ω) s} and A2 = A ∩ {T ≤ s}. Certainly, A1 ∈ Fs , while if u < s, ∅ ∈ Fu A1 ∩ {s ∧ T ≤ u} = A ∩ {T > s} ∩ {s ∧ T ≤ u} = A1 ∈ Fs if u ≥ s. Now Fs ⊆ Fu , so by definition of Fs∧T , we have A1 ∈ Fs∧T , and (A3.4.5) holds for A1 . On A2 , t ≥ s ≥ T so X(t ∧ T ) = X(s ∧ T ) there, and (A3.4.5) holds for A2 . By addition, we have shown (A3.4.5). Finally, we quote the form of the Doob–Meyer decomposition theorem used in Chapter 14; see e.g. Liptser and Shiryayev (1977) for proof. Theorem A3.4.IX (Doob–Meyer). Let F be a history and X(·) a bounded F -submartingale with right-continuous trajectories. Then, there exists a unique (up to equivalence) uniformly integrable F -martingale Y (·) and a unique F -predictable cumulative process A(·) such that X(t) = Y (t) + A(t).

(A3.4.6)

A3.4.

Martingales

431

For nondecreasing processes A(·) with right-continuous trajectories, it can be shown that F -predictability is equivalent to the property that for every bounded F -martingale Z(·) and positive u, u u Z(t) A(dt) = E Z(t−) A(dt) . E 0

0

Since for -adapted u cumulative process ξ and any F -martingale Z, uany F E Z(u) 0 ξ(dt) = E 0 Z(t) ξ(dt) , the property above is equivalent to u E[Z(u)A(u)] = E Z(t−) A(dt) . 0

A cumulative process with this property is referred to in many texts as a natural increasing process. The theorem can then be rephrased thus: every bounded submartingale has a unique decomposition into the sum of a uniformly integrable martingale and a natural increasing function. The relation between natural increasing and predictable processes is discussed in Dellacherie and Meyer (1980). The boundedness condition in Theorem A3.4.IX is much stronger than is really necessary, and it is a special case of Liptser and Shiryayev’s (1977) ‘Class D’ condition for supermartingales; namely, that the family {X(T )} is uniformly integrable for all F -stopping times. More general results, of which the decomposition for point processes described in Chapter 13 is in fact a special case, relax the boundedness or uniform integrability conditions but weaken the conclusion by requiring Y (·) to be only a local martingale [i.e. the stopped processes Y (· ∧ Tn ) are martingales for a suitable increasing sequence {Tn } of F -stopping times]. The Doob–Meyer theorem is often stated for supermartingales, in which case the natural increasing function should be subtracted from the martingale term, not added to it. Given an F -martingale S, it is square integrable on [0, τ ] for some τ ≤ ∞ if sup0 0 FIX !!! 24, 290 + doubly infinite sequences of positive numbers S ∞ ∞ 14 {t0 , t±1 , t±2 , . . .} with n=1 tn = n=1 t−n = ∞ U linear space; complex-valued Borel measurable functions ζ on X with |ζ| ≤ 1 52; 57 U ⊗V product topology on product space X × Y of topological spaces (X , U), (Y, V) 378 V = V(X ) [0, 1]-valued measurable functions h(·) with 1 − h(·) of bounded support in X 59 = {h ∈ V(X ): inf x h(x) > 0}, i.e., − log h ∈ BM+ (X ) 59 V0 (X ) V(X ) limits of monotone sequences from V(X ) 59 W = X × M# product space supporting Campbell measure C 269 P X

General Unless otherwise specified, A ∈ BX , k and n ∈ Z+ , t and x ∈ R, h ∈ V(X ), and z ∈ C. ˘ #

F n∗ a, g µ a.e. µ, µ-a.e. a.s., P-a.s. A(·), AF (·) A(n) A ck , c[k]

reduced measure (by factorization) 160, 183 extension of concept from totally finite to boundedly finite measure space 158, viii n-fold convolution power of measure or d.f. F 55 suffixes for atomic measure, ground process of MPP 4, 3 variation norm of (signed) measure µ 374 almost everywhere with respect to measure µ 376 almost sure, P-almost surely 376 F -compensator for ξ on R+ 358 n-fold product set A × · · · × A 130 family of sets generating B; semiring of 31, 368 bounded Borel sets generating BX kth cumulant, kth factorial cumulant, 116 of distribution {pn }

xiv c(x) = c(y, y + x)

Principal Notation

covariance density of stationary mean square 160, 69 continuous process on Rd cumulant, factorial cumulant measure 147, 69 Ck (·), C[k] (·) covariance measure of ξ 191, 69 C2 (A × B) = cov(ξ(A), ξ(B)) reduced covariance measure of stationary N or ξ 292, 238 C˘2 (·) Campbell measure, modified Campbell measure 269, 270 CP , CP! reduced Campbell measure (= Palm measure) 287, 331 C˘P (·) δ(·) Dirac delta function Dirac measure, = A δ(u − x) du = IA (x) 382, 3 δx (A) Dirichlet process 12 Dα ∆F (x) = F (x) − F (x−) jump at x in right-continuous function F 107 left- and right-hand discontinuity operators 376 ∆L , ∆R F (· ; ·) finite-dimensional (fidi) distributions 158, 26 history on R+ ; R 236, 356; 394 F; F† Φ(·) characteristic functional 15, 54 G[h] (h ∈ V) probability generating functional (p.g.fl.) of N 144, 59 p.g.fl. of cluster centre process Nc 178 Gc [·] p.g.fl. of cluster member process Nm (· | x) 178, 192 Gm [· | x] G expected information gain of stationary N on R 280, 442 304, 205 Γ(·) Bartlett spectrum for stationary ξ on Rd H(t) integrated hazard function (IHF) [Q(t) in Vol.I] 109, 361 H(P; µ) generalized entropy 277, 441 internal history of ξ on R+ ; R 236, 358; 395 H; H† IA (x) = δx (A) indicator function of element x in set A 194 I σ-field of events invariant under shift operator Su Janossy measure 124 Jn (A1 × · · · × An ) local Janossy measure 137, 73 Jn (· | A) 125, 119, 506 jn (x1 , . . . , xn ) Janossy density K compact set; generic Borel set in mark space K 371, 8 31 (·) Lebesgue measure in B(Rd ) reference measure on mark space 401 K (·) Laplace functional of ξ 161, 57 L[f ] (f ∈ BM+ (X )) p.g.fl. of Cox process directed by ξ 170 Lξ [1 − h] λ(·), λ intensity measure of N , intensity of stationary N 44, 46 conditional intensity function 231, 390 λ∗ (t, ω) complete intensity function for stationary MPP on R 394 λ† (t, κ, ω) 136 mk (·) (m[k] (·)) kth (factorial) moment density ˘2 reduced second-order moment density, measure, m ˘ 2, M of stationary N 289 mean density of ground process Ng of MPP N 198, 323 mg M (A) expectation measure E[ξ(A)] 65 kth-order moment measure E[ξ (k) (·)] 66 Mk (·) N (A) number of points in A 42 N (a, b], = N ((a, b]) number of points in half-open interval (a, b] 19, 42

Principal Notation

N (t) Nc Nm (· | x) Ng N∗ {(pn , Πn )} P = (pij )

xv

= N (0, t] = N ((0, t]) 42 cluster centre process 176 cluster member or component process 176 ground process of MPP 194, 7 support counting measure of N 4 probability measure elements for finite point process 123 matrix of one-step transition probabilities pij of discretetime Markov chain on countable state space X 96 P (z) probability generating function (p.g.f.) of 10, 115 distribution {pn } avoidance function 31, 33 P0 (A) P probability measure of N or ξ on c.s.m.s. X 158, 6 Palm distribution for stationary N or ξ on R 288 P0 (·) P0 averaged (= mean) Palm measure for stationary MPP 319 Palm measure for κ ∈ K 318 P(0,κ) (·) local Palm measure for (x, κ) ∈ X × K 318 Px,κ (·) stationary distribution for (pij ) 96 {πk } ∅, ∅(·) empty set; null measure 17; 88, 292 Q-matrix of transition rates qij for continuous-time Q = (qij ) Markov chain on countable state space X 97 ρ(x, y) metric for x, y in metric space 370 ρ(y | x) Papangelou (conditional) intensity 120, 506 nested bounded sets, Si ↑ X (i → ∞) 16 {Si } random walk, sequence of partial sums 66 {Sn } sphere of radius r, centre x, in metric space X 371, 5 Sr (x) = Sr (0) 459 Sr successive points of N on R, t−1 < 0 ≤ t0 15 {ti (N )}, {ti } intervals between points of N on R, τi = ti − ti−1 15 {τi } 382, 10 T = {Tn } = {{Ani }} dissecting system of nested partitions tiling 16 tail σ-algebra of process on Rd 208 T∞ U (A) = E[N (A)] renewal measure 67


The table below lists the identifying number of formal statements of the first edition (1988) of this book and their identification in both volumes of this second edition. 1988 edition

this edition

1988 edition

this edition

2.2.I–III 2.3.III 2.4.I–II 2.4.V–VIII

2.2.I–III 2.3.I 2.4.I–II 2.4.III–VI

3.2.I–6.V

3.2.I–6.V

4.2.I–6.V

4.2.I–6.V

5.2.I–VII 5.3.I–III 5.4.I–III 5.4.IV–VI 5.5.I 6.1.I 6.1.II and 7 6.1.III–IV and 7 6.1.V–VII and 7 6.2.I–IX 6.3.I–VI 6.3.VII–IX 6.4.I–II 6.4.III 6.4.IV–V 6.4.VI 6.4.VII–IX

5.2.I–VII 5.3.I–III 5.4.I–III 5.4.V–VII 5.5.I 9.1.I 9.1.VI 9.1.VIII–IX 9.1.XIV–XVI 9.2.I–IX 9.3.I–VI 10.1.II–IV 9.4.I–II 9.5.I 9.5.IV–V 9.4.III 9.4.VI–VIII

7.1.I–II 7.1.III 7.1.IV–VI 7.1.VII 7.1.VIII and 6 7.1.IX–X 7.1.XI 7.1.XII–XIII 7.2.I 7.2.II 7.2.III 7.2.IV 7.2.V–VIII 7.3.I–V 7.4.I–II 7.4.III 7.4.IV–V 7.4.VI 7.4.VII

9.1.II–III 9.1.VII 9.1.IV 9.1.V 9.1.VIII 9.1.XV, XII 9.2.X 6.4.I(a)–(b) 9.3.VII 9.3.VIII–IX 9.3.VIII 9.3.X 9.3.XII–XV 9.2.XI–XV 9.4.IV–V 9.5.VI 9.4.VII–VIII 9.5.II 9.4.IX

8.1.I 8.1.II 8.2.I 8.2.II 8.2.III

(6.1.13) 6.1.II, IV 6.3.I 6.3.II, (6.3.6) Ex.12.1.6

xvi

xvii


1988 edition 8.2.IV 8.3.I–III 8.4.I–VIII 8.5.I–III 9.1.I–VII 9.1.IX 9.1.X 9.1.XI Ex.9.1.3 9.2.I–VI (9.2.12) 9.2.VII 9.2.VIII 9.3.I–V 9.4.I–V 9.4.VI–IX 10.1.I–III 10.1.IV 10.1.V–VI 10.2.I–V 10.2.VI 10.2.VII–VIII 10.3.I–IX 10.3.X–XII 10.4.I–II 10.4.III 10.4.IV–VII 10.5.II–III 10.6.I–VIII 11.1.I–V 11.2.I–II 11.3.I–VIII 11.4.I–IV 11.4.V–VI 12.1.I–III 12.1.IV–VI 12.2.I 12.2.II–V

this edition (6.3.6) 6.3.III–V 10.2.I–VIII 6.2.II 11.1.II–VIII 11.1.IX 11.1.XI 12.3.X 11.1.X 11.2.I–VI 11.2.VII 10.2.IX 11.2.VIII 11.3.I–V 11.4.I–V 13.6.I–IV 12.1.I–III 12.1.VI 12.4.I–II 12.2.I–V 12.2.IV 12.2.VI–VII 12.3.I–IX 12.4.III–V 12.6.I–II 12.1.IV 12.6.III–VI 15.2.I–II 15.3.I–VIII 8.6.I–V 8.2.I–II 8.4.I–VIII 8.5.I–IV 8.5.VI–VII 13.1.I–III 13.1.V–VII 13.2.I 13.2.III–VI

1988 edition

this edition

12.2.VI–VIII 12.3.I 12.3.II–VI 12.4.I 12.4.II 12.4.III 12.4.IV–VI

13.4.I–III 13.1.IV 13.3.I–V 13.4.IV Ex.13.4.7 13.4.V 13.4.VII–IX

13.1.I–III 13.1.IV–VI 13.1.VII 13.2.I–II Ex.13.2(b) 13.2.III–IV 13.2.V 13.2.VI 13.2.VII–IX 13.3.I 13.3.II–IV 13.3.V–VIII 13.3.IX–XI 13.4.I–III 13.4.III 13.4.IV 13.4.V 13.4.VI 13.5.I–II 13.5.III 13.5.IV–V 13.5.VI 13.5.VII–IX 13.5.X 13.5.XI

7.1.I–III 7.2.I–III 7.1.IV 14.1.I–II 14.1.III 14.1.IV–V 14.1.VII 14.2.I 14.2.VII 14.2.II 14.3.I–III 14.4.I–IV 14.5.I–III 14.6.I–III 7.4.I 14.2.VIII 14.2.VII 14.2.IX pp. 394–396 14.3.IV 14.3.V–VI 14.8.I 14.8.V–VII (14.8.9) 14.8.VIII

14.2.I–V 14.2.VI–VII 14.3.I–III

15.6.I–III, V–VI 15.7.I–II 15.7.III–V

Appendix identical except for A2.1.IV A1.6.I A2.1.V–VI A2.1.IV–V

CHAPTER 8

Second-Order Properties of Stationary Point Processes

Second-order properties are extremely important in the statistical analysis of point processes, not least because of the relative ease with which they can be estimated in both spatial and temporal contexts. However, there are several shortcomings when compared with, for example, the second-order properties of classical time series. There are ambiguities in the point process context as to just which second-order aspects of the process are in view. The second-order properties of the intervals, in a point process on R, are far from equivalent to the second-order properties of the counts, as already noted in Chapter 3 and elsewhere. In this chapter, our concern is solely with random measure or counting properties, broadly interpreted. A more important difficulty, however, is that the defining property of a point process—that its realizations are integer-valued measures—is not clearly reflected in properties of the moment measures. It does imply the presence of diagonal singularities in the moment measures, but this property is shared with other random measures possessing an atomic component. Nor does there seem to exist a class of tractable point processes, analogous to Gaussian processes, whose second-order properties are coextensive with those of point processes in general. Indeed, there are still open questions concerning the class of measures that can appear as moment measures for point processes or for random measures more generally. Gibbs processes defined by point–pair interactions come close to the generality required for a Gaussian process analogue but have neither the same appeal nor the same tractability as the Gaussian processes. Other examples, such as Hawkes processes, also come close to this role without fulfilling it entirely. Ultimately, these problems are related to the nonlinearity of key features of point processes such as positivity and integer counts. Thus, the second-order theory, with its associated toolkit of linear 288

8.1.

Second-Moment and Covariance Measures

289

prediction and filtering methods, although still important, is of less general utility for point processes than for classical time series. Nevertheless, it seems worthwhile to set out systematically both the aspects of practical importance and their underpinning mathematical properties. Such a programme is the aim of the present chapter, which includes a discussion of both time-domain and frequency-domain techniques for secondorder stationary point processes and random measures. Deeper theoretical issues, such as ergodicity, the general structure of moment measures for stationary random measures, and invariance under wider classes of transformations, are taken up in Chapter 12. Spatial processes are treated briefly here, reappearing in Chapters 12 and 15. To avoid encumbering the main text with tools and arguments that are hardly used elsewhere in the book, the main technical arguments relating to the Fourier transforms of second-moment measures are placed in the final section, Section 8.6. We shall assume throughout the chapter that the basic point processes are simple. For multivariate and marked point processes, we take this to mean that the ground process is simple. As we have already remarked in Chapter 6, there is no significant loss of generality in making this assumption since the batch size in a nonsimple point process can always be treated as an additional mark and the properties of the original process derived from those for marked point processes.

8.1. Second-Moment and Covariance Measures Second-order properties of stationary processes have already made brief appearances in Section 3.5 and Proposition 6.1.I. Here we take as our starting point the second and third properties listed in Proposition 6.1.I. For the purposes of this chapter, these can be restated as follows. Proposition 8.1.I (Stationary random measure: Second-order moment structure). Let ξ be a stationary random measure on X = Rd for which the second-order moment measure exists. (a) The first-moment measure M1 (·) is a multiple of Lebesgue measure (·); i.e. M1 (dx) = m (dx) for a nonnegative constant m, the mean density. (b) The second-moment measure M2 (·) is expressible as the product of a Lebesgue component (dx) along the diagonal x = y and a reduced mea˘ 2 (du) say, along u = x − y, or in integral form, for bounded sure, M measurable functions f of bounded support, ˘ 2 (du). (8.1.1a) f (s, t) M2 (ds × dt) = f (x, x + u) (dx) M X (2)

X

X

In particular, by taking f (x, y) = IUd (x)IB (y − x), ˘ 2 (B) = E ξ(x + B) ξ(dx) . M Ud

(8.1.1b)

290

8. Second-Order Properties of Stationary Point Processes

A point process or random measure for which the first- and second-moment measures exist and satisfy (a) and (b) of Proposition 8.1.I will be referred to as being second-order stationary. We should note, however, that a point process for which the first- and second-order moments satisfy the stationarity assumptions above is not necessarily stationary: nonstationary processes can have stationary first and second moments (see Exercises 8.1.1 and 8.1.2). We retain the accent ˘ to denote reduced measures formed by dropping one component from the moment measures of stationary processes as a con˘ [2] (·), C˘2 (·), and sequence of a factorization of the form (8.1.1). Thus, M ˘ C[2] stand, respectively, for the reduced forms of the second factorial moment measure, covariance measure, and factorial covariance measure. A proof of such factorization can be based on the observation that, under stationarity, M2 (dx, d(x+u)) is independent of x and so should have the form (dx)×Q(du) for some measure Q(·); see Chapter 12 and Proposition A2.7.III for details and background. Our principal aim in this section is to study the properties of these reduced measures and the relations between their properties and those of the point processes or random measures from which they derive. We start with a dis˘ 2 , which is arguably the most fundamental if not always the most cussion of M convenient of the various forms. ˘ 2 (·) be the reduced second-moment measure of Proposition 8.1.II. Let M a nonzero, second-order stationary point process or random measure ξ on Rd ˘ 2 is with mean density m. Then M ˘ ˘ (i) symmetric: M2 (A) = M2 (−A) ; ˘ 2 (A) ≥ 0, with strict inequality at least when 0 ∈ A and (ii) positive: M either ξ has an atomic component or A is an open set; (iii) positive-definite: for all bounded measurable functions ψ of bounded support, ˘ 2 (dx) ≥ 0 , (ψ ∗ ψ ∗ )(x) M (8.1.2) Rd

where ψ ∗ φ(x) =

Rd

ψ(y)φ(x − y) dy,

ψ ∗ (x) = ψ(−x);

(iv) translation-bounded: for every bounded Borel set A in Rd , there exists a finite constant KA such that ˘ 2 (x + A) ≤ KA (all x ∈ Rd ). (8.1.3) M If also ξ is ergodic and the bounded convex Borel set A increases in such a way that r(A) = sup{r: A ⊇ Sr (0)} → ∞, where Sr (0) denotes the ball in Rd of radius r and centre at 0, then in this limit, for all bounded Borel sets B, ˘ 2 (A) / (A) → m2 (8.1.4) M and 1 (A)

˘ 2 (B) ξ(x + B) ξ(dx) → M A

a.s.

(8.1.5)

8.1.


291

Proof. Symmetry follows from the symmetry of M2 so that, in shorthand form, ˘ 2 (du) (dx) = M2 dx × d(x + u) M = M2 d(x + u) × dx ˘ 2 (−du) (dy), = M2 dy × d(y − u) = M ˘2 (A) follows directly from (8.1.1b). which establishes (i). Nonnegativity of M Positivity for A 0 when ξ has an atomic component follows from Proposition 8.1.IV below, while for the other case, since A is open so that A ⊇ S2 (0) for some sphere of radius 2 > 0, we can choose < 12 and then ˘ 2 S2 (0) = E ˘2 A ≥ M ξ x + S2 (0) ξ(dx) M Ud ≥E ξ S2 (x) ξ(dx) since Ud ⊃ S (0), S (0)

≥E

ξ S (0) ξ(dx)

since S2 (x) ⊃ S (0) for x ∈ S (0),

S (0)

2 = M2 S (0) × S (0) ≥ m S (0) > 0

since > 0.

Positive-definiteness is a consequence of &2 & & & ˘ 2 (du)ψ(x)ψ(x + u) (dx) 0 ≤ E && ψ(x) ξ(dx)&& = M X X X ˘ 2 (du) = ψ ∗ (u − w)ψ(w) (dw). M X

X

˘ 2 is a positive, positive-definite Properties (ii) and (iii) together show that M (p.p.d.) measure; (iv) is then a consequence of general properties of p.p.d. measures, as set out in Section 8.6. The final two assertions follow from the ergodic theorems developed in Chapter 11. In particular, a simple form of ergodic theorem for point processes and random measures ξ on Rd asserts that, for sets A satisfying the conditions outlined in (v), as r(A) → ∞, ξ(A)/(A) → m a.s. and in L1 -norm. If second & &2 moments exist, then also E&ξ(A)/(A)−m& → 0. From these results, it is easy to show that provided both r(A) and r(B) → ∞, M2 (A × A)/[(A)]2 → m2 and, more generally, M2 (A×B)/[(A)(B)] → m2 . Approximating further, we find that M2 (U )/( × )(U ) → m2 for a wide class of sets U ∈ X (2) including cylinder sets such as U (A, r) = {(x, u): x ∈ A, y ∈ x + Sr (0)}. But ˘ 2 (A), ˘ 2 (dv) = Sr (0) M M2 (ds × dt) = (du) M U (A,r)

Sr (0)

A

and so (8.1.4) follows after dividing by ( × ) U (A, r) = (A) Sr (0) . Equation (8.1.5) can be established by similar arguments and is a simple special case of the higher-order ergodic theorems described in Chapter 11.

292


Most of the results above transfer directly or with minor modifications to the other reduced second-order measures. The most important of these is the reduced covariance measure, which can be defined here through the relation ˘ 2 (du) − m2 (du). C˘2 (du) = M

(8.1.6)

The covariance measure itself can be regarded as the second-moment measure of the mean-corrected random signed measure ˜ ξ(A) ≡ ξ(A) − m (A);

(8.1.7)

note that ξ˜ is a.s. of bounded variation on bounded sets. The reduced form ˘ 2 (·). inherits the following properties from M Corollary 8.1.III. The reduced covariance measure C˘2 (·) of a second-order stationary random measure ξ is symmetric, positive-definite, and translationbounded but in general is a signed measure rather than a measure. If ξ is ergodic, then for A, B and r(A) → ∞ as for (8.1.5), and ξ˜ in (8.1.7),

1 (A)

C˘2 (A)/(A) → 0,

˜ + B) ξ(dx) ˜ ξ(x → C˘2 (B) = E

˜ + B) ξ(dx) ˜ ξ(x .

(8.1.8) (8.1.9)

Ud

A

For point processes, a characteristic feature of the reduced forms of both the moment and covariance measures is the atom at the origin. For a simple point process, this is removed by transferring to the corresponding reduced ˘ [2] (·) and C˘[2] (·). This is not the case, however, for more factorial measures M general point processes and random measures. The situation is summarized in the proposition below and its corollary (see also Kallenberg, 1983, Chapter 2). Proposition 8.1.IV. Let ξ be a stationary second-order random measure or point process on Rd with mean density m and reduced covariance measure C˘2 . Then C˘2 (du) has a positive atom at u = 0 if and only if ξ has a nontrivial ˘ 2 ({0}) and both equal atomic component, in which case C˘2 ({0}) = M + *

2 E (8.1.10) ξ({x}) ξ(dx) = E [ξ({xi })] . Ud

i:xi ∈Ud

Moreover, there exists a σ-finite measure µ(·) on R+ such that (i) µ has finite mass outside any neighbourhood of the origin, and for every b > 0, the atoms of ξ with mass greater than b can be represented as a stationary marked point process on X × R+ with ground rate µ(b, ∞) and stationary mark distribution Πb (dκ) = µ(dκ]/µ(b, ∞) on κ > b; (ii) µ(·) integrates κ on R+ , and R+ κ µ(dκ) ≤ m; (iii) ξ is purely atomic a.s. if and only if m = R+ κ µ(dκ); and ˘ 2 ({0}) = C˘2 ({0}). (iv) µ(·) integrates κ2 on R+ , and κ2 µ(dκ) = M R+

8.1.


293

Proof. Choose any monotonically decreasing sequence of nonempty sets An with diam An ↓ 0 and An ↓ {0}. Then, for any x ∈ X , ξ(x + An ) ↓ ξ({x}) a.s. From (8.1.1b) and monotone convergence, we obtain ˘ 2 (An ) = E ξ(x + An ) ξ(dx) ↓ E ξ({x}) ξ(dx) M d Ud *U +

2 = E ξ({xi }) . xi ∈Ud

˘ 2 and C˘2 are In particular, if ξ is a.s. continuous, it follows that both M continuous at the origin, and conversely. Suppose next that b > 0 is given, and consider the atoms from ξ with masses ξ({x}) > b. If ξ is second-order stationary, there can be at most a finite number of such atoms in any finite interval. The set of such atoms is therefore denumerable and can be represented as an ordered sequence of pairs {(xi , κi )}, where xi < xj for −∞ < i < j < ∞ and b < κi = ξ({xi }). As in Section 6.4, equation (6.4.6), the set of pairs therefore constitutes a marked point process, which we denote by ξb (·). Let mgb and Πb (·) denote, respectively, the mean density of the ground process for ξb and its stationary mark distribution. Consistency of the ergodic limits requires that for b < b and B ⊆ (b, ∞), mgb Πb (B) = mgb Πb (B) ≡ µ(B).

(8.1.11)

This relation therefore defines µ consistently and uniquely as a σ-finite measure on all of R+ . Taking B = (b, ∞) in (8.1.11) then implies that µ(b, ∞) = mgb < ∞, establishing (i). Moreover, the mean density of ξb , mb say, is given by ∞

mb = mgb

∞

κ Πb (dκ) = b

∞

κ µ(dκ) =

κI{κ>b} µ(dκ) . 0

b

Since mb ≤ m < ∞ and for any A, ξb (A) ↑ ξa (A) as b → 0, whereξa denotes the atomic component of ξ, we must have mb = E ξb (Ud ) ↑ E ξa (Ud ) ≡ ma ≤ E(ξ(Ud )) ≡ m as b → 0. Hence, ∞ ∞ ma = lim κI{κ>b} µ(dκ) = κ µ(dκ), b→0

0

0

establishing (ii). Assertion (iii) is the same as the diffuse measure ξ − ξa having zero mean, implying that it is a.s. null. Finally, for any b > 0, consideration of the second moment of ξb yields the equations + * ∞ ∞

g 2 2 2 mb κ Πb (dκ) = κ µ(dκ) = E [ξ({xi })] . b

b

xi ∈Ud : ξ({xi })>b

˘ 2 ({0}) < ∞ and converges Since the right-hand side is bounded above by M ˘ to M2 ({0}) as b → 0, (iv) follows.

294


Condition (iii) above identifies purely atomic stationary random measures (see also Kallenberg, 1983). We would like to be able to use some property of µ to identify point processes (i.e. integer-valued random measures) and then simple point processes. The former identification is tantamount to a version of the moment problem: when do the moments of a measure [here µ(·)] suffice to identify the measure? This has no easy solution for our present purposes. The latter is much simpler. Corollary 8.1.V. A second-order stationary point process N with density ˘ 2 ({0}) = m, which m is a simple point process if and only if C˘2 ({0}) = M is equivalent to the reduced second-order factorial moment and covariance measures having no atom at the origin. Proof. A stationary random measure ξ is a simple point process if and only if it is integer-valued and all its atoms have mass 1. The latter condition ∞ ∞ is satisfied if and only if 1 κ µ(dκ) = 1 κ2 µ(dκ); i.e. µ has all its mass ˘ 2 ({0}). The equivalent form of the latter on {1}, or equivalently, m = M ˘ [2] ({0}) = M ˘ 2 ({0}) − m. condition follows from the relation M Analytical derivations of the relations for κr µ(dκ) for positive integers r and stationary point processes have been given in Propositions 3.3.VIII and 3.3.IX. In Chapter 12, there is an analogue of Corollary 8.1.V for a higherorder reduced factorial measure of a stationary point process to vanish at {0} as a condition for the process to have a bounded batch-size distribution or equivalently the factorial moment of the same order of µ(·) to vanish. Returning to more general properties, results such as (8.1.4) and (8.1.8) can be rephrased in further equivalent ways. When X = R, for example, they reduce respectively to E[ξ 2 (0, x)] ∼ m2 x2 ,

(x → ∞),

var ξ(0, x) = o(x2 )

results already discussed for ergodic point processes in Section 3.4. Other useful results follow as special cases of the general representations (8.1.1). These imply, for example, that

h(y) ξ(dy) =

cov

g(x) ξ(dx), Rd

Rd

Rd

C˘2 (du)

g(x)h(x + u) (dx). Rd

(8.1.12) In particular, (8.1.12) leads to the following expressions for the variance: V (A) ≡ var ξ(A) =

IA (x)IA (x + u) (dx) C˘2 (du) IA (x) (dx) IA−x (u) C˘2 (du)

Rd Rd

=

R

d

Rd

C˘2 (A − x) (dx).

= A

(8.1.13a)

8.1.


295

When X = R and A = (0, x], this becomes x (x − |u|) C˘2 (du) = 2 V (x) ≡ var ξ(0, x] = −x

x

Fc (u) du,

(8.1.13b)

0−

where for x > 0, Fc (x) = 12 C˘2 ({0}) + C˘2 (0, x] = 12 C˘2 [−x, x] is a symmetrized form of the distribution function corresponding to the reduced covariance measure. Properties of V (x) can be read off rather simply from this last representation: for example, it is absolutely continuous with a density function of which there exists a version that is continuous except perhaps for a countable number of finite discontinuities. Further details and an alternative approach in the point process case are outlined in Exercise 8.1.3. Note that, when it exists, the covariance density is a second derivative in (0, ∞) of V (x). See Exercise 8.1.4 for an analogue of (8.1.13b) in the case of a stationary isotropic point process in R2 . The variance function V (A) is widely used in applications, often in the form of the ratio to the expected value M (A); for a simple point process, this is just C˘ (A − x) (dx) C˘[2] (A − x) (dx) V (A) A 2 =1+ A . (8.1.14) = M (A) M (A) m(A) This ratio equals 1 for a Poisson process, while values larger than 1 indicate clustering and values less than 1 indicate repulsion or some tendency to regular spacing. For suitably small sets, for which diam A → 0, V (A)/M (A) → 1; that is, locally the process is like a Poisson process in having the variance-tomean ratio ≈ 1 (see Exercise 8.1.5). As (A) → ∞, various possibilities for the behaviour of V (A)/M (A) exist and are realizable (see Exercise 8.1.6), but most commonly, the covariance measure is totally finite, in which case V (A)/M (A) → 1 + m−1 C˘[2] (X )

(A ↑ X ).

A stationary random measure is of bounded variability if V (A) itself remains bounded as (A) → ∞ as for (8.1.5) [see Exercises 7.2.10(a) and 8.1.6]. [This terminology is preferred to controlled variability (Cox and Isham, 1980, p. 94).] Example 8.1(a) Stationary Poisson cluster processes. For a stationary Poisson cluster process and all values of the cluster centre x, monotone convergence shows that the cluster member process satisfies M[2] (An × An | x) → E[Z(Z − 1)] as (An ) → ∞ through a convex averaging sequence {An }, where Z ≡ Nm (X | 0) denotes a generic r.v. for the total number of points in a cluster. Then, since (6.3.12) for large A gives C[2] (A × A) ∼ E[Z(Z − 1)] M c (A), we have C˘[2] (X ) = E[Z(Z − 1)] and thus V (A)/M (A) → 1 + E[Z(Z − 1)]/EZ = EZ 2 /EZ.

(8.1.15)

296


Characteristically, therefore, the variance-to-mean ratio for a Poisson cluster process increases from a value approximately equal to 1 for very small sets to a limiting value equal to the ratio of the mean square cluster size to the mean cluster size for very large sets [see the formula for the compound Poisson process in Exercise 2.1.8(b)]. The region of rapid growth of the ratio occurs as A passes through sets with dimensions comparable to those of (the spread of) individual clusters. These comments provide the background to diagnostic procedures such as plotting the ratio V (A)/M (A) against M (A) or (A) as (A) → ∞ and to the Greig-Smith method of nested quadrats, which uses a components-of-variance analysis to determine the characteristic dimensions at which clustering effects or local inhomogeneities begin to influence the variance [see Greig-Smith (1964) for further discussion]. The representation (8.1.1b) has important interpretations when ξ is a point process rather than a general random measure, and for the discussion in this section we assume that the process is orderly. In particular, it follows in this case that ˘2 (A) = E #{point-pairs (xi , xj ): xi ∈ Ud and xj ∈ x1 + A} M (8.1.16a) = E rate of occurrence of point-pairs (xi , xj ): xj − xi ∈ A . (8.1.16b) Dividing by the mean density (= intensity = average rate of occurrence) m ˘ 2 in terms of the expectation measure of the yields an interpretation of M Palm process (see Section 3.4 and the discussion in Chapter 13) obtained by conditioning on the presence of a point at the origin: ˘ 2 (A) / m. E #points xi ∈ A | point at x = 0 = M

(8.1.17)

It is even more useful to have density versions of (8.1.17), assuming (as we ˘ [2] (A) = ˘ [2] is absolutely continuous, so M m[2] (x) dx. This now do) that M A density is related to the corresponding covariance density by m ˘ [2] (x) = c˘[2] (x) + m2 .

(8.1.18)

When the density exists, the ratio m ˘ [2] /m has been called the intensity of the process (e.g. Cox and Lewis, 1966, p. 69) or the conditional intensity function (e.g. Cox and Isham, 1980, Section 2.5). We call it the second-order intensity ˘ 2 (·) so that and denote it by h ˘ 2 (x) = m h ˘ [2] (x)/m = m + c˘[2] (x)/m. ˘ 2 (x) can also be interpreted as the intensity at x of the process conditional h on a point at the origin; this is an interpretation taken up further in the discussion of Palm measures in Chapter 13. Notice that, in d = 1, we have for

8.1.


297

a renewal process as in Chapter 4 with renewal function U (x) (x > 0) that is ˘ 2 (x) = h ˘ 2 (−x) = U (|x|). We call the ratio absolutely continuous, h r2 (x) ≡

˘ 2 (x) m ˘ [2] (x) h = m m2

(8.1.19)

the relative second-order intensity [but note that in Vere-Jones (1978a) it is called the relative conditional intensity]. It equals 1 for a stationary Poisson process, while for other stationary processes it provides a useful indication of the strength and character of second-order dependence effects between pairs of points at different separations x ∈ Rd : for example, when r2 (x) > 1, pointpairs separated by the vector x are more common than in the purely random (Poisson) case, while if r2 (x) < 1 such point-pairs are less common. ˘ 2 (A) and related functions, spheres In considering the reduced measures M Sr (0) constitute a natural class of sets to use for A in dimension d ≥ 2; define ˘ 2 (Sr (0) \ {0}) = M ˘ [2] (Sr (0)), ˘ 2 (r) = M K

(8.1.20)

the equivalent formulation here being a consequence of orderliness. Ripley (1976, 1977) introduced this function, though what is now commonly called Ripley’s K-function (including Ripley, 1981) is the density-free version K(r) =

˘ 2 (Sr (0) \ {0}) ˘ 2 (r) M K = , 2 m m2

(8.1.21)

so, since λ = m because of orderliness, λK(r) = E(# of points within r of the origin | point at the origin), (8.1.22) where on the right-hand side the origin itself is excluded from the count. The function K(r) is monotonically nondecreasing on its range of definition r > 0 and converges to 0 as r → 0. As can be seen from the examples below and is discussed further in Chapter 12, this function is particularly useful in studying stationary isotropic point processes because it then provides a succinct summary of the second-order properties of the process. For a Poisson process, K(r) = (Sr (0)). Recall the definition of K(r) in terms of the sphere Sr (0). Noting the ˘ 2 (r) = K (r) interpretation in (8.1.22), we see that the derivative (d/dr)K gives the conditional probability of a point on the surface of a spherical shell of radius r, conditional on a point at the centre of the shell. Consequently, for an isotropic process in R2 , the probability density that a point is located at distance r from a given point of the process and in the direction θ equals K (r)/(2πr), independent of θ because of isotropy. In dimension d ≥ 3, the same equality holds on replacing the denominator 2πr by the surface area of Sr (0).

298


For stationary isotropic processes in R2 , the relative second-order intensity r2 (x), which → 1 as x → 0 when it is continuous there, is a function of |x| alone, and ρ(r) = r2 (x) − 1, where r = |x|, has been called the radial correlation function (see e.g. Glass and Tobler, 1971), though it may lack the positive-definiteness property of a true correlation function. The same quantity can be introduced, irrespective of isotropy, as a derivative of Ripley’s K-function K(r) in (8.1.21): write dK(r) K (r) ρ(r) = − 1 = − 1. (8.1.23) d(πr2 ) 2πr Examples of the use of m ˘ [2] (·) and ρ(r) are given in Vere-Jones (1978a), Chong (1981) and Ohser and Stoyan (1981), amongst many other references. Example 8.1(b) A two-dimensional Neyman–Scott process. By using the general results of Example 6.3(a), it can be shown that the reduced second factorial cumulant measure is given by ˘ F (u + A) F (du) = µc m[2] G(A), C[2] (A) = µc m[2] R2

where F is the probability distribution for the location of a cluster member about the cluster centre, G is the probability distribution for the difference of two i.i.d. random vectors with distribution F , µc is the Poisson density of cluster centres, and m[2] is the second factorial moment of the number of cluster members. For the K-function, we find K(r) = πr2 + [m[2] /(µc m21 )]G1 (r), where G1 (r) is the d.f. for the distance between two ‘offspring’ from the same ‘parent’, while ρ(r) = [m[2] /(µc m21 )]g1 (r), where g1 (r) = G1 (r) is the probability density function for the distance between two offspring from the same parent. Note that ρ is everywhere positive, an indication of overdispersion or clustering relative to the Poisson process, at all distances from an arbitrarily chosen point of the process. Some particular results for the case where F is a bivariate normal distribution are given in Exercise 8.1.7. Example 8.1(c) Matérn’s Model I for underdispersion (Matérn, 1960). Let {xn } denote a realization of a stationary Poisson process N on the line with intensity λ. Identify the subset {xn } of those points of the realization that are within a distance R of another such point, i.e. # $ {xn } = x ∈ {xn }: |x − y| < R for some y ∈ {xn } with y = x , and let {xn }\{xn } ≡ {xn } constitute a realization of a new point process N (note that N = {xn } is defined without using any Poisson properties of N ). The probability that any given point x of N will be absent from N is then the probability, 1 − e−2λR , that at least one further point of N is within a distance R of x. While these events are not mutually independent, they have

8.1.


299

the same probability, so the mean density m for the modified process equals m = λe−2λR ≤ e−1 /(2R)

for all λ;

the inequality is trict except for λR = 12 . To find the second-order properties of N , consider the probability q(v) that for a given pair of points distance v apart in N , both are also in N . Then ⎧ (0 < v ≤ R), ⎪ ⎨0 exp − λ 2R + v) (R < v ≤ 2R), q(v) = ⎪ ⎩ exp − 4λR (v > 2R). The factorial moment density of N is thus m ˘ [2] (x) = λ2 q(x), and the relative second-order intensity [see (8.1.19)] is given by r2 (x) =

0 eλ(2R−x)+

(0 < x ≤ R), (x > R).

Thus, the process shows complete inhibition (as for any hard-core model) up to distance R and then a region of overdispersion for distances between R and 2R before settling down to Poisson-type behaviour for distances beyond 2R. The process is in fact of renewal type: the results above and others can be deduced from the renewal function for the process [see Exercise 8.1.9(a) for further details]. The model can readily be extended to point processes in the plane or space, but the analogues of the explicit expressions above become more cumbersome as the expression for the area or volume of the common intersection of circles or spheres becomes more complex (see Exercise 8.1.8). The set of rejected points {xn } is ‘clustered’ in the sense that every point has a nearest neighbour within a distance R [see Exercise 8.1.9(c)]. We conclude this section with some notes on possible estimates for reduced moment measures, being guided by the interpretations of the model-defined quantities and their interpretation described above. Assume, as is usually the case, that we observe only a finite part of a single realization of an ergodic process. Let B denote a suitable test set, such as an interval on the line or a rectangle or disk in the plane, and A a (larger) observation region. Then, replacing Ud by A in the right-hand side of (8.1.1b) and allowing for the change to the second factorial moment, we obtain * +

1 ∗ ˘ [2] (B) = E N (xi + B) , (8.1.24) M (A) i: xi ∈A

where N ∗ (x + B) = N (x + B) − δ0 (B), so that N (x + B) is reduced by 1 when B contains the origin.

300


The corresponding na¨ıve estimate is obtained by dropping the expectation sign in the expression above (i.e. by taking each point xi in A in turn as origin, counting the number of points in sets xi + B having a common relative position to xi but ignoring xi itself if it happens to lie within the test region, and then dividing by the Lebesgue measure of the observation region); we denote it by

, [2] (B; A) = 1 N ∗ (xi + B). (8.1.25) M (A) i:xi ∈A

Note that in the case of a process with multiple points, the points at each (1) (n ) xi should be labelled xi , . . . , xi i , and the definition of N ∗ implies that we (j) (j) (j) (k) omit pairs (xi , xi ) but not any pair (xi , xi ) with j = k. In principle, (8.1.1b) implies that this estimate is unbiased, while the assumed ergodicity of the process and the first assertion of (8.1.5) imply that it is consistent. In practice, however, difficulties arise with edge effects since N ∗ (xi + B) may not be observable if xi lies near the boundary of B. Replacing it by N ∗ [(xi + B) ∩ A] introduces a bias that may be corrected in a variety of ways. For example, we may subtract an explicit correction factor [see Exercise 8.1.11(b)], or we may take observations over an extended region A + B (plus sampling), thereby ensuring that all necessary information is available but at the expense of the fullest use of the data. One commonly used correction replaces (8.1.25) by the form

c M[2] (B; A)

N (A)(B) = (A)

N ∗ [A ∩ (xi + B)] xi ∈A xi ∈A [A ∩ (xi + B)]

(8.1.26)

so that each observation count N ∗ (xi + B) is given a relative weight equal to that fraction of (xi + B) that remains inside A; see also Exercise 8.1.10(a). Estimates of the reduced covariance measure, and hence of the variance function, can be obtained by subtracting appropriate multiples of (B) as noted in Exercise 8.1.11(c). These comments are included to suggest a basis for the systematic treatment of moment estimation for point processes; Krickeberg (1980) and Jolivet (1978) discuss some further issues and special problems, while applications are discussed by Ripley (1976, 1981), Diggle (1983), Vere-Jones (1978a), and many others.

Exercises and Complements to Section 8.1 8.1.1 Consider a nonstationary Poisson cluster process on R with cluster centres having intensity µc (t) and a cluster with centre t having either a single point at t with probability p1 (t) or two points, one at t and the other at t + X, where the r.v. X has d.f. F . Show that p1 (·) and µc (·) can be chosen so that the process is first-order stationary but not second-order stationary.

8.1.


301

8.1.2 Construct an example of a point process that has stationary covariance measure but nonstationary expectation measure. [Hint: Such a process is necessarily not simple: consider a compound Poisson process in which the rate of occurrence of groups and mean square group size are adjusted suitably.] 8.1.3 Let V (x) = var(N (0, x]) denote the variance function of a second-order stationary point process N (·) on the line, and write M2 (x) = E([N (0, x]]2 ) = V (x) + (mx)2 , where m = EN (0, 1]. (a) Show that M2 (x) is superadditive in x > 0 and hence that V (0+) ≡ limx↓0 V (x)/x exists, with V (0+) ≥ m. (b) Show that (M2 (x))1/2 is subadditive and hence that limx→∞ V (x)/x2 exists and is finite. (c) When N (·) is crudely stationary (see Section 3.2), show that V (0+) = m if and only if the process is simple. (d) Construct an example of a second-order stationary point process for which the set of discontinuities of the left and right derivatives of V (·) is countably dense in (0, ∞). x (e) Writing M2 (x) = λ 0 (1 + 2U (y)) dy, where λ is the intensity of N (·), show that limx→∞ U (x)/λx exists and is ≥ 1. (f) Show that supx>0 (U (x + y) − U (x)) ≤ 2U (y) + m/λ. x (g) Use (8.1.13) to show that V (x) = 2 0 Fc (u) du where, in terms of the ˘2 (0, u] = 1 C ˘2 , Fc (u) = 1 C ˘ ({0})+C ˘ [−u, u]. reduced covariance measure C 2 2 2 2 Deduce that, when it exists, the covariance density is a second derivative in R+ of V (x). [Hint: See Daley (1971) for (a)–(e) and Berbée (1983) for (f).] 8.1.4 Suppose N (·) is a simple stationary isotropic point process in R2 with intensity λ, finite second-moment measure, and second-order intensity [see (8.1.18)] ˘ 2 (x) = h(|x|), ˘ h say, for points distance |x| apart. Show that for a sphere Sr of radius r, V (Sr ) ≡ var N (Sr ) equals 2

λπr + λ

r

2πu du 0

r+u

arcos 0+

max

u2 + v 2 − r 2 − 1, 2uv

˘ v h(v) dv

˘ Supposethat h(u) → 0 monotonically for u large enough. Deduce that when r ˘ limr→∞ 1 uh(u) du < ∞, limr→∞ V (Sr )/M (Sr ) exists [see below (8.1.14)]. 8.1.5 (a) If {In } is a nested decreasing sequence of intervals with (In ) → 0 as n → ∞, show that for any second-order stationary simple point process on R, V (In )/M (In ) → 1. (b) Show that replacing {In } by more general nested sets {An } may lead to V (An )/M (An ) → 1. [Hint: Consider a stationary deterministic process at j unit rate, and for some fixed integer j ≥ 2, let An = i=1 (i, i + 1/n].] (c) Let {An } be a nested decreasing sequence of sets in Rd with diam(An ) → 0 as n → ∞. Show that V (An )/M (An ) → 1 as n → ∞ for second-order stationary simple point processes on Rd . 8.1.6 Processes of bounded variability. Show that for a nontrivial stationary cluster point process on R with finite second-moment measure to be of bounded variability, the cluster centre process must be of bounded variability and all clusters must be of the same size.

302

8. Second-Order Properties of Stationary Point Processes As a special case, suppose the cluster centre process is deterministic and that points are randomly jittered with jitter distribution F , say. What conditions on F are needed for the jittered process to be of bounded variability? [See Cox and Isham (1980, Section 3.5) for more discussion.]

8.1.7 Isotropic Neyman–Scott process. In Example 8.1(b), suppose that the d.f. F is the bivariate normal distribution with zero mean and covariance matrix

⎧ 2 ⎩ σ1 Σ=⎪

ρσ1 σ2

⎫

ρσ1 σ2 ⎪ ⎭. σ22

Then, the symmetrized d.f. G for the vector distance between two offspring from the same parent is bivariate normal also with zero mean vector and covariance matrix 2Σ. When σ12 = σ22 = σ 2 , say, and ρ = 0, the process is isotropic and K(r) = πr2 + [m[2] /(µc m21 )](1 − e−r

2

/4σ 2

).

8.1.8 Rd -analogue of Matérn’s Model I. Let v(R, a) denote the volume of the intersection of two Rd hyperspheres of radius R whose centres are distance a apart. Construct a point process in Rd analogous to the process in R of Example 8.1(b) and show that this Rd analogue has M (A) = λe−λv(R,0) (A), ˘ 2 (x) = h

⎧ ⎨0 ⎩

(0 < |x| ≤ R),

λ exp ( − λ[v(R, 0) − v(R, |x|)])

(R < |x| ≤ 2R),

λ exp ( − λv(R, 0))

(2R < |x|).

[Hint: See Cox and Isham (1980, Exercise 6.3) for the case d = 2.] 8.1.9 Matérn’s Model I: Further properties. (a) Renewal process. Let {tn : n = 1, 2, . . .} be the successive epochs in (1, ∞) of a Poisson process on R+ at rate λ, and attach marks I(tn ) = 0 or 1 successively as follows, starting with tn initially unmarked. If tn is unmarked, then I(tn ) = 0 if tn+1 < tn + 1, in which case I(tn+1 ) = 0 also, or else tn+1 > tn + 1, I(tn ) = 1, and tn+1 is initially unmarked. If I(tn ) = 0, then I(tn+1 ) = 0 if tn+1 < tn + 1, or else tn+1 > tn + 1 and tn+1 is initially unmarked. Show that {tn : n = 0, 1, . . .}, defined by t0 = 0 and tn+1 = inf{tj > tn : I(tj ) = 1} (n = 0, 1, . . .), are the epochs of a renewal process with a renewal density function h(·) that is ultimately constant, namely 0 (0 < x ≤ 1), h(x) dx = λe−λ min(x,2) (x > 1). (b) Show that Example 8.1(c) is a version of the corresponding stationary renewal process. (c) The complementary set. Every point in the complementary set {xn } of ‘rejected points’ in the construction of Matérn’s Model I in Example 8.1(c) shows clustering characteristics: for one thing, the nearest-neighbour distance of any xn is at most R. Investigate other properties of this process.

8.2.

The Bartlett Spectrum

303

[Hint: Consider first the case d = 1; find its density, cluster structure, nearest-neighbour distribution, and covariance density. Which of these are accessible when d ≥ 2? What properties of {xn } can be deduced by complementarity with respect to a Poisson process of the underdispersed process of Example 8.1(c)?] 8.1.10 Matern’s Model II for underdispersion. Consider an independent marked Poisson process with realization {(xi , κi )} in which the points {xi } have intensity λ, say, and the independent marks have a common uniform distribution on (0, 1) (any absolutely continuous distribution will do). A point xi is rejected if there is any other point within distance R and with mark larger than κi . Show that the retained points {xi }, say, have density (1 − e−2λR )/(2R) and that the relative second-order intensity r2 (x) vanishes for |x| < R, equals 1 for |x| > 2R, and for R < |x| < 2R, r2 (x) =

2R + (3R + x)e−λ(R+x) − (5R + x)e−λ(3R+x) > 1. R(R + x)(3R + x)

Examine the Rd analogues of the model (see Exercise 8.1.8). c (B) in (8.1.26) is unbiased. 8.1.11 (a) Show the weighted estimate M[2] (b) A simpler but cruder correction subtracts from (8.1.25) the expected bias when the observed process is Poisson with the same mean rate. Express c this as a correction to M[2] (B). [Hint: See e.g. Miles (1974) and Vere-Jones (1978a, p. 80) who give explicit forms.] (c) Although the cumulative forms given above admit consistent estimates, they are less easy to interpret than smoothed estimates of the corresponding densities. For example, in R2 , estimates of the radial correlation function and related quantities can be obtained by counting the number of points in an annulus about a given point of the realization, dividing by the area of the annulus, subtracting the appropriate mean, and regarding the resultant value as an estimate of ρ(r) at a distance r corresponding to the mid-radius of the annulus. Fill out the details behind these remarks. [Hint: See e.g. Vere-Jones (1978a) and Chong (1981) for applications.]

8.2. The Bartlett Spectrum The spectral theory of point processes has two origins. On the theoretical side, the results can be derived from specializations of Doob’s (1949, 1953) theory of processes with stationary increments and related treatments of generalized stochastic processes by Bochner (1955) and Yaglom (1961). The key features relevant to the practical analysis of point process data were identified by Bartlett (1963) and followed up by several authors, as summarized for example in Cox and Lewis (1966) and Brillinger (1972, 1978). The treatment given in this chapter is based on developments of the theory of Fourier transforms of unbounded measures (see e.g. Argabright and de Lamadrid, 1974). As such, it requires an extension, not quite trivial, of the classical Bochner theorem and related results used in standard time series analysis. We describe

304


this extension, concerned with properties of positive, positive-definite (p.p.d.) measures, in Section 8.6. Here in this section, we summarize and illustrate the properties that are most relevant to the practical analysis of point process models. ˘2 We saw in Proposition 8.1.II that the reduced second-moment measure M of a stationary random measure is a p.p.d. measure so that all the proper˘ 2 is ties developed for such measures in Section 8.6 apply. In particular, M transformable so that it possesses a well-defined Fourier transform (in the sense of generalized functions), which is again a measure, and for which the explicit versions of the Parseval relation and the inversion theorem, derived in that section, are valid. The reduced covariance measure C˘2 is not itself a ˘ 2 only by the term m2 , which is also a p.p.d. measure, but it differs from M p.p.d. measure [its Fourier transform is the multiple (m2 /2π)δ0 of the measure consisting of a single atom at the origin]. Thus, C˘2 can be represented as a difference of two p.p.d. measures, so that the same results (existence of a Fourier transform that is a difference of two p.p.d. measures, Parseval relations, etc.) hold for it also. A similar remark applies to the reduced second factorial moment measure and the corresponding factorial cumulant measure, where it is a matter of subtracting an atom at the origin. Any one of these four measures could be taken as the basis for further development of the spectral theory. It is convenient, and consistent with the standard convention in time series analysis, to choose as the spectrum of the process ξ the inverse Fourier transform of the (ordinary) covariance measure. The proposition below summarizes the main results pertaining to this transform; (8.2.1) and (8.2.2) are examples of Parseval relations. Proposition 8.2.I. Let ξ be a second-order stationary point process or random measure on Rd with reduced covariance measure C˘2 . Then (a) there exists a symmetric, translation-bounded measure Γ on BRd such that, for all ψ in the space S of functions of rapid decay defined below (8.6.1), ˜ ˘ ψ(x) C2 (dx) = ψ(ω) Γ(dω), (8.2.1)

Rd

Rd

˜ where ψ(ω) = Rd ei(ω·u) ψ(u) du (ω ∈ Rd ); (b) the inversion relations (8.6.6–10) and (8.6.12) hold, with µ identified as Γ and ν as C˘2 ; and (c) for bounded measurable φ with bounded support and also for φ ∈ S, if ζφ = Rd φ(x) ξ(dx), then 2 ˜ var ζφ = |φ(ω)| Γ(dω) = (φ ∗ φ∗ )(u) C˘2 (du) ≥ 0, (8.2.2) Rd

Rd

∗

where φ (u) = φ(−u). Proof. The statements all follow from the p.p.d. properties noted in the opening paragraph and the results for p.p.d. measures outlined in Section 8.6. In particular, (8.2.2) follows from Proposition 8.6.IV.

8.2.


305

Definition 8.2.II. The Bartlett spectrum of a second-order stationary point process or random measure ξ on Rd is the measure Γ(·) associated with the reduced covariance measure C˘2 of ξ in Proposition 8.2.I. Equations (8.2.1), usually in the form of (8.2.4) below, and (8.2.2) are generally the most convenient results to use in establishing the form of the Bartlett spectrum for a given process. Note in particular the special case for X = R and ψ the indicator function for (0, t], var ξ(0, t] = R

sin 12 ωt 1 2ω

2 Γ(dω),

(8.2.3)

which is essentially Daley’s (1971) representation for the variance function of a stationary point process or random measure [Daley uses a measure defined on R+ , while in (8.2.3), Γ(·) is a symmetric measure on R]. An alternative route to (8.2.3) exploiting a skeleton process, the standard Bochner representation and weak convergence, is sketched in Exercise 8.2.1. It is clear from Proposition 8.2.I that while the spectral measure Γ is positive, it is not in general a p.p.d. measure. However, since the reduced second˘ 2 is positive and is the Fourier transform of the positive moment measure M 2 measure Γ(·) + [m /(2π)d ]δ0 (·), Γ(·) can be made into a p.p.d. measure by the addition of a sufficiently large atom at the origin. In the point process case, the reduced covariance measure has an atom at the origin that transforms into a positive multiple of Lebesgue measure, and consequently the Bartlett spectrum of a point process is never totally finite. On the other hand, the factorial covariance measure is often both absolutely continuous and totally finite, and then Γ(·) is absolutely continuous with a density γ(·), which can be written (for the case d = 1)

∞

2πγ(ω) = m + −∞

e−iωx c[2] (x) dx

= m + c˜[2] (−ω) = m + c˜[2] (ω).

(8.2.4)

It was in this form that the spectral measure was originally introduced by Bartlett (1963). It is not known whether every p.p.d. measure can arise as the secondmoment measure of some random measure nor, when it does, how to construct a process yielding the given measure as its second-moment measure. The standard construction using Gaussian processes or measures is not available here, as such processes do not have nonnegative trajectories (see Wiener’s homogeneous chaos example in Chapter 9). Some partial results arise from the examples considered below and from Exercises 8.2.11–12 and 8.4.6–7. Davidson (1974) provided a construction for identifying the second-moment measures of stationary random measures on the circle (see the further discussion in Chapter 12), but it relies on the finiteness of the invariant measure on a circle, and

306


it is not obvious how it might be extended to either point processes or random measures on the line. In the very special case of a discrete point process on the four points of the compass (NESW), with translation interpreted as rotation through π/2, the family of second-moment measures can be identified explicitly and is strictly contained in the class of p.p.d. measures; see Exercise 8.2.5 for details. We now discuss the Bartlett spectrum for some basic point processes on Rd . Example 8.2(a) Poisson process with constant intensity on Rd . Here C˘2 consists only of the atom mδ0 (·) so Γ is absolutely continuous with density m/(2π)d . This ‘white-noise’ spectrum is consistent with the completely random character of the process. Note that the Parseval relations (8.2.1) and (8.2.2) take, respectively, the special forms, with ζφ = Rd φ(x) N (dx), m ˜ ψ(ω) dω mψ(0) = (2π)d Rd and

m var ζφ = m |φ(x)| dx = d (2π) d R

2

Rd

2 ˜ |φ(ω)| dω.

Example 8.2(b) Stationary renewal process. If the renewal density u(t) exists and the process is stationary with mean rate λ = 1/µ, where µ is the mean lifetime, we have from Example 5.4(b) that m ˘ [2] (x) = λu(|x|) and hence

c˘2 (x) = δ0 (x) + λ u(|x|) − λ .

If further the difference u(x) − λ is integrable on (0, ∞), (8.2.4) yields for ω = 0 F(ω) 1 F(−ω) λ 1 λ 1+ + + −1 , = γ(ω) = 2π 2π 1 − F(ω) 1 − F(−ω) 1 − F(ω) 1 − F(−ω) (8.2.5) ∞ iωx where F (ω) = 0 e dF (x) is the characteristic function of the lifetime distribution. For ω = 0, we obtain from the above or Exercise 4.4.5 ∞ σ 2 + µ2 λ λ 1+ 1+2 = u(x) − λ dx . γ(0) = 2π µ2 2π 0 Special cases, when lifetime distributions are of ‘phase type’ for example, yield rational polynomials for F and hence rational spectral densities (see e.g. Neuts, 1979). Exercise 8.2.6 gives a simple nontrivial example. Since a stationary renewal process has moment measures of all orders whenever it exists, the Bartlett spectrum exists for all such processes, but without the additional restriction it may not be absolutely continuous or (even if it is) γ(0) need not be finite as above. The extreme case described in the next example is worth particular mention.

8.2.


307

Example 8.2(c) Stationary deterministic process. Here, points occur on a regular lattice of span a, the whole lattice being randomly shifted so that the first point to the right of the origin is uniformly distributed on (0, a]. The ˘ 2 (·) has an infinite sum, with mass 1/a at each of the points ka, measure M (k = 0, ±1, . . .). Its Fourier transform has mass 1/a2 at each of the points 2πj/a, (j = 0, ±1, . . .). Moving to the Fourier transform of the covariance measure deletes the atom at j = 0 so that Γ(·) can be written in terms of Dirac measures as ∞ 1 Γ(A) = 2 (8.2.6) δ2πj/a (A) + δ−2πj/a (A) . a j=1 Example 8.2(d) Cluster processes. For a general cluster process N in Rd , the variance of an integral Rd φ(x) N (dx) can be written (see Exercise 6.3.4) φ(x) N (dx) =

var Rd

Vφ (u) M c (du) + mφ (u)mφ (v) C2c (du × dv),

Rd

(8.2.7)

(Rd )(2)

where mφ (u) =

Rd

φ(x) M1 (dx | u),

Vφ (u) =

(Rd )(2)

φ(s)φ(t) C2 (ds × dt | u),

and we use the notation M c (·) and C2c (·) from (6.3.4–5). In the stationary case, M c (du) = mc du, where mc is the mean density of the cluster centre process, while C2c has a reduced form that can be written in terms of the Bartlett spectrum Γc of the cluster centre process. Since also C2 (ds × dt | y) depends only on the differences s − y and t − y, the first term in (8.2.7) can be written in terms of the measure B defined via bounded measurable h by h(y) B(dy) = h(s − t) C2 (ds × dt | 0). (Rd )(2)

Rd

Here the measure B is both positive-definite and totally finite (since the mean square cluster size is necessarily finite); it has therefore an ordinary Fourier transform B(ω) = (2π)−d Rd e−i(ω·x) B(dx), which can be written in the symmetric form B(ω) = var e−i(ω·x) Nm (dx | 0) , Rd

where, it should be recalled, var Z = E |Z|2 − |EZ|2 for a complex-valued r.v. Z. Thus, writing 1 (ω | 0) = e−i(ω·x) M1 (dx | 0) = E e−i(ω·x) Nm (dx | 0), M Rd

Rd

308


we obtain from (8.2.7) var

mc 2 ˜ ˜ 1 (ω | 0)|2 Γc (dω) . B(ω) dω + | M |φ(ω)| (2π)d Rd

φ(x) N (dx) =

Rd

This relation shows that the Bartlett spectrum of the cluster process N can be identified with the measure −d 1 (ω | 0)|2 Γc (dω). Γ(dω) = B(ω)m dω + |M c (2π)

(8.2.8)

The first term can be regarded as the contribution to the spectrum from the internal cluster structure; the second term is a filtered version of the spectrum of the cluster centre process with the filtering reflecting the mean distribution of the cluster, as in Daley (1972b). For a stationary Poisson cluster process, further simplification occurs. Letting µc denote the intensity of the Poisson process of cluster centres, we find that Γ has a density γ, which has the simple alternative forms µc γ(ω) = (2π)d

!R& & µc = E && d (2π)

M1 (dx | 0) +

d

R

−i(y·ω)

e &2 " & i(x·ω) e Nm (dx | 0)&& , d Rd

Rd

M[2] (dx × dy | 0) (8.2.9)

which is easily recognized as the transformed version of (6.3.5). Specific results for the Neyman–Scott and Bartlett–Lewis processes follow readily from these equations (see Exercises 8.2.9 and 8.2.10). We shall see in Section 8.3 that, for filtering and prediction purposes, a particularly important role is played by point processes having a rational spectral density. Many common and useful examples fall into this class. By suitable specification of the components, both renewal and cluster processes can give rise to spectral measures with rational spectral densities. For example, it is clear from (8.2.5) that this will occur whenever the interval distribution of a renewal process has a rational Laplace transform, that is, whenever the distribution is expressible as a finite convolution or mixture of exponentials. Several types of cluster processes, as well as Cox processes, have rational spectral densities, in particular the Neyman–Scott process with an exponential or Erlang distribution for the distances of the cluster elements from the cluster centre [see also Exercise 8.2.7(b)]. The wide choice of such examples shows not only the richness of the class but also the relative lack of discrimination in the spectrum as a means of distinguishing between processes that in other respects may be quite dissimilar. One of the most important examples is the Hawkes process with suitably restricted response function (i.e. infectivity measure) as described below.

8.2.


309

Example 8.2(e) Hawkes process with rational spectral density. From Example 6.3(c) and the results on branching processes in Exercise 5.5.6, we see 1 of the first-moment measure of the total offthat the Fourier transform M spring process is a rational function of the Fourier–Stieltjes transform µ ˜ of the infectivity measure, namely 1 (ω | 0) = 1/[1 − µ M ˜(ω)],

where

µ ˜(ω) =

∞

eiωx µ(dx).

0

Combining this result with the expressions for the mean rate and covariance density given by (6.3.26) and (6.3.27) and with the general form (8.2.8) for cluster processes, we obtain the spectral density for the Hawkes process in the form λ/(2π) γ(ω) = . (8.2.10) (1 − ν) |1 − µ ˜(ω)|2 1 (ω). Consequently, when µ ˜(ω) is a rational function of ω, so too is M Because the form of (8.2.10) is similar to that of the spectral density of an autoregression in continuous time, one might hope that the Hawkes model could play a role similar to that of autoregressive models in the context of mean square continuous processes. This hope is frustrated by the special probabilistic structure of the Hawkes model, which requires that µ(·) ≥ 0. If this condition is violated, it is not clear that there exists any point process with the spectral form (8.2.10), and if such a process does exist, it certainly will not have the Poisson branching structure of a Hawkes process. Despite this difficulty, the possibility of using the Hawkes process to approximate general point process spectra was explored by Hawkes (1971b), Hawkes and Adamopoulos (1973), Ozaki (1979) and, more deliberately, by Ogata and Akaike (1982), with an application in Ogata et al. (1982). Ogata and Akaike (1982) suggest taking for µ a measure on [0, ∞) with density function µ(t) = K eαt k=0 bk Lk (t) for α > 0 and Laguerre polynomials Lk (t). This form leads automatically to processes with rational spectral densities since the Fourier transforms of the Laguerre polynomials are themselves rational. The simplest case occurs when K = 0 and b0 = αν for 0 < ν < 1, so that µ ˜(ω) = να/(α−iω) and λ ω 2 + α2 . γ(ω) = · 2 2π(1 − ν) ω + α2 (1 − ν)2 Note the characteristic feature for point processes with rational spectral density that the numerator and denominator are of equal degree. Further examples are given in the papers cited and in Vere-Jones and Ozaki (1982). To yield a valid model, the parameters should be constrained to ensure that the density of the infectivity measure (and hence the conditional intensity) is everywhere nonnegative; for stationarity, the infectivity measure should have total mass < 1. These conditions are relatively stringent and quite difficult to impose in estimation procedures. Within these constraints,

310


however, the Hawkes model is one of the most flexible models available in that it allows both the calculation of the form of the spectrum and the investigation of probabilistic aspects of the process. The basic results described so far apply to stationary (translation-invariant) point processes in any general Euclidean space Rd . When d > 1, however, additional symmetries such as isotropy (invariance under rotations) become possible and have important implications for the structure of the spectral measures. As an illustration, we conclude this section with a brief discussion of isotropic random measures in R2 , this time looking at the Fourier transforms. In the stationary, isotropic case, the second-order properties of a random ˘ 2 (·) measure in R2 are fully defined by the mean density m and the function K defined in (8.1.20). We examine the constraints on the Bartlett spectrum in R2 implied by this isotropy condition and show how to represent the spectrum ˘ 2 (·). in terms of m and K Consider first the effect of the double Fourier transform on a function h: R2 → R which, in addition to being bounded, measurable, and of bounded support, is circularly symmetric, i.e. h(x, y) = h(r cos θ, r sin θ) = g(r)

(all r ∈ S)

for some function g. The transform is given by ˜ h(ω, φ) ≡

ei(ωx+φy) h(x, y) dx dy =

R2 ∞

=

0

2π

eir(ω cos θ+φ sin θ) dθ

rg(r) dr 0

2π

eirρ cos(θ−ψ) dθ

rg(r) dr 0

∞

0

using (ρ, ψ) as polar coordinates in the (ω, φ) plane. Now the integral over θ 2π is simply a Bessel function J0 (u) = 1/(2π) 0 eiu cos θ dθ, so ˜ h(ω, φ) = 2π

∞

rJ0 (rρ)g(r) dr ≡ g˜B (ρ),

where ρ = (ω 2 + φ2 )1/2 .

0

(8.2.11) ˜ Consequently, h(ω, φ) is again circularly symmetric, reducing to the function g˜B (·), which we call the Bessel transform of g(·) (we have included the factor 2π—this is a departure from the usual definition) and is also called a Hankel transform—see e.g. Copson (1935, p. 342). By arguing analogously from the inverse Fourier transform 1 ˜ ei(ωx+φy) h(ω, h(x, y) = φ) dω dφ, (2π)2 R2 it follows that the Bessel transform is inverted as in ∞ 1 g(r) = ρ˜ g B (ρ)J0 (rρ) dρ. 2π 0

(8.2.12)

8.2.


311

From this discussion, we should expect the Bartlett spectral density of a stationary isotropic process to be circularly symmetric in frequency space and ˘ 2 (r). To cover to be related to the inverse Bessel transform of the density of K the situation where densities may not exist, the Bessel transform relation needs to be put into the form of a Parseval relation so that it can be extended to measures, as follows. Proposition 8.2.III. Let Γ(·) be the Bartlett spectrum on R2 associated with a simple stationary isotropic point process in R2 . Then Γ(·) is circularly symmetric and is expressible via (ω1 , ω2 ) = (ρ cos ψ, ρ sin ψ) as dψ dρ + m2 κ(dρ) + 2πm2 δ0 (dρ) , (8.2.13) Γ(dρ × dψ) = mρ 2π 2π ˘ 2 (·) of (8.1.20) by the Parseval– where κ is related to the radial measure K Bessel equation ∞ ∞ B ˘ 2 (dr) g˜ (ρ) κ(dρ) = g(r) K (8.2.14) 0

0

for all bounded measurable g of finite support on R+ and g˜B is defined by (8.2.11). Proof. Recall that the Bartlett spectrum is the Fourier transform in R2 of the complete covariance measure C˘2 , which for disks Sr (0) takes the form ˘ 2 (r), C˘2 Sr (0) = m − m2 πr2 + m2 K where the first term arises from the diagonal concentration associated with a simple point process; the second, the term involving the square of the mean, must be subtracted from the second moment to yield the covariance; and the third is the form of the reduced second factorial moment measure. Using mixed differential notation, this can be rewritten as ˘ 2 (dr) dθ . C˘2 (dx × dy) = m δ0 (dx × dy) − m2 dx dy + m2 K 2π The first and second terms have the following inverse Fourier transforms, respectively: m dω1 dω2 mρ dρ dψ dρ dψ = = mρ · , (2π)2 (2π)2 2π 2π dψ 4π 2 m2 δ0 (dω1 × dω2 ) = 2πm2 δ0 (dρ) · . (2π)2 2π ˘ 2 (dr) dθ/(2π) by Denoting the double Fourier transform of the measure K L(dω1 × dω2 ), the Parseval relation for such transforms implies that, with h ˜ as earlier, and h ∞ 2π dθ ˜ 1 , ω2 ) L(dω1 × dω2 ) = ˘ 2 (dr) . h(ω h(r cos θ, r sin θ) K 2π R2 0 0

312


Now

2π

h(r cos θ, r sin θ) 0

dθ 2π ∞

∞ 2π 1 ˜ cos ψ, ρ sin ψ) dψ dθ dρ e−iρr cos(θ−ψ) ρ h(ρ = (2π)2 0 0 0 ∞ 2π 1 ˜ cos ψ, ρ sin ψ) dψ, dρ ρJ0 (ρr)h(ρ = 2π 0 0

where as before the invariance of integrating θ over any interval of length 2π ˜ 1 , ω2 ) to have the product form has been used. If, in particular, we take h(ω g˜B (ρ)f (ψ), we obtain from this relation and the Bessel transform equation (8.2.12) that

g˜ (ρ)f (ψ) L(dρ × dψ) = B

0

(0,∞)×(0,2π)

2π

dψ f (ψ) 2π

∞

˘ 2 (dr). g(r) K

0

Since the integral here depends on f only through its integral over (0, 2π), a uniqueness argument implies that L(·) has a disintegration of the form L(dρ × dψ) = κ(dρ) [dψ/(2π)], where κ(·) satisfies (8.2.14). ˘ 2 (dr) dr (and not K ˘ 2 ), in the sense of Note that (8.2.14) defines (1/r)K generalized functions, as the Bessel transform of (1/ρ) κ(dρ) dρ. Example 8.2(f) An isotropic Neyman–Scott process. Consider the circularly symmetric case from Example 8.1(b) and Exercise 8.1.7, for which we have ˘ 2 (dr) = 2πr dr + m[2] re−r2 /4σ2 dr . K µm21 2σ 2 It is easy to check from (8.2.14) that the measure 2πr dr on R+ is the Parseval– Bessel transform of the measure consisting of a unit atom at the origin. The second term is a density, and it can be derived (via the Fourier transform in R2 or otherwise) as the Parseval–Bessel transform of the density κ(ρ) =

µm[2] 2 2 ρe−σ ρ . 2 2πµm1

Consequently, for this isotropic Neyman–Scott model, the Bartlett spectrum is absolutely continuous with spectral density γ(ω, φ) =

µm[2] −σ2 (ω2 +φ2 ) β(ρ) µm1 e + ≡ , 2 4π 2π 2π

where the function β(·) as just defined exhibits the Bartlett spectrum in the polar form β(ρ) dρ [dψ/(2π)].

8.2.


313

Exercises and Complements to Section 8.2 8.2.1 Given a second-order stationary point process N , the relation {Xh (n)} = {N (nh, (n + 1)h]} defines a second-order stationary discrete time series. Express var N (0, nh] in terms of the second-moment structure of {Xh (n)}. Use the standard spectral representation of the second moments of a discrete-time process to give a spectral representation for var N (0, nh], and argue that for h → 0 there is a weak limit as in (8.2.3). 8.2.2 Superposition. Show that if ξ1 , ξ2 are independent second-order stationary random measures with Bartlett spectra Γ1 , Γ2 , respectively, then ξ1 + ξ2 has spectrum Γ1 + Γ2 . More generally, if ξ1 , ξ2 , . . . are independent second-order stationary random measures such that the L2 limit ξ = ξ1 + ξ2 + · · · exists, then ξ has Bartlett spectrum Γ1 + Γ2 + · · · . 8.2.3 Cox process. Let ξ be a second-order stationary random measure on Rd with Bartlett spectrum Γ and mean density m. Show that the Cox process directed by ξ has Bartlett spectrum Γ(·) + m(2π)−d (·), where (·) denotes Lebesgue measure on Rd . 8.2.4 Quadratic random measure [see Example 6.1(c) and Exercise 6.1.3]. (a) Let Zi (t)(i = 1, 2) be independent mean square continuous second-order stationary random processes on R with respective spectral d.f.s Fi and zero mean. Show that the product Z1 Z2 is a mean square continuous second-order stationary process with spectral measure F1 ∗ F2 . (b) If Z is a mean square continuous stationary Gaussian process with spectral d.f. F and zero mean, then the quadratic random measure whose sample paths have density Z 2 (·) has covariance density 2|c(·)|2 and Bartlett spectrum 2F ∗ F, where c(x) = cov(Z(0), Z(x)). (c) Investigate what changes are needed in (a) and (b) when the zero mean assumption is omitted. 8.2.5 Cyclic point process on four points. Consider a {0, 1}-valued process on the four compass points NESW that is stationary (i.e. invariant under cyclic permutations). Denote the probabilities of the six basic configurations 0000, 1000, 1100, 1010, 1110, and 1111 by {p0 , p1 , . . . , p5 }, respectively. (i) Show that the mean density and reduced second-moment measure are given respectively by m = 14 p1 + 12 (p2 + p3 ) + 34 p4 + p5 , ˘ 2 = {a, b, c, d}, M where a = m, b = d = 14 p2 + 12 p4 + p5 , c = 12 p3 + 12 p4 + p5 . Show that ˘ 2 is a p.p.d. measure with Fourier transform proportional to (a + c + 2b, M a − c, a + c − 2b, a − c). (ii) Renormalize the probabilities so that m = 1 (equivalent to looking at the Palm measure and its first moment) and the second-moment measure has

314

8. Second-Order Properties of Stationary Point Processes standardized form {1, β, γ, β}. Show that this is a p.p.d. measure if and only if β, γ are nonnegative and γ ≤ 1, 1 + γ ≥ 2β. However, this is the second-moment measure of a point process on NESW if and only if, in addition, 1 + β ≥ 2γ. [Hint: Write x = 12 p4 + p5 , y = 14 p1 + 14 p4 , so that x < min(β, γ) and (x, y) lies on the line y = 3x−K, where K = 2β +2γ −1. Nonnegative solutions x, y exist if and only if 13 K ≤ min(β, γ), which yields both the p.p.d. condition and the additional condition.]

8.2.6 Stationary renewal process. Let the lifetime d.f. F (·) of the process as in Example 8.2(b) be the convolution of two exponentially distributed random variables with means 1/µj (j = 1, 2). Evaluate (8.2.5) explicitly. 8.2.7 Random translations. Let the point process N be second-order stationary with Bartlett spectrum Γ and mean density m. If the points of N are subjected to independent random translation with common d.f. F, show that the resultant point process NT has Bartlett spectrum [see (8.2.8)] ΓT (dω) = |F(ω)|2 Γ(dω) + m(2π)−d (1 − |F(ω)|2 ) (dω). 8.2.8 Iterated random translations. Let the independent translation of points of N as in Exercise 8.2.7 be iterated n times. Show that the Bartlett spectrum Γn of the resulting process satisfies Γn (dω) = |F(ω)|2 Γn−1 (dω) + m(2π)−d (1 − |F(ω)|2 ) (dω) = |F(ω)|2n Γ(dω) + m(2π)−d (1 − |F(ω)|2n ) (dω) and hence give conditions for Γn (·) to converge weakly to m(2π)−d (·). (See Chapter 11). 8.2.9 Neyman–Scott process [continued from Example 6.3(a)]. (a) Show that the Bartlett spectrum for a Neyman–Scott process on R, with (Poisson) cluster centre process at rate µc , m[1] and m[2] for the first two factorial moments of the cluster size distribution, and common d.f. F for the distances of the points of a cluster from their centre, has density γNS (ω) given by γNS (ω) = (µc /2π)[m[1] + m[2] |F(ω)|2 ], where F(ω) =

∞

−∞

eixω F (dx).

(b) In the particular case where F (x) = 1 − e−αx (x ≥ 0), deduce that γNS (·) is the rational function γNS (ω) =

µc m[1] α2 m[2] /m[1] 1+ . 2π α2 + ω 2

(c) When the Neyman–Scott process is as above on Rd , show that γNS (ω) = (µc m[1] /(2π)d )[1 + (m[2] /m[1] )|F(ω)|2 ]

with F(ω) = Rd eix·ω F (dx). Deduce that when d = 2 and F (·) is a bivariate normal d.f. with zero mean and the usual second-moment parameters σ12 , σ22 and ρσ1 σ2 , the spectrum has density γNS (ω1 , ω2 ) =

µc m[1] m[2] 1+ exp(−σ12 ω12 − 2ρσ1 σ2 ω1 ω2 − ρ22 ω22 ) . 2 4π m[1]

8.2.


315

(d) Show that if in (a) the cluster structure is modified to include the cluster centre, then γNS (ω) = (µc /2π)[1 + m[1] (1 + F(ω) + F(−ω)) + m[2] |F(ω)|2 ]. (e) Show that if in (a) the cluster centre process is a general stationary point process with mean intensity µc and Bartlett spectrum Γc (·), then the Bartlett spectrum ΓNS (·) of the cluster process is given by µc ΓNS (dω) = |m[1] F(ω)|2 Γc (dω) + [m[1] + (m[2] − m2[1] )|F(ω)|2 ] (dω). 2π [Hint: Except for (d), the results can be derived, first by compounding and then by using random translations as in Exercise 8.2.7; otherwise, see (8.2.8).] 8.2.10 Bartlett–Lewis model [continued from Example 6.3(b)]. (a) Use (6.3.23) to show that the Bartlett spectrum has density γBL (·) given by ∞ ∞ ∞

µc (j + 1)qj + (k + 1 − j)qk (Fj (ω) + Fj (−ω)) . γBL (ω) = 2π j=0

j=1 k=j

Observe that γBL (ω) = γNS (ω) as in Exercise 8.2.9(d) in the cases q1 = 1 and m[1] = 1, m[2] = 0, respectively. (b) Show that when qj = (1 − α)αj (j = 0, 1, . . .) with 0 < α < 1, so that each cluster is a transient renewal process,

1 1 µc + −1 , γBL (ω) = 2π(1 − α) 1 − αF(ω) 1 − αF(−ω) while when q0 = 0, qj = (1 − α)αj−1 (j = 1, 2, . . .),

1 1 µc γBL (ω) = + − 1 − (1 − α)2 . 2πα(1 − α) 1 − αF(ω) 1 − αF (−ω) (c) The formulae in parts (a) and (b) assume that the cluster centre is included in the cluster process. Show that omitting the cluster centres leads to

*

+

∞ ∞

µc γBL (ω) = jqj + (k − j)qk (Fj (ω) + Fj (−ω)) 2π

*

j=1

k=j+1

+

j−1 ∞

µc = jqj + qj (j − k)(Fk (ω) + Fk (−ω)) . 2π ∞

j=1

j=2

k=1

8.2.11 Let M2 be a p.p.d. measure on BR with density m2 . Show that if 0 < a ≤ m2 (x) ≤ b < ∞ (all x) then there exists a zero-mean Gaussian process X(t) such that m2 (x) = E[X 2 (t)X 2 (t + x)] andhence that M2 is the reduced second-moment measure of the process ξ(A) = A X 2 (t) dt (A ∈ BR ). Deduce that any p.p.d. function c2 (·) can be a reduced covariance density; i.e. there is some a > 0 such that a + c2 (x) is the second-moment density of some second-order stationary random measure. 8.2.12 Let F be any totally bounded symmetric measure Rd . Show that F can be a covariance measure. [Hint: Construct a Gauss–Poisson process and refer to Proposition 6.3.IV. See Milne and Westcott (1972) for further details.]

316


8.3. Multivariate and Marked Point Processes This section provides a first introduction to the wide range of extensions of the previous theory, incorporating both time-domain and frequency-domain aspects. We look first at multivariate and marked point processes, with stationarity in time (i.e. translation invariance) still playing the central role. The results given thus far for second-order stationary random measures and point processes on Rd extend easily to multivariate processes on Rd , though for convenience we discuss mostly the case d = 1. The first-moment measure in Proposition 8.1.I(a) becomes a vector of first-moment measures Mi (A) = E[ξi (A)]

(i = 1, . . . , K; A ∈ BR ),

one for each of the K components. Under stationarity, which means translation invariance of the joint probability structure, not just of each component separately, this reduces to a vector of mean densities {mi , i = 1, . . . , K}. Similarly, the second-order moment and covariance measures in the univariate case are replaced by matrices M and C of auto- and cross-moment (or covariance) measures with elements for i, j = 1, . . . , K and A, B ∈ BR , Mij (A × B) = E[ξi (A)ξj (B)], Cij (A × B) = Mij (A × B) − Mi (A)Mj (B). Under stationarity, the diagonal components Mii are invariant under simulta˘ ii , which neous shifts in both coordinates and so possess reduced forms M inherit the properties of the reduced moment measures listed in Proposition 8.1.II. More than this is true, however. Since every linear combinak tion i=1 αi ξi (Ai ) is again stationary, we find on taking expectations of the k k squares that the quadratic forms i=1 j=1 αi αj Mij (Ai × Aj ) are all stationary under diagonal shifts and therefore possess diagonal factorizations. ˘ ij (·), C˘ij (·), say, for From this there follows the existence of reduced forms, M the off-diagonal as well as the diagonal components of the matrices. ˘ ij , C˘ij (i = j) In the point process case, the off-diagonal components M will not have the atom at the origin characteristic of the diagonal components unless there is positive probability of pairs of points occurring simultaneously in both the i and j streams. In particular, if the ground process Ng (·) = K i=1 Ni (·) is orderly, both the matrix of reduced factorial moment measures ˘ ij (A) − [δij δ0 (A)mi ](A) ˘ [i,j] (A) = M ˘ M(A) = M and the corresponding matrix of reduced factorial covariance measures with elements ˘ [i,j] (A) − mi mj (A) C˘[i,j] (A) = M will be free from atoms at the origin. ˘ enjoys matrix versions of Whether or not such atoms exist, the matrix M the properties listed in Proposition 8.1.II; we state them for clarity.

8.3.

Multivariate and Marked Point Processes

317

Proposition 8.3.I (Stationary multivariate random measure: Second-order moment properties). ˘ ii (A) > 0 if A 0 and either Ni has an atomic ˘ (i) M(A) ≥ 0, with M component or A is an open set; ˘ ˘ T (−A); (ii) M(A) =M ˘ is positive-definite: for all finite sequences {fi } of bounded measurable (iii) M complex functions of bounded support, K K

˘ ij (du) ≥ 0; (8.3.1) fi (x)fj (x + u) M i=1 j=1

R

˘ is translation-bounded: for given A, there exists a constant KA such (iv) M ˘ ij (x + A)| < KA ; ˘ + A)|| = K |M that ||M(x i,j=1 (v) If also the process is ergodic as for equations (8.1.4–5), then as r(A) → ∞, ˘ ˘ ∞ ≡ (mi mj ), and for all bounded Borel sets B, M(A)/(A) →M 1 ˘ ij (B). ξi (x + B) ξj (dx) → M (A) A The properties follow readily from the same device of applying the univariate results to linear combinations of the components (see Exercise 8.3.1). Note that property (ii) implies that the diagonal measures are symmet˘ ij (A) = M ˘ ji (−A), confirming the ric, while for the off-diagonal measures M importance of order in specifying the cross-moments. The spectral theory also extends easily to multivariate processes on R. For any linear combination of the components, the basic p.p.d. properties (i) and (iii) above are interchanged by the Fourier transform map, implying that the moment measures can be represented by a matrix of spectral measures, which again enjoys the properties listed above (see Exercise 8.3.2). For practical purposes, the multivariate extension of the Bartlett spectrum (Definition 8.2.II) is of greatest importance. This comprises the matrix Γ of auto- and cross-spectral measures Γij (·) in which the diagonal elements Γii (·) have the properties described in Section 8.2 and the matrix as a whole has the positive-definiteness property in (8.3.1). Indeed, (8.3.1) can be regarded as being derived from the filtered form k ∞

X(t) = fi (t − u) ξi (du) (8.3.2) i=1

−∞

for which the spectral measure ΓX has the form ΓX (dω) =

k k

f˜i (ω)f˜j (ω) Γij (dω).

(8.3.3)

i=1 j=1

In the generality considered here, the components ξi at (8.3.2) may be point processes or random measures. If the latter are absolutely continuous, the appropriate components of the matrix Γ then reduce to the usual spectra

318


and cross-spectra of the stationary processes formed by their densities. In this way, the theory embraces both point and continuous processes as well as mixed versions. If the continuous process has varying sign, as occurs with a Gaussian process, or is given in the wide sense only, then the appropriate framework is the matrix extension of the wide sense theory summarized after Definition 8.4.VII. From the practical viewpoint, these remarks mean that the interaction of point process systems, or mixtures of point process and continuous systems, can be studied in the frequency domain very much as if they were all continuous systems. The essential difference is that each point process component leads to a δ-function component in the diagonal term C˘ii (·) to which there is then a corresponding nonzero constant contribution in the spectral measure Γii (·). Bearing this in mind, all the standard concepts of multivariate spectral theory, such as coherence and phase, or real and quadratic spectra, carry over with minor variations to this more general context and provide valuable tools for the descriptive analysis of multivariate point processes and mixed systems. Brillinger (1975a, b, 1978, 1981) outlines both differences and similarities; for an example studied in depth, see Brillinger (1992). The next two examples illustrate simple special cases of these ideas. Example 8.3(a) A bivariate Poisson process [continued from Example 6.3(e)]. The stationary bivariate point process described earlier is determined by three parameters: rates µ1 and µ2 for the occurrence of single points in processes 1 3 (du) = µ3 G(du) on R, and 2, respectively, and a boundedly finite measure Q in which µ3 is the rate of occurrence of pairs of points, one in each process and G(du) is a probability distribution for the signed distance u from the process 1 point to the other point. It is convenient for the rest of the example to have G(du) = g(u) du for some probability density function g(·) on R. Since the two component processes are both Poisson, the only nonzero second-order factorial cumulant measure is in the cross-covariance term, with g(u) du = C˘[21] (−A).

C˘[12] (A) = µ3 A

˘ C ˘ of reduced secondThe matrices m(u), ˘ ˘ c(u) of densities for the matrices M, moment measures are given respectively by ⎧ ⎫ 0 ⎪ ⎩ µ1 + µ3 ⎭ δ0 (u) + m(u) ˘ =⎪ 0 µ2 + µ3 ⎧ (µ1 + µ3 )2 ⎪ ⎩ (µ1 + µ3 )(µ2 + µ3 ) + µ3 g(−u)

⎫ (µ1 + µ3 )(µ2 + µ3 ) + µ3 g(u) ⎪ ⎭ (µ2 + µ3 )2

and ⎧ ⎩ µ1 + µ3 ˘ c(u) = ⎪ 0

⎫ ⎧ 0 0 ⎪ ⎭ δ0 (u) + ⎪ ⎩ µ2 + µ3 µ3 g(−u)

⎫ µ3 g(u) ⎪ ⎭. 0

8.3.


319

The corresponding Bartlett spectra are all absolutely continuous, the densities γij (ω) of the matrix Γ being given by ⎧ ⎫ 1 ⎪ µ +µ µ3 G(ω) ⎪ ⎪ ⎪ (8.3.4) ⎩ 1 3 ⎭, 2π µ3 G(−ω) µ2 + µ3 where G(ω) = R e−iuω g(u) du. The coherence of the two processes, at frequency ω, is the ratio µ3 |G(ω)| , ρ12 (ω) = (µ1 + µ3 )(µ2 + µ3 ) while their phase at the same frequency is Im(G(ω)) . θ12 (ω) = arctan Re(G(ω)) Example 8.3(b) System identification: a special case. In the previous example, the spectral densities completely determine the parameters of the process. This leads to the more general problem of determining the characteristics of a point process system, meaning some mechanism for producing a point process output from a point process input. Deletions (or thinnings), delays (or translations), and triggering of clusters can all be regarded as examples of point process systems. The problem of system identification then consists of determining the mechanism, or at least its main features, from measurements on its input and output. The two components of the previous example can be regarded as the input and output of a system specified as follows: a proportion π1 = µ1 /(µ1 + µ3 ) of the input points are randomly deleted while each of the points in the remaining proportion π2 = 1 − π1 is transmitted after independent delays with d.f. G [such a specification requires G(·) to be concentrated on a half-line], with this transmitted output being contaminated with ‘noise’ consisting of the points of a Poisson process at rate µ2 . It is evident from the spectral representation in (8.3.4) that the three system parameters π1 , G and µ2 can be identified by measuring the response of the system to a Poisson input process and finding the joint first- and second-order properties of the input and output. It is equally evident that this identification is impossible on the basis of separate observations of the input and output. Suppose now that the Poisson input process is replaced by any simple stationary input process with mean density m and spectral density γ(·) in place of (µ1 + µ3 )/(2π). Then, in place of the matrix with components at (8.3.4), we would have the matrix ⎫ ⎧ γ(ω)G(ω) ⎪ ⎪ ⎪ ⎪ γ(ω) ⎪ ⎪ ⎪ ⎪ . (8.3.5) ⎪ ⎪ m + mπ µ ⎪ ⎪ 2 2 2 2 ⎭ ⎩ γ(ω)G(ω) γ(ω) − + π1 |G(ω)| 2π 2π Once more it is evident that in principle the parameters π1 , G and µ can be identified from this matrix of spectral densities.

320


Many applications of multivariate point process models arise as extensions of contingency table models when more precise data become available concerning the occurrence times of the registered events. Typical examples arise in the analysis of medical or epidemiological data collected by different local authorities. If the only data available represent counts of occurrences for each region and within crude (e.g. yearly) time intervals, then methods of categorical data analysis may help to uncover and interpret spatial and temporal dependences. If, however, the data are extended to record the times of each individual occurrence, then marked point process methods may be more appropriate. Several recent books, such as Cressie (1991), Ripley (1988) and Guttorp (1995), provide useful introductions to and examples of such studies. The interpretation of the marks, however, is by no means restricted to such spatial examples. Examples abound in neurophysiology, geology, physics, astronomy, and so on in which interest centres on the evolution and interdependencies of sequences of events involving different types of events. The first stages in the point process analysis of such data are likely to involve descriptive studies, which have the aim of mapping basic characteristics and dependences. Here, while they may be followed later by model-fitting and testing exercises, nonparametric estimates of the first- and second-order characteristics are of particular importance. Such estimates closely follow the univariate forms described earlier [see in particular (8.1.4–5) and (8.1.16–17)]. They take their cue from (8.1.5) in Proposition 8.1.II. Since we are considering MPPs with time as the underlying dimension, estimates such as (8.1.16) for the reduced moment measures here take the form

, ˘ jk ((0, τ ]) = 1 M Nj (tik , tik + τ ] . (8.3.6) T i:0≤tik 0. The second-order moment density has the form

8.3.


327

m ˘ 2 (u; k1 , k2 ) = λ2 α2 sk1 sk2 ∞ Hk1 (K)Hk2 (K)Nm (Nm − 1) + λE f˘(x)f˘(x + u) dx, (8.3.18) K2 0 where Hk (j) = 1 if j ≥ k, 0 otherwise, and the integral follows the notation of equation (6.3.19). The first term here represents the product of the means, while the second is the contribution to the second moment from pairs belonging to the same cluster. Note that Hk1 (K) Hk2 (K) = Hmax(k1 ,k2 ) (K); taking expectations with respect to the parent cluster mark in the second term yields m ˘ 2 (u; k1 , k2 ) = λ2 α2 sk1 sk2 + λα(α + 1)smax(k1 ,k2 ) φ(u),

(8.3.19)

where φ(u) denotes the integral in (8.3.18). This quantity exists for the marked process without any further restrictions, but the second-moment mea s sure does not exist for the ground process unless the sum k1 k2 max(k1 ,k2 ) = k (2k + 1)sk converges, equivalent to the existence of a second moment for the parent mark distribution. When this condition is satisfied, the bivariate mark kernel at separation u, Π2 (k1 , k2 | u), can be found by renormalizing [i.e. by dividing (8.3.19) by the double sum just described]. Even if we sum out one variable, the marginal distribution of the other does not reduce to the stationary mark distribution because of the intervention of the second term. Expressions for the mark covariance and mark correlation at separation u can be found from the bivariate mark kernel: details are left to the reader. The assumption of i.i.d. marks within a cluster implies that there is no dependence on the separation u except through the term φ(u). This implies in particular that the bivariate mark kernel is symmetric in u. It would, however, be quite natural in some modelling situations to incorporate an explicit dependence of the mark distribution on the distance from the cluster centre, in which case a further dependence on u would arise, causing the bivariate distribution to be asymmetric in general. MPPs can give rise to a diverse range of second-order characteristics (see e.g. Stoyan, 1984; Isham, 1985): the ‘simple’ case of a finite mark space in Proposition 8.3.I bears this out. Schlather (2001) gives a valuable survey. From a theoretical viewpoint, some of the most interesting applications of stationary MPPs are to situations where the marks are not merely statistically dependent on the past evolution of the process but are direct functions of it. As an extreme case, the mark at time t can be taken as the whole past history of the point process up to time t. This idea lies behind one approach to the Palm theory of Chapter 13. The following elementary example gives some insight into this application. Example 8.3(f) Forward recurrence times. Assume there is given a simple stationary point process on R, and associate with any point ti of the process the length Li = inf{u: N (ti − u, ti ) ≥ 1} of the previous interval. Then, the MPP consisting of the pairs (ti , Li ) is stationary. Assuming that N has a

328


finite mean density m, it follows from Proposition 8.3.II and (8.3.16) that a stationary probability distribution ΠL (·) exists for the interoccurrence times. The integral relation (8.3.14) then leads to important relations involving ΠL (·) as for example in the following deduction of the distribution of the stationary forward recurrence time random variable. The distance of the point nearest to the right of the origin, t1 say, has this distribution, with t1 = inf{ti : ti > 0}. If i is the index of this point, then 0 < t1 = ti ≤ Li . Take any bounded measurable function g(·) of bounded support and define h(t, κ) = g(t) if 0 ≤ τ ≤ κ, h(t, κ) = 0 otherwise. The left-hand side of (8.3.14) equals R×R+

h(t, κ) M1 (dt × dκ) = E * =E

R×R+

h(t, κ) N (dt × dκ) +

h(ti , κi ) = E[g(t1 ]

i:t)i>0

since h(t, κ) = 0 for t > t1 ; evaluating the right-hand side as below gives E[g(t1 )]

=m

∞

g(u) du 0

∞

ΠL (dκ) = m u

∞

[1 − FL (u)]g(u) du,

0

t where FL (t) = 0 ΠL (du) is the distribution function for the interval length. Since g is an arbitrary measurable function of bounded support, we can for example choose g(t) = I(0,x] (t) and obtain Pr{t1 ≤ x} on the left-hand side, x equal to m 0 [1 − FL (u)] du from the right-hand side; thus, the distribution for the point t1 immediately following the origin (i.e. the distribution for the forward recurrence time) has the density f1 (x) = m[1 − FL (x)] = [1 − FL (x)]/µL , where µL is the mean interval length [see (4.2.3) and Proposition 4.2.I]. This simple derivation of a Palm–Khinchin relation uses an argument similar to the original work of Palm (1943). Example 8.3(g) Vehicles on a road. We consider a spatially stationary distribution of cars along a long straight road, the car at xi having a (constant) velocity vi , with vi = vj in general. Our aim is to determine the evolution in time, if any, of characteristics of the process. The family of transformations that concerns us is given by (xi , vi ) → (xi + tvi , vi )

(real t).

Denote by mt , Πt (·), and ct (u, v1 , v2 ) the mean density, the stationary (in space) velocity distribution, and the spatial covariance density at time t. We can refer moments at time t to moments at time 0 on account of the following

8.3.


329

reasoning. From (8.3.14), we have for the space–velocity mean density at time t, Mt (dx × dv) say, h(x, v) Mt (dx × dv) = h(x + tv, v)m0 dx Π0 (dv) R×R+ R×R+ h(y, v)m0 dy Π0 (dv), = R×R+

so that the mean vehicle density and velocity distribution remain constant in time whatever their initial forms. Applying a similar argument to the second-order integrals implies that if the covariance densities ct (u, v1 , v2 ) exist for t = 0, they exist for all t > 0 and are given by ct (u, v1 , v2 ) = c0 u + t(v2 − v1 ), v1 , v2 . The asymptotic covariance properties of ct (·) at t → ∞ thus depend on the behaviour of c0 (u, v1 , v2 ) for large u. In most practical cases, a mixing condition holds and implies that for all v1 , v2 , c0 (u, v1 , v2 ) → 0 as |u| → ∞. Under these conditions, any correlation structure tends to die out, this being an illustration of the ‘Poisson tendency’ of vehicular traffic (Thedeen, 1964). This example can also be treated as a line process and extended in various ways (see e.g. Bartlett, 1967; Solomon and Wang, 1972).

Exercises and Complements to Section 8.3 8.3.1 Detail the argument that establishes Proposition 8.3.I by applying Proposition 8.1.I to the linear combinations ai ξi (·). ˘ ij (·)) of nonnegative measures be positive-definite as in 8.3.2 Let the matrix (M (8.3.1). Show that the matrix of Fourier transforms (Fij (·)) consists of nonnegative measures with the same positive-definite property. 8.3.3 Consider a multivariate Neyman–Scott process in which cluster centres occur in time at rate µc and cluster members may be of different types with joint density p(k, u) = πk fk (u), πk = 1 = fk (u)du (k = 1, . . . , K). Find expressions, generalizing those of Example 6.3(c), for the means and covariance densities of the different component streams and the corresponding multivariate Bartlett spectra. 8.3.4 Consider a cluster process in which the cluster centres form a simple stationary point process with mean density λc and Bartlett spectrum with density γ11 (·), while the clusters have the Hawkes branching structure of Example 8.3(c). Regard the resultant process as the output of a system with the cluster centre process the input and the generation of cluster members representing a type of positive feedback with the linear structure characteristic of a Hawkes process. (a) Arguing from the general relations for the second-order properties of a cluster process, show that the output process here has the spectral density γ22 (ω) =

[λc /(2π)]((1 − ν)−1 − 1) + γ11 (ω) , |1 − µ (ω)|2

330


(0), which [see (8.3.11)] is a different generalization of (8.2.10). where ν = µ The only contributions to the cross-covariance terms are from the cluster centre to cluster members, leading to c12 (u) = λc m1 (u | 0) (see the notation in Exercise 5.5.6), and thus γ12 (ω) =

λc /(2π) −1

(1 − µ (ω))

= γ21 (−ω).

(b) By specializing γ11 (·), more specific examples of input/output systems are obtained. For example, the input may be a Cox process directed by a continuous nonnegative process X(·), in which case we have a continuous input process X(·) causally affecting an output point process. If, moreover, X(·) is itself a shot-noise process generated by some primary point process, we recover a somewhat more general case of mutually exciting point processes. 8.3.5 Explicitly state the mappings and show their use in applying the factorization Lemma A2.7.II to prove Proposition 8.3.II. 8.3.6 MPPs with infinite mean ground density. Suppose given a countable infinity of stationary (R × K)-valued MPPs Nj , j = 1, 2, . . . , defined on some common probability space and K ⊆ R+ . Suppose that Nj has finite mean density mj and each point of Nj has the positive-valued mark κj , say, and there is a probability distribution {πj } with πj > 0 for j = 1, 2, . . . such that π m = ∞. j j j (a) Let the MPP N equal Nj with probability πj for j = 1, 2, . . . . Then N is nonergodic: limT →∞ N ((0, T ] × K)/T = limT →∞ Nj ((0, T ] × K)/T = mj with probability πj . Since each Nj is well defined, so is N , and its mean ground density equals π m = ∞. Denoting a realization of N j j j by {(xi , κi )}, consider the stationary random measure ξ(A) = κ. xi ∈A i κ is independent of j a.s., and that Show that ξ(·) is nonergodic unless m j j its mean density equals π m κ , which can be finite or infinite. j j j j (b) Now suppose that the Nj are mutually independent marked Poisson processes. (i) Show that the superposition of any specified finite collection of the Nj is an MPP with finite mean density. (ii) Let J be a countably infinite subset of {1, 2, . . .}, and consider N = j∈J Nj . Then, N is not an MPP because N ((0, 1]×K) = ∞ a.s., contradicting the finiteness condition in Definition 6.4.I(a). (c) Suppose in (b) that the Nj are mutually independent simple stationary MPPs (not necessarily Poisson). Do the conclusions (i) and (ii) continue to hold? 8.3.7 Let the bivariate simple Poisson process model of Example 8.3(a) be stationary so that it can be described in terms of three rate functions µ1 , µ2 , µ3 and a distribution function G(·) of the signed distance between a pair of related points, taking a type 1 point as the initial point. Show that in terms of these quantities, m 1 = µ1 + µ3 , m 2 = µ2 + u 3 , ˘[2] (−du; 2, 1). ˘ C[2] (du; 1, 2) = µ3 G(du) = C

8.4.

Spectral Representation

331

Use the p.g.fl. or otherwise to show that when X = R, the joint distribution of the distances T1 and T2 from an arbitrary origin to the nearest points of types 1 and 2, respectively, is given by log Pr{T1 > x, T2 > y}

x+y

= −2m1 x − 2m2 y + µ3

( min(x, y − v) − max(−x, −y − v)) G(dv), −x−y

while the joint distribution of the forward recurrence times T1+ , T2+ from the origin to the nearest points in the positive direction is given by log Pr{T1+ > x, T2+ > y}

y

= −m1 x − m2 y + µ3

( min(x, y − v) − max(0, −v)) G(dv). −x

Consider extensions to the case X = Rd . 8.3.8 Gauss–Poisson process with asymmetric bivariate mark distribution. In a marked process of correlated pairs (marked Gauss–Poisson process), suppose that the joint distribution of the marks corresponding to the two points in a pair depends on the separation of the two points and that the mark of the first occurring point in the pair is (say) always the larger. Construct an explicit example for which the bivariate mark distribution at separation u depends explicitly on u and is asymmetric. 8.3.9 Bivariate forward recurrence time. Extend the argument of Example 8.3(f) to the case of a bivariate point process by using an MPP in which the mark at a point ti of the process is of the form (ji ; L1i , L2i ), where ji is the type of the point and L1i , L2i are the backward occurrence times to the last points of types 1 and 2, respectively. Obtain a bivariate extension of the Palm– Khinchin equations, and compare these with the extensions to nonorderly point processes discussed in (3.4.14). Hence or otherwise, obtain expressions for the joint distributions of the intervals between an arbitrary point of type i (i = 1, 2) and the next occurring points of types 1 and 2 in Example 8.3(a). [Daley and Milne (1975) use a different approach that exploits methods similar to those of Chapter 3].

8.4. Spectral Representation We take up next the possibility of developing a Cramér-type spectral representation for stationary point processes and random measures. In R, such a representation is essentially a corollary of the spectral representation for processes with stationary increments given by Doob (1949) and for stationary interval functions given by Brillinger (1972). No essentially new points arise, although minor refinements are possible as a result of the additional properties available for p.p.d. measures. We give a brief but essentially selfcontained account of the representation theory for random measures in Rd following the general lines of the approach in Vere-Jones (1974). The relation to spectral representations for stationary generalized processes is discussed in Daley (1971) and Jowett and Vere-Jones (1972).

332


In order to be consistent with the representation theory for continuous-time processes, we work throughout with the mean-corrected process ξ 0 (dx) = ξ(dx) − m dx

(8.4.1)

with zero mean, where ξ is a second-order stationary random measure with mean density m. Thus, we are concerned with properties of the Bartlett spectrum. An equivalent and perhaps slightly more direct theory could be built up from the properties of ξ(·) and the second-moment measure: the differences are outlined in Exercise 8.4.1. The essence of the Cramér representation is an isomorphism between two Hilbert spaces, one of random variables defined on a probability space and the other of functions on the state space X = Rd . In the present context, we use the notation L2 (ξ 0 ) to denote the Hilbert space of (equivalence classes of) random variables formed from linear combinations of the second-order random variables ξ 0 (A) (bounded A ∈ BX ) and their mean square limits, while L2 (Γ) denotes the Hilbert space of (equivalence classes of) measurable functions square integrable with respect to Γ. Since Γ is not in general totally finite, we cannot apply directly the theory for mean square continuous processes. Rather, there are two possible routes to the required representations: we can exploit the results already available for continuous processes by means of smoothing techniques such as those used in Section 8.5, or we can develop the theory from first principles, using appropriate modifications of the classical proofs where necessary. We adopt the latter approach, although we only sketch the arguments where they directly mimic the standard theory. A convenient starting point is the following lemma in which S again denotes the space of functions of rapid decay in Rd . Lemma 8.4.I. Given any boundedly finite measure Γ in Rd , the space S is dense in L2 (Γ). Proof. The result is a minor modification of standard results [see e.g. Kingman and Taylor (1966, p.131) and Exercise 8.4.2]. The key step in establishing the isomorphism between the spaces L2 (ξ 0 ) and L2 (Γ) is a special case of Proposition 8.6.IV, which, with the notation f (x) ξ 0 (dx), (8.4.2) ζf = Rd

where f is a bounded Borel function of bounded support, can be stated in the form f˜ = |f˜(ω)|2 Γ(dω) = f (x)f (x + u) du C˘2 (dx) L2 (Γ)

Rd

= var(ζf ) = ζf L2 (ξ0 ) .

Rd

Rd

(8.4.3)

A first corollary of this equality of norms is the following counterpart of the lemma above.

8.4.


333

Lemma 8.4.II. For ψ ∈ S, the random integrals ζψ = dense in L2 (ξ 0 ).

Rd

ψ(x) ξ 0 (dx) are

Proof. It is enough to show that for any given bounded A ∈ B(Rd ), ξ 0 (A) can be approximated in mean square by elements ζψn with ψn ∈ S. Working from the Fourier transform side, it follows from (8.4.3) that IÃ ∈ L2 (Γ) and thus by Lemma 8.4.I that IÃ can be approximated by a sequence of functions in S. Now S is invariant under the Fourier transform map, so this sequence can be written as ψñ with ψn ∈ S. Applying (8.4.3) with ψ = IA ψn leads to IÃ − ψñ L2 (Γ) = ξ 0 (A) − ζψn L2 (ξ0 ) . By construction, the left-hand side → 0 as n → ∞, and hence also the righthand side → 0, which from our opening remark is all that is required. Lemmas 8.4.I and 8.4.II show that for ψ ∈ S there is a correspondence ψ˜ ↔ ζψ between elements ψ˜ of a set dense in L2 (Γ) and elements ζψ of a set dense in L2 (ξ 0 ). The correspondence is one-to-one between equivalence classes of functions and is norm-preserving. From this last fact, it follows that the correspondence can be extended to an isometric isomorphism between the full Hilbert spaces L2 (Γ) and L2 (ξ 0 ) (see Exercise 8.4.3 for details), thus establishing the following proposition. Proposition 8.4.III. There is an isometric isomorphism between L2 (Γ) and L2 (ξ 0 ) in which, for ψ ∈ S, the integral ζψ in (8.4.2) ∈ L2 (ξ 0 ) and the Fourier transform ψ˜ ∈ L2 (Γ) are corresponding elements. The main weakness of this proposition is that it does not give an explicit Fourier representation of the random measure and associated integrals ζψ . To overcome this deficiency, we adopt the standard procedure of introducing a mean square integral with respect to a certain wide-sense random signed measure with uncorrelated values on disjoint sets. For any bounded A ∈ B(Rd ), let Z(A) denote the random element in L2 (ξ 0 ) ˜ corresponding to ψ(ω) ≡ IA (ω) in L2 (Γ). For disjoint sets A1 , A2 , it follows from the polarized form of (8.4.2) (obtained by expressing inner products in terms of norms) that E Z(A1 ) Z(A2 ) =

Rd

IA1 (ω)IA2 (ω) Γ(dω) = 0,

(8.4.4)

so that the Z(·) are indeed uncorrelated on disjoint sets (or, in the setting of the real line, have orthogonal increments). The definition of a mean square integral with respect to such a family is a standard procedure (see e.g. Doob, 1953; Cramér and Leadbetter, 1967) and leads to the conclusion that for every g ∈ L2 (Γ) the integral g(ω) Z(dω) Rd

334


can be defined uniquely as a mean square limit of integrals of simple functions and can be identified with the unique random variable associated with g in the isomorphism theorem described by Proposition 8.4.III. In particular, for g = ψ˜ ∈ S, the integral below can be identified with the random element ζψ ; that is, ˜ ψ(ω) Z(dω) = Rd

ψ(x) ξ 0 (dx). Rd

Also, referring to the convergence property displayed in the proof of Lemma 8.4.II (and this defines an equivalence relation as noted), the limit relation can be written as ξ 0 (A) = l.i.m. ζψn n→∞

(see e.g. Doob, 1953, p. 8). More generally, it follows from Proposition 8.6.IV and (8.4.3) that the same conclusion holds for any bounded ψ of bounded support. Thus, we have the following result, which is a slight strengthening, as well as an extension to Rd , of the corresponding result in Vere-Jones (1974). Theorem 8.4.IV. Let ξ be a second-order stationary random measure or point process in Rd with Bartlett spectrum Γ. Then, there exists a secondorder wide-sense random measure Z(·) defined on bounded A ∈ B(Rd ) for which (i) EZ(A) = 0 = E[Z(A)Z(B) ] for bounded disjoint A, B ∈ B(Rd ); (8.4.4 ) (ii) var Z(A) = E(|Z(A)|2 ) = Γ(A);

(8.4.5)

to g in the isomor(iii) for all g ∈ L2 (Γ), the random variable ζ corresponding phism of Proposition 8.4.III is expressible as ζ = Rd g(ω) Z(dω); and (iv) for all ψ ∈ S and all bounded measurable ψ of bounded support, ζψ ≡

˜ ψ(ω) Z(dω)

0

ψ(x) ξ (dx) = Rd

a.s.

(8.4.6)

Rd

Observe that in the Parseval relation in (8.4.6) the left-hand side represents the usual random integral defined on a realization by realization basis, whereas the right-hand side is a mean square integral that does not have a meaning in this sense. The two most important classes of functions ψ are covered by the theorem. In Exercise 8.4.4, we indicate how (8.4.6) can be extended to somewhat wider classes of functions and, in particular, (8.4.6) continues to hold whenever ψ is Lebesgue integrable and ψ˜ ∈ L2 (Γ). An alternative approach to the substance of part (iv) of this theorem is simply to define the integral on the left-hand side of (8.4.6) to be equal to the right-hand side there for all ψ˜ ∈ L2 (Γ), but this begs the question as to when this definition coincides with the a.s. definition of the integral used until now. More explicit representation theorems can be obtained as corollaries to (8.4.6). In particular, taking ψ(x) = IA (x), we have the following.

8.4.


335

Corollary 8.4.V. For all bounded A ∈ B(Rd ), IÃ (ω) Z(dω) ξ 0 (A) =

a.s.

(8.4.7)

Rd

We cannot immediately obtain an inversion theorem for Z(·) in this form because the corresponding integral (2π)−d Rd I˜B (−x) ξ 0 (dx) need not exist. The finite integral over UdT presents no difficulties, however, and leads to the second corollary. Corollary 8.4.VI. For all bounded A ∈ Rd that are Γ-continuity sets, 1 Z(A) = l.i.m. (8.4.8) IÃ (−x) ξ 0 (dx). T →∞ (2π)d Ud T Proof. From the theorem, the finite integral in (8.4.8) can be transformed into the expression [for θ = (θ1 , . . . , θd ) and ω = (ω1 , . . . , ωd ) ∈ Rd ] + * d sin T (ωi − θi ) dθ. Z(dω) ωi − θi Rd A i=1 Provided A is a continuity set for Γ, the integrand convolved with IA (ω) converges in L2 (Γ) to IA (ω) as T → ∞ (see Exercise 8.4.5: the proof is straightforward for intervals A but not so direct for general bounded A), and hence the integral converges in mean square to Z(A). In very simple cases, Corollary 8.4.VI can be used to calculate directly the process Z(·) having orthogonal increments. Such an example is given below, partly to illustrate the potential dangers of using the second-order representation for anything other than second-order properties. Example 8.4(a) The Fourier transform of the Poisson process. Let ξ be a Poisson process on R with constant rate λ. Then, it follows from (8.4.8) that T ixa 1 e − eixb N (dx) − λ dx . Z((a, b]) = l.i.m. T →∞ 2πi −T x Consider in particular the process Ua (ω) ≡ Z(ω + a) − Z(ω − a) = l.i.m. T →∞

1 π

T

−T

e−iωx sin ax N (dx) − λ dx . x

Using standard results from Chapter 9 for the characteristic functional of the Poisson process, we find Φ(ω, s) ≡ E exp(isUa (ω) ∞ ise−iωx sin ax ise−iωx sin ax −1− dx exp = exp λ x x −∞ ∞ sin ax 2 sin ax 3 = exp λ − 12 s2 cos ωx dx + O(s3 ) x x −∞ $ # = exp − 12 πλas2 + O(s3 )

336


uniformly in ω [see e.g. Copson (1935, p. 153) for evaluation of the integral]. It follows that the variance of Ua (ω) is proportional to the length of the interval and independent of its location, corresponding to the presumption that Z(·) in this case must be a process with orthogonal and second-order stationary increments. On the other hand, Z(·) clearly does not have strictly stationary increments, for the full form of the characteristic function depends nontrivially on ω. Similarly, it can be checked from the joint characteristic function that Z does not have independent increments. Indeed, as follows from inspecting its characteristic function, Ua (ω) has an infinitely divisible distribution of pure jump type, with a subtle dependence of the jump distribution on a and ω that produces the requisite characteristics of the second-order properties. The spectral representation for stationary random measures and point processes plays a similar role in guiding intuition and aiding computation as it does for classical time series. We illustrate its use below by establishing basic procedures for estimating the Bartlett spectrum in two practically important cases: simple point processes and random (point process) sampling of a stationary continuous process. Further examples arise in Section 8.5, where we examine linear filters and prediction. Example 8.4(b) Finite Fourier transform and point process periodogram. Estimates of the Bartlett spectrum provide a powerful means of checking for periodicity in point process data as well as for investigating other features reflected in the second-order properties. The basic tool for estimating the spectrum is the point process periodogram, defined much as in the continuous case through the finite Fourier transform of the realization of a point process on a finite time interval (0, T ), namely T N (T )

1 − e−iωT JT (ω) = e−iωt [N (dt) − m dt] = e−iωtk − m , (8.4.9) iω 0 k=1

in terms of which the periodogram is then defined as 1 (ω ∈ R). (8.4.10) |JT (ω)|2 IT (ω) = 2πT Express JT (ω) in the form of the left-hand side of (8.4.6) by setting ψ(t) = e−iωt I(0,T ) (t), which is certainly bounded and of bounded support. Then, it follows from Proposition 8.4.IV(iv) that iT (ω −ω) e −1 JT (ω) = Z(dω ) a.s. − ω) i(ω R The orthogonality properties of Z now imply that &2 & iT (ω −ω) &e − 1 && 1 & Γ(dω ) (8.4.11a) E[IT (ω)] = 2πT R & i(ω − ω) & sin 12 T (ω − ω) 2 T = Γ(dω ). (8.4.11b) 1 − ω) 2π R T (ω 2

8.4.


337

If Γ(·) has an atom at ω, then it follows from (8.4.11a) that IT (ω) ∼ T Γ({ω}). On the other hand, if Γ(·) has a continuous density γ(ω ) in a neighbourhood of ω, then it follows from (8.4.11b) that E[IT (ω)] → γ(ω). Thus, the periodogram is an asymptotically unbiased estimate of the spectral density wherever the density exists. The contrast between the two cases is the basis of tests for periodic effects, meaning here some periodic fluctuation in the rate of occurrence of events. Consistency is another story, however, and some degree of smoothing must be introduced to obtain consistent estimates of the spectral density. The theory here parallels the standard theory except insofar as the observations are not Gaussian and some spectral mass is carried at arbitrarily large frequencies. The latter feature is a consequence of assuming that the points {tk } of the process are observed with complete precision, which is a fiction in any real context: in reality, only limited precision is possible, amounting to some smoothing or rounding of the observations, which then induces a tapering of the spectrum at very high frequencies. Nevertheless, the lack of any natural upper bound to the observed frequency range, even from a finite set of observations, causes difficulties in tackling questions such as the detection and estimation of an unknown periodicity modulating the occurrence times of the observed points. Indeed, the very definition of such a modulation, except for specific models such as the Poisson process (when it can appear as a periodic modulation of the intensity), is a matter of some difficulty. The crux of the matter for the spectral theory is that, whatever the form of modulation may be, it should induce a periodic variation in the reduced covariance measure. Vere-Jones and Ozaki (1982) discuss some of these issues in simple special contexts; the general problem of testing for unknown frequencies in point process models appears to lack any definitive treatment. Brillinger (1978, 1981) gives a systematic overview of the differences between ordinary time series and point process analogues. Example 8.4(c) Random sampling of a random process. A situation of some practical importance arises when a stationary continuous-time stochastic process X(t) is sampled at the epochs {ti } of a stationary point process. The resultant process can be considered in two ways, either as a discrete-time process Yi = X(ti ) or as a random measure with jump increments ξ(dt) = X(t) N (dt). Neither operation is linear, but the second equation is just a multiplication of the two processes and leads to the more tractable results. Neither N (·) nor ξ(·) is a process with zero mean; to express the latter as a process with zero mean, suppose for simplicity that X(·) has zero mean, and then write ξ(dt) = X(t) N 0 (dt) + mX(t) dt, where N 0 (dt) = N (dt) − m dt and m = EN (0, 1] is the mean rate of the

338


sampling process. Proceeding formally leads to ˜ − v) ZX (du) N (dv) + m φ(u) ˜ ZX (du), φ(t) ξ(dt) = φ(u R

R

R

R

corresponding to a representation of the measure Zξ as a convolution of ZX and ZN with an additional term for the mean. Leaving aside the general case, suppose that the processes X(·) and N (·) are independent. Then we find var

φ(t) ξ(dt) R 2 ˜ − v)|2 γ (du) γ (dv) + m2 |φ(u)| ˜ = |φ(u γN (du), X N R

R

R

from which we deduce that γX (dω − u) γN (du) + m2 γX (dω). γξ (dω) = R

Hence, for the covariance measures we have ˘2 (du). C˘ξ (du) = c˘X (u) m2 du + C˘N (du) = c˘X (u) M Of course, the last result can easily be derived directly by considering E X(t) N (t, t + dt] X(t + u) N (t + u, t + u + du] . In practice, one generally must estimate the spectrum γX (·) given a (finite portion of a) realization of ξ(·). When N is a Poisson process at rate m, γξ (dω) = (m/2π)(var X) dω + m2 γX (dω), so γX can be obtained quite easily from γξ . In general, however, a deconvolution procedure may be needed, and the problem is complicated further by the fact that the spectral measures concerned are not totally finite. Consequently, numerical Fourier transform routines cannot be applied without some further manipulations [see Brillinger (1972) for further details]. Only partial results are available for the extension of the spectral theory to random signed measures. One approach, which we outline briefly below, follows Thornett (1979) in defining a second-order random measure as a family of random variables {W (A)}, indexed by the Borel sets, whose first and second moments satisfy the same additivity and continuity requirements as the firstand second-moment measures of a stationary random measure. The resulting theory may be regarded as a natural generalization to Rd of the theory of random interval functions developed by Bochner (1955) and extended and applied to a statistical context by Brillinger (1972).

8.4.


339

Definition 8.4.VII. A wide-sense second-order stationary random measure on X = Rd is a jointly distributed family of real- or complex-valued random variables {ξ(A): A ∈ BX } satisfying the conditions, for bounded A, {An } and B ∈ BX , (i) Eξ(A) = m(A), var ξ(A) < ∞; (ii) var((Sx ξ)(A)) = var ξ(Tx A) = var ξ(A); (iii) ξ(A ∪ B) = ξ(A) + ξ(B) a.s. for disjoint A, B; and (iv) ξ(An ) → 0 in mean square when An ↓ ∅ as n → ∞. If the random variables ξ(·) here are nonnegative, then (iii) reduces to the first part of (6.1.2) and implies that in (iv) the random variables ξ(An ) decrease monotonically a.s.; that is, ξ(An+1 ) ≤ ξ(An ) a.s., so that (iv) can be strengthened to ξ(An ) → 0 a.s. when An ↓ ∅ as n → ∞ [see the second part of (6.1.2)]. We then know from Chapter 9 that there exists a strict-sense random measure that can be taken as a version of ξ(·) so that nothing new is obtained. Thus, the essence of the extension in Definition 8.4.VII is to random signed measures. For the sequel, we work only with the mean corrected version, taking m = 0 in the definition. Given such a family then, we can always find a Gaussian family with the same first- and second-moment properties: the construction is standard and needs no detailed explanation (see Doob, 1953; Thornett, 1979). For example, the Poisson process, corrected to have zero mean, has var ξ(A) = λ(A), where λ is the intensity; this function is the same as the variance function for the Wiener chaos process in Chapter 9. While the definition refers only to variances, covariances are defined by implication from the relation, valid for real-valued ξ(·), 2 cov ξ(A), ξ(B) = var ξ(A ∪ B) + var ξ(A ∩ B) − var ξ(A \ B) − var ξ(B \ A), which is readily verified first for disjoint A and B and then for general A and B by substituting in the expansion of cov ξ(A), ξ(B) = cov ξ(A ∩ B) + ξ(A\B), ξ(A ∩ B) + ξ(B\A) . Although we can obtain in this way a covariance function C(A × B) defined on products of bounded A, B ∈ BX , it is not obvious that it can be extended to a signed measure on B(X (2) ). Consequently, it is not clear whether or not a covariance measure exists for such a family. When it does, the further theory can be developed much as earlier. Irrespective of such existence, it is still possible to define both a spectrum for the process and an associated spectral representation. Thus, for any bounded Borel set A, consider the process XA (x) ≡ ξ(Tx A). Mean square continuity follows from condition (iv), so XA (·) has a spectral measure ΓA (·), and we can define Γ(dω) = |IÃ (ω)|−2 ΓA (dω)

340


for all ω such that |IÃ (ω)| = 0. Since we cannot ensure that |IÃ (ω)| = 0 for all ω, some care is needed in showing that the resultant measure Γ(·) can in fact be consistently defined for a sufficiently rich class of sets A [one approach is outlined by Thornett (1979) and given as Exercise 8.4.6]. Just as before, the measure Γ is translation-bounded and hence integrates (1 + ω 2 )−1 , for example. On the other hand, it is not positive-definite in general and not all the explicit inversion theorems can be carried over. Nevertheless, for all bounded A ∈ BX , we certainly have var ξ(A) = |IÃ (ω)|2 Γ(dω) (8.4.12) and its covariance extension

cov ξ(A), ξ(B) =

IÃ (ω) I˜B (ω) Γ(dω).

(8.4.13)

Since the indicator functions are dense in L2 (Γ), more general integrals of the form φ(x) ξ(dx) can be defined as mean square limits of linear combinations of the random variables ξ(A), at least when φ˜ ∈ L2 (Γ). For such integrals, the more general formulae 2 ˜ var φ(x) ξ(dx) = |φ(ω)| Γ(dω)

and cov

φ(x) ξ(dx),

˜ ψ(ω) ˜ ψ(x) ξ(dx) = φ(ω) Γ(dω)

are available, but it is not clear whether the integrals make sense other than in this mean square sense. As noted earlier, it is also an open question as to whether Γ is necessarily the Fourier transform of some measure, which we could then interpret as a reduced covariance measure. The isomorphism result in Proposition 8.4.III can be extended to this wider context with only minor changes in the argument: it asserts the isomorphism between L2 (X) and L2 (Γ) and provides a spectral representation, for bounded A ∈ BX , a.s. (8.4.14) ξ(A) = IÃ (ω) Z(dω) just as in the previous discussion. To summarize, we have the following theorem of which further details of proof are left to the reader. Theorem 8.4.VIII. Let {ξ(·)} be a wide-sense second-order stationary random measure as in Definition 8.4.VII. Then, there exists a spectral measure Γ(·) and a process Z(·) of orthogonal increments with var Z(dω) = Γ(dω) such that (8.4.12–14) hold.

8.4.


341

Exercises and Complements to Section 8.4 8.4.1 Representation in terms of the second-moment measure. Show that the effect of working with the Fourier transform of the second moment rather than the Bartlett spectrum would be to set up an isomorphism between the spaces L2 (ξ) generated by all linear combinations of the r.v.s ξ(A) and L2 (ν), where ˘ 2 . Show that the representation ν is the inverse Fourier transform of M

˜ φ(ω) Z1 (dω)

φ(x) ξ(dx) = Rd

Rd

holds for functions φ in a suitably restricted class, where Z1 (A) = mδ0 (A) + Z(A), and Z and Z1 differ only by an atom at ω = 0. 8.4.2 Let Γ be a nontrivial boundedly finite measure. Show the following: (a) Simple functions of the form a I [bounded Ak ∈ B(Rd )] are dense k k Ak in L2 (Γ). (b) For bounded A ∈ B(Rd ), there exist open sets Un ∈ B(Rd ) with Un ⊇ A, Γ(Un ) ↓ Γ(A). (c) Any such Un is the countable union of hyper-rectangles of the form {αi < xi ≤ βi , i = 1, . . . , d}. (d) Indicator functions on such hyper-rectangles can be approximated by sequences of infinitely differentiable functions of bounded support. Now complete the proof of Lemma 8.4.I. 8.4.3 Given ψ˜ ∈ L2 (Γ), choose ψn ∈ S such that ψ˜ − ψ L2 (Γ) → 0 (n → ∞), and deduce that {Zψn } is a Cauchy sequence in L2 (ξ 0 ). Show that there is a unique r.v. ζ ∈ L2 (ξ 0 ) such that Zψn → ζ in mean square. Interchange the roles of L2 (Γ) and L2 (ξ 0 ) and deduce the assertion of Proposition 8.4.III. 8.4.4 Show that (8.4.6) can be extended to all L1 functions φ such that φ˜ ∈ L2 (Γ). [Hint: The left- and right-hand sides can be represented, respectively, as an a.s. limit of integrals of bounded functions of bounded support and as a mean square limit. When both limits exist, they must be equal a.s. This argument establishes a conjecture in Vere-Jones (1974).] 8.4.5 Establish the following properties of the function hT (ω) = ω −1 sin ωT (they are needed in a proof of Corollary 8.4.IV). ∞ (a) −∞ hT (ω) dω = π. (b) For any continuous function φ with bounded support, the function

∞

φT (ω) ≡

φ(ω − u)hT (u) du → φ(ω)

pointwise as T → ∞

−∞

[this is an application of Fourier’s single integral (see Zygmund, 1968, Section 16.1)]. Show that the result still holds if only φ ∈ L1 (ξ) and φ is of bounded variation in any closed interval contained in its support. (c) φT (ω) → φ(ω) in L2 (Γ) for any p.p.d. measure (or for any Bartlett spectrum) Γ. [Hint: |φT (ω)| ≤ constant/|ω| for large |ω| while supω |φT (ω)| < ∞; these properties are enough to ensure that |φT (ω)|2 ≤ g(ω) for some Γ-integrable function g.]

342

8. Second-Order Properties of Stationary Point Processes (d) Interpret the convergence in (c) as

|φT (ω)|2 Γ(dω) =

R

hT (ω − v)φ(v) dv Γ(dω)

hT (ω − u)φ(u) du

R R

hT (ω − u)hT (ω − v) Γ(dω)

φ(u) φ(v) du dv

=

R

R

= R2

→

R2

=

R

R

φ(u) φ(v) Γ∗T (du × dv) φ(u) φ(v) Γ∗ (du × dv)

|φ(ω)|2 Γ(dω),

R ∗ ∗ 2 where ΓT (du × dv) and Γ are measures in B(R ), the former with density R hT (ω − u) hT (ω − v) Γ(dω), while the latter reduces to Γ along the diagonal u = v. These results are enough to establish that Γ∗T → Γ∗ vaguely in R2 and hence that a similar result holds when φ(·) is replaced by the indicator function of a bounded Borel set in R1 that is a continuity set for Γ.

8.4.6 Show that for Γ to be the spectral measure of a wide-sense second-order stationary random measure, it is necessary and sufficient that Γ integrate all functions |IÃ (ω)|2 for bounded Borel sets A. Deduce that any translation-bounded measure can be a spectral measure. [Hint: Use a Gaussian construction for the sufficiency; then use Lin’s lemma. See also Thornett (1979).] 8.4.7 (a) Show that if a wide-sense second-order stationary process has a reduced co˘ ˘ variance measure C(·), then C({0}) = limT →∞ Γ((−T, T ])/(2T ) continues to hold (see Theorem 8.6.III). (b) Use Exercise 8.2.4 to show that not all spectral measures are transforms; that is, not all wide-sense processes have an associated reduced covariance measure (see also Exercise 8.6.3).

8.5. Linear Filters and Prediction One of the most important uses of spectral representation theory is to obtain spectral characteristics of processes acted on by a linear filter, meaning here any time-invariant linear combination of values of the process or any mean square limit of such combinations. This use carries over formally unchanged from mean square continuous processes to second-order point processes and random measures and includes the procedures for developing optimal linear predictors for future values of the process. Obtaining the precise conditions for these extensions and their character requires some care, however, and forms the main content of the present section.

8.5.

Linear Filters and Prediction

343

Let ξ(·) be a second-order stationary random measure and ψ ∈ L1 a smoothing function; consider the smoothed process defined by ∞ X(t) = ψ(t − u) ξ(du). (8.5.1) −∞

Substituting from the Parseval relation (8.4.6) and recalling that the Fourier iωt ˜ transform of the shifted function ψ(t − u) is ψ(−ω)e , we find ∞ ˜ X(t) = eiωt ψ(−ω) Z(dω). (8.5.2) −∞

The spectrum ΓX (·) of the transformed process is 2 ˜ Γ(dω). ΓX (dω) = |ψ(−ω)|

(8.5.3)

This will be totally finite, which implies that X(·) is a mean square continuous process, provided ψ˜ ∈ L2 (Γ). The relation (8.5.1) can be interpreted even more broadly; for example, if A(·) is a totally finite measure, the convolution A ∗ ξ still defines a.s. a random measure and (8.5.2) and (8.5.3) continue to hold. Thus, (8.5.1) continues to make sense, with a generalized function interpretation of ψ, provided the outcome defines a.s. a random measure. However, the situation becomes decidedly more complex when, as is often necessary in applications to prediction, signed measures intervene; then at best the wide-sense theory can be used, and the character of the filtered process, in a realization-by-realization sense, has to be ascertained post hoc. Example 8.5(a) Binning. A special case of practical importance arises when X = R and the measure ξ is ‘binned’; that is, integrated over intervals of constant length ∆, say. Considering first the continuous-time process X(t) ≡ ξ t − 12 ∆, t + 21 ∆], (8.5.2) yields

∞

X(t) = −∞

eiωt

sin 12 ω∆ Z(dω), 1 2ω

hence

ΓX (dω) =

sin 12 ω∆ 1 2ω

2 Γ(dω).

It is commonly the case that the binned process is sampled only at the lattice points {n∆: n = 0, ±1, . . .}. The sampled process can then be represented in the aliased form sπ/∆ ∞ 2kπ

einθ ZX Y (n) ≡ X(n∆) = + dθ . ∆ 0 k=−∞

Taking ∆ as the unit of time, we see from this representation that the discretetime process {Y (n)} has spectral measure GY (·) on (0, 2π] given by GY (dθ) = sin2 θ

∞

Γ(2kπ + dθ) . (θ + 2kπ)2

k=−∞

(8.5.4)

344


In the simplest case of a Poisson process, Γ(dω) = [µ/(2π)] dω, so that GY (dθ) = sin2 θ

∞

µ [µ/(2π)] dθ dθ = (θ + 2kπ)2 2π

k=−∞

since the infinite series is just an expansion of cosec2 θ. This reduction reflects the fact that the random variables Y (n) are then independent with common variance µ. Binning is widely used in practical applications of time series methods to point process data, and even where it is not explicitly invoked, it is present implicitly in the rounding of observations to a fixed number of decimal places. Indeed, the point process results themselves can be regarded as the limit when the binsize approaches zero and the character of the process Y (n) approaches that of a sequence of δ-functions in continuous time. See e.g. Vere-Jones and Davies (1966) and Vere-Jones (1970), where these ideas are applied in the earthquake context. Perhaps the most important examples of linear filtering come in the form of linear predictions of a timeseries or point process. By a linear predictor we t mean a predictor of the form −∞ f (t − u) ξ(du); that is, a linear functional of the past, with the quantity to be predicted a linear functional of the future. In the point process case, the problem commonly reduces to predicting, as a linear functional of the past, the mean intensity at some time point in the future. When the process has a mean square continuous density, this corresponds exactly to the classical problem of predicting a future value of the process as a linear functional of its past. Thus, our task is essentially to check when the classical procedures can be carried over to random measures and to write out the forms that they take in random measure terms. It is important to contrast the linear predictors obtained in this way with the conditional intensity functions we described in Chapter 7. The conditional intensity function comprises the best nonlinear predictor of the mean rate at a point just ahead of the present. It is best out of all possible functionals of the past, linear or nonlinear, subject only to the measurability and nonanticipating characteristics described in Chapter 7. The linear predictors are best out of the more restricted class of linear functionals of the past. They are difficult to use effectively in predicting nonlinear features such as a maximum or the time to the next event in a point process. On the other hand, they perform well enough in predicting large-scale features where the law of large numbers tilts the distributions toward normality. They are generally easy to combine and manipulate and can sometimes be obtained when the full conditional intensity is inaccessible. The Wold decomposition theorem plays an important role in finding the best linear predictor for mean square continuous processes, and we start with an extension of this theorem for random measures. As in Section 8.4, we use ξ and ξ 0 to denote a second-order stationary random measure and its zero mean

8.5.


345

form, respectively, with the additional understanding that X = R. Since the results to be developed depend only on the spectral representation theorems, ξ can be either a strict- or wide-sense random measure. We continue to use L2 (ξ 0 ) to denote the Hilbert space of equivalence classes of random variables formed from linear combinations of ξ 0 (A) for bounded A ∈ B and their mean square limits. Similarly, L2 (ξ 0 ; t) denotes the Hilbert space formed from ξ 0 (A) with the further constraint that A ⊂ (−∞, t]. Definition 8.5.I. The second-order strict- or wide-sense stationary random measure ξ is deterministic if L (ξ 0 ; t) = L2 (ξ 0 ) and purely nondeter2 t∈R 0 ministic if t∈R L2 (ξ ; t) = {0}. The following extension of Wold’s theorem holds (Vere-Jones, 1974). Theorem 8.5.II. For any second-order stationary random measure ξ, the zero mean process ξ 0 can be written uniquely in the form ξ 0 = ξ10 + ξ20 , where ξ10 and ξ20 are mutually orthogonal, stationary, wide-sense zero-mean random measures, and ξ10 is deterministic and ξ20 purely nondeterministic. Proof. Again we start from the known theorems for mean square continuous processes [see e.g. Cramér and Leadbetter (1967), especially Chapters 5–7] and use smoothing arguments similar to those around (8.5.1) to extend them to the random measure context. To this end, set t X(t) = e−(t−u) ξ 0 (du), (8.5.5) −∞

where the integral can be understood, whether ξ 0 is a strict- or wide-sense random measure, as a mean square limit of linear combinations of indicator functions. These indicator functions can all be taken of sets ⊆ (−∞, t], so we have X(t) ∈ L2 (ξ 0 ; t), and more generally, X(s) ∈ L2 (ξ 0 ; t) for any s ≤ t, so L2 (X; t) ⊆ L2 (ξ 0 ; t). To show that we have equality here, we write t+h X(t + h) − e−h X(t) − ξ 0 (t, t + h] = [e−(t+h−u) − 1] ξ 0 (du), t iωh ∞ − e−h e eiωh − 1 = Z(dω), eiωt − 1 + iω iω −∞ where Z is the process of orthogonal increments associated with ξ 0 as in Theorem 8.4.IV. Subdividing any finite interval (a, a + ∆] into n subintervals of length h = ∆/n, we obtain n

X(a + kh) − e−h X a + (k + 1)h − ξ 0 (a, a + ∆] k=1

∞

= −∞

!

n

k=1

" e

iω(a+kh)

eiωh − 1 eiωh − e−h − 1 + iω iω

Z(dω).

346


The variance of the left-hand side therefore equals &2 & ∞ sin 12 ω∆ 2 && eiω − 1 && Γ(dω) −h 1 − e − . iω & 1 + ω 2 sin 12 ωh & −∞ The measure (1 + ω 2 )−1 Γ(dω) is totally finite (see Exercise 8.6.5), the term | · |2 is uniformly bounded in ω by 4h2 and for fixed ω it is o(h2 ) as h → 0, and the term in braces is bounded by (∆/h)2 and for fixed ω equals const. × h−2 (1 + o(1)) as h → 0. The dominated convergence theorem can therefore be applied to conclude that this variance → 0 as h → 0 and hence that ξ 0 (a, b] can be approximated in mean square by linear combinations of {X(t): t ≤ b}. This shows that L2 (ξ 0 ; t) ⊆ L2 (X; t), and thus L2 (ξ 0 ; t) = L2 (X; t) must hold. The Wold decomposition for X(t) takes the form X(t) = X1 (t) + X2 (t), where X1 (·) is deterministic and X2 (·) purely nondeterministic. The decomposition reflects an orthogonal decomposition of L2 (X), and hence of L2 (ξ 0 ) also, into two orthogonal subspaces such that X1 (t) is the projection of X(t) onto one and X2 (t) the projection onto the other. Then ξ10 (A) and ξ20 (A) may be defined as the projections of ξ 0 (A) onto these same subspaces. Furthermore, ξ10 (a, b] and ξ20 (a, b] can be expressed as mean square limits of linear combinations of X1 (t) and X2 (t) in exactly the same way as ξ 0 (a, b] is expressed above in terms of X(t): the deterministic and purely nondeterministic properties of X1 (·) and X2 (·), respectively, carry over to ξ10 (·) and ξ20 (·). Uniqueness is a consequence of the uniqueness of any orthogonal decomposition. To verify the additivity property of both ξ10 (·) and ξ20 (·), take a sequence {An } of disjoint bounded Borel sets with bounded union. From the a.s. countable additivity of ξ 0 , which is equivalent to property (iv) of Definition 8.4.VII, we have ! ∞ " ∞

0 ξ An = ξ 0 (An ) a.s.; n=1

hence,

! ξ10

∞ n=1

" An

−

∞

n=1

n=1

! 0

ξ (An ) =

ξ20

∞ n=1

" An

−

∞

ξ(An )

a.s.

n=1

Since the expressions on the two sides of this equation belong to orthogonal subspaces, both must reduce a.s. to the zero random variable. Properties (i)– (iii) of Definition 8.4.VII are readily checked, so it follows that both ξ10 (·) and ξ20 (·) are wide-sense second-order stationary random measures. But note that even when ξ 0 is known to be a strict-sense random measure, the argument above shows only that ξ10 and ξ20 are wide-sense random measures. The classical results that relate the presence of a deterministic component to properties of the spectral measure can also be carried over from X(·) to the random measure ξ(·). They are set out in the following theorem.

8.5.


347

Theorem 8.5.III. Let ξ(·) be a strict- or wide-sense second-order stationary random measure with Bartlett spectrum Γ. Then ξ(·) is purely nondeterministic if and only if Γ is absolutely continuous and its density γ satisfies the condition ∞ log γ(ω) dω > −∞. (8.5.6) 1 + ω2 −∞ This condition is equivalent to the existence of a factorization γ(ω) = |˜ g (ω)|2 ,

(8.5.7)

where g˜(·) is the Fourier transform of a (real) generalized function with support on [0, ∞) and can be written in the form g˜(ω) = (1 − iω)˜ g1 (ω), where g˜1 (·) is the Fourier transform of an L2 (R) function with its support in R+ . The function g˜(·) can be characterized uniquely among all possible factorizations by the requirement that it have an analytic continuation into the upper half-plane Im(ω) > 0, where it is zero-free and satisfies the normalization condition ∞ log γ(ω) 1 dω . (8.5.8) g˜(i) = exp 2π −∞ 1 + ω 2 Proof. Since ξ is purely nondeterministic if and only if X defined at (8.5.5) is purely nondeterministic, the results follow from those for the continuous-time process X(·) as set out, for example, in Hannan (1970, Section 3.4). From Sections 8.2 and 8.6, it follows that the spectral measure ΓX of X(·) is related to the Bartlett spectrum Γ of ξ by ΓX (dω) = (1 + ω 2 )−1 Γ(dω), so ΓX has a density γX if and only if Γ has a density, andthe density γ satisfies (8.5.6) if ∞ and only if γX does because the discrepancy −∞ (1 + ω 2 )−1 log(1 + ω 2 ) dω is finite. Similarly, if γX (ω) = |˜ gX (ω)|2 , where g˜X (·) is the Fourier transform of an L2 (R) function with support in R+ , we can set g1 = gX so that (8.5.7) holds together with the assertions immediately following it. Finally, (8.5.8) follows from the corresponding relation for g1 since ∞ log γX (ω) 1 dω g˜(i) = 2˜ g1 (i) = 2 exp 2π −∞ 1 + ω 2 ∞ log γ(ω) 1 dω = exp 2π −∞ 1 + ω 2 using the identity

∞

−∞

log(1 + ω 2 ) dω = 2π log 2. 1 + ω2

These extensions from ΓX to Γ are to be expected because the criteria are analytic and relate to the factorization of the function γ rather than to its behaviour as ω → ±∞. We illustrate the results by two examples.

348


Example 8.5(b) Two-point Poisson cluster process. Suppose that clusters occur at the instants of a Poisson process with parameter µ and that each cluster contains exactly two members, one at the cluster centre and the other at a fixed time h after the first. Then, the reduced covariance measure has just three atoms, one of mass 2µ at 0 and the others at ±h, each of mass µ. The Bartlett spectrum is therefore absolutely continuous with density γ(ω) = µ(1 + cos ωh)/π = µ(2 cos2 12 ωh)/π. In seeking a factorization of the form (8.5.7), it is natural to try (2µ/π)1/2 × cos 12 ωh as a candidate, but checking the normalization condition (8.5.8) reveals a discrepancy: using the relation

∞

−∞

(1 + ω 2 ) log cos2 12 ωh dω = 2π log 12 (1 + e−h )

leads to (2µ/π)1/2 (1 + e−h )/2 for the right-hand side of (8.5.8), while the candidate gives g˜(i) = (2µ/π)1/2 cosh 12 ωh. It is not difficult to see that the correct factorization is . . 2µ 1 + eiωh 2µ iωh/2 g˜(ω) = cos 12 ωh. = e π 2 π In this form, we can recognize g˜(·) as the Fourier transform of a measure with atoms [µ/(2π)]1/2 at t = 0 and t = h, whereas the unsuccessful candidate function is the transform of a measure with atoms of the same mass but at t = ± 12 h; that is, the support is not contained in [0, ∞). Example 8.5(c) Random measures with rational spectral density. When the spectral density is expressible as a rational function, and hence of the form !

m j=1

"/! 2

(ω +

αj2 )

n

" 2

(ω +

βj2 )

j=1

for nonnegative integers m, n with m ≤ n, and real αj , βj , the identification of the canonical factorization is much simpler because it is uniquely determined (up to a constant of unit modulus) by the requirements that g˜(ω) be analytic and zero-free in the upper half-plane. Two situations commonly occur according to whether m < n or m = n. In the former case, the process has a mean square continuous density x(·) and Γ(·) is a totally finite measure. The problem reduces to the classical one of identifying the canonical factorization of the spectrum for the density of the process. For point processes, however, the δ-function in the covariance measure produces a term that does not converge to zero as |ω| → ∞, implying that m = n; the same situation obtains whenever the random measure has a purely atomic component.

8.5.


349

As an example of the latter form, recall the comments preceding Example 8.2(e) concerning point process models with spectral densities of the form γ(ω) =

A2 (α2 + ω 2 ) . β 2 + ω2

The canonical factorization here takes the form (with A, α, and β real and positive) A(α − iω) α−β g˜(ω) = =A 1+ β − iω β − iω corresponding to the time-domain representation g(t) = A δ0 (t) + (α − β)I[0,∞) (t)e−βt . Similar forms occur in more general point process models, with polynomial a sum of products of exponential and polynomial factors in place of the exponential. The main thrust of these factorization results is that they lead to a timedomain representation that can be used to develop explicit prediction formulae. The fact that the canonical factor g˜(ω) is in general the transform not of a function but only of a generalized function leads to some specific difficulties. However, much of the argument is not affected by this fact, as we now indicate. Let Z(·) be the process of orthogonal increments arising in the spectral representation of ξ 0 , and g˜(·) the canonical factor described in Theorem 8.5.III. Introduce a further process U (·) with orthogonal increments by scaling the Z(·) process to have stationary increments as in Z(dω) = g˜(ω) U (dω),

(8.5.9)

where the invertibility of g˜ implies that for all real ω E|U (dω)|2 = |˜ g (ω)|−2 E|Z(dω)|2 = dω. Note that the use of the complex conjugate of g˜ in (8.5.9) is purely for convenience: it simplifies the resulting moving average representation in the time domain. Corresponding to U in the frequency domain, we may, in the usual way, define a new process V in the time domain through the Parseval relations, so ∞ ∞ ˜ φ(t) V (dt) = φ(ω) U (dω), (8.5.10) −∞

−∞

which in this case can be extended to all functions φ ∈ L2 (R). It can be verified that V (·) also has orthogonal and stationary increments, with E|V (dt)|2 = 2π dt,

350


corresponding to the more complete statement ∞ ∞ φ(t) V (dt) = 2π |φ(t)|2 dt var −∞ −∞ ∞ 2 |φ(ω)| dω = var = −∞

∞

˜ φ(ω) U (dω) .

−∞

On the other hand, from the Parseval relation for the ξ 0 process, we have for integrable φ, for which φ˜ ∈ L2 (Γ),

∞

φ(t) ξ 0 (dt) =

−∞

∞

˜ φ(ω) Z(dω) =

−∞

∞

˜ g˜(ω) U (dω). φ(ω)

(8.5.11)

−∞

Thus, if we could identify φ˜g¯˜ with the Fourier transform of some function φ ∗ g ∗ in the time domain, it would be possible to write

∞

0

∞

φ(t) ξ (dt) = −∞

−∞

∗

(φ ∗ g )(s) V (ds) =

∞

t

φ(t) dt −∞

−∞

g(t − s) V (ds),

corresponding to the moving average representation

t

ξ 0 (dt) = −∞

g(t − s) V (ds) dt.

Because g(·) is not, in general, a function, these last steps have a purely formal character. They are valid in the case of a process ξ 0 having a mean square continuous density, but in general we need to impose further conditions before obtaining any meaningful results. In most point process examples, the generalized function g(·) can be represented as a measure, but it is an open question as to whether this is true for all second-order random measures. We proceed by imposing conditions that, although restrictive, are at least general enough to cover the case of a point process with rational spectral density. They correspond to assuming that the reduced factorial cumulant ,[2] is totally finite, so that the spectral density can be written in measure C the form γ(ω) = (2π)−1 m + c˜[2] (ω) . Specifically, assume that

g˜(ω) = A 1 + c˜(ω)

(8.5.12)

for some positive constant A and function c˜ ∈ L2 (R). Then, the generalized function aspect of g(·) is limited to a δ-function at the origin, and there exists an L2 (R) function c(·) such that A δ0 (t) + c(t) (t ≥ 0), g(t) = 0 (t < 0).

8.5.


351

Under the same conditions, the reciprocal 1/˜ g (ω) can be written ˜ 1/˜ g (ω) = A−1 1 − d(ω) , ˜ where d(ω) = c˜(ω)/(1 + c˜(ω)), and from ∞ 2 2 ˜ |d(ω)| γ(ω) dω = A −∞

∞

−∞

|˜ c(ω)|2 dω < ∞

it follows that d˜ ∈ L2 (γ). Often, we have L2 (γ) ⊆ L2 (R), in which case d˜ ∈ L2 (R), implying the existence of a representation of a Fourier inverse of 1/˜ g (ω) as −1 δ0 (t) − d(t) (t ≥ 0) A (8.5.13) 0 (t < 0) for some function d ∈ L2 (R). Proposition 8.5.IV (Moving Average and Autoregressive (ARMA) Representations). Suppose (8.5.12) holds for some c˜ ∈ L2 (R). Then, using the notation of (8.5.12–13), for φ ∈ L1 (R) such that φ˜ ∈ L2 (R), the zero-mean process ξ 0 (·) is expressible as 0 ˜ ˜ φ(t) ξ (dt) = φ(t) V (dt) + φ(t)X(t) dt a.s., (8.5.14) R

R

R

where V (·) is a zero-mean process with stationary orthogonal increments such that E|V (dt)|2 = 2πA2 dt (8.5.15) and X(·) is a mean square continuous process that can be written in the moving average form t X(t) = c(t − u) V (du) a.s. (8.5.16) −∞

or, if furthermore d˜ ∈ L2 (R), in the autoregressive form t d(t − u) ξ 0 (du) a.s. X(t) =

(8.5.17)

−∞

Proof. Under the stated assumptions, it follows from (8.5.11) that ˜ ˜ c(ω) U (dω) a.s. (8.5.18) φ(t) ξ 0 (dt) = A φ(ω) U (dω) + A φ(ω)˜ R

R

R

Consider now the process X(·) defined by the spectral representation X(t) = eitω c˜(ω) U (dω) = eitω ZX (dω) a.s., (8.5.19) R

R

352


where ZX has orthogonal increments and satisfies E |Z(dω)|2 = γX (ω) dω = |˜ c(ω)|2 dω. To ensure that R X(t)φ(t) dt can be validly interpreted as a mean square integral, it is enough to show that φ˜ ∈ L2 (γX ), as in the discussion ˜ around (8.5.3). But φ ∈ L1 (R) implies that |φ(ω)| is bounded for ω ∈ R, and then the assumption that c˜ ∈ L2 (R) implies that 2 2 ˜ ˜ |φ(ω)| |˜ c(ω)|2 dω = |φ(ω)| |γX (ω) dω < ∞, R

R

as required. The terms on the right-hand side of (8.5.18) can now be replaced by their corresponding time-domain versions. Thus, we have ˜ A φ(ω) U (dω) = φ(t) V (dt), R

R

absorbing the constant A into the definition of the orthogonal-increment process V as in (8.5.10), while the discussion above implies that the last term in ˜ (8.5.18) can be replaced by R φ(t)X(t) dt, with X(t) defined as in (8.5.16). This establishes the representation (8.5.14). To establish the autoregressive form in (8.5.17), observe that itω ˜ ˜ Y (t) ≡ e d(ω) Z(dω) = A eitω d(ω) 1 + c˜(ω) U (dω) R R itω = A e c˜(ω) U (dω) = X(t), R

the integrals being well defined and equal a.s. from the assumption that c˜ ∈ L2 (R), from which it follows that d˜ ∈ L2 (Γ). If ξ 0 is a strict-sense random measure, then the time-domain integral (8.5.17) is well defined for φ ∈ L

1 (R) and can be identified a.s. with its frequency-domain version Y (t) above. If ξ 0 is merely a wide-sense process, then (8.5.17) can be defined only as a mean square limit, which will exist whenever d˜ ∈ L2 (Γ). In either case, therefore, X(t) = Y (t) a.s. Equation (8.5.14) can be combined with equations (8.5.16) and (8.5.17) to yield the abbreviated but suggestive forms set out below; they embody the essential content of the moving average and autoregressive representations in the present context. Corollary 8.5.V. With the same assumptions and notation as in Proposition 8.5.IV, t− c(t − u) V (du) dt a.s., (8.5.20) ξ 0 (dt) = V (dt) + −∞ t−

ξ 0 (dt) = V (dt) +

−∞

d(t − u) ξ 0 (du) dt

a.s.

(8.5.21)

8.5.


353

There is a close analogy between (8.5.20) and the martingale decomposition of the cumulative process outlined in the previous chapter: the first term in (8.5.20) corresponds to the martingale term, or innovation, while the second corresponds to the conditional intensity. The difference lies in the fact that the second term in (8.5.20) is necessarily representable as a linear combination of past values, whereas the conditional intensity, its analogue in the general situation, is not normally a linear combination of this type. Finally, we can use the results of the proposition to establish the forms of the best linear predictors when the assumptions of Proposition 8.5.IV hold. Consider specifically the problem of predicting forward the integral Q≡ φ(s) ξ 0 (ds) a.s. (8.5.22) R

from observations on ξ 0 (·) up to time t. The best linear predictor, in the mean square sense, is just the projection of φ onto the Hilbert space L2 (ξ 0 ; t). From equations (8.5.14) and (8.5.20), we see that it can be written as t ∞ ,t (s) ds a.s., ,t = φ(s) ξ 0 (ds) + φ(s)X (8.5.23) Q −∞

t

where for s > t, ,t (s) = X

t

−∞

c(s − u) V (du)

The truncated function cst (u) =

c(u) 0

a.s.

(8.5.24)

(u > s − t), (u ≤ s − t),

is in L2 (R) when c is, and the same is therefore true of its Fourier transform. ,t (s) and Q , t are well Consequently, the random integrals in the definitions of X defined by the same argument as used in proving Proposition 8.5.IV. Equation (8.5.24) already gives an explicit form for the predictor, but it is not convenient for direct use since it requires the computation of V (·). In ,t (s) is more useful. To find it, practice, the autoregressive representation of X observe that t ˜ ,t (s) = cst (s − u) V (du) = c˜st (ω) U (dω) = c˜st (ω)[1 − d(ω)] Z(dω) X −∞ t

=

−∞

c(s − u) −

R

t−u

R

c(s − u − v)d(v) dv ξ 0 (du)

a.s.

(8.5.25)

0

The integral is well defined not only in the mean square sense but also in the a.s. sense if d ∈ L1 (R). In this case, the integrand in (8.5.25) can also be written in the form s−u d(s − u) + c(s − u − v)d(v) dv, t−u

354


which is then the sum of two L1 (R) functions, both of which can be integrated against ξ 0 . These arguments are enough to establish the validity of the autoregressive form (8.5.25) as an alternative to (8.5.24). It is important to emphasize that ,t (s) is to be interpreted as the predictor of the intensity of the ξ 0 process X at time s > t, or in abbreviated notation, ,t (s) ds = E[ξ 0 (ds) | Ht ] = E[λ(s) | Ht ], X

(8.5.26)

where both expectations are to be understood only in the sense of Hilbertspace projections. Thus, the assumptions of Proposition 8.5.IV imply that the intensity is predicted forward as a mean square continuous function of the past. In contrast to the case where the process itself is mean square continuous, when the predictors may involve differentiations, here they are always smoothing operators. The discussion can be summarized as follows. Proposition 8.5.VI. Under the conditions of Proposition 8.5.IV, the best linear predictor of the functional Q in (8.5.22), given the history Ht of the ξ 0 process on (−∞, t], is as in (8.5.23), in which the mean square continuous ,t (s) may be regarded as the best linear predictor of the ‘intensity’ process X ξ 0 (ds)/ds for s > t and has the moving average representation (8.5.24) and the autoregressive representation t , ht (s − u) ξ 0 (du), Xt (s) = −∞

where

t−u

ht (s − u) = c(s − u) −

c(s − u − v)d(v) dv 0

s−u

c(s − u − w)d(w) dw.

= d(s − u) +

(8.5.27)

t−u

Returning to the original random measure ξ (as distinct from ξ 0 ), we obtain the following straightforward corollary, stated in the abbreviated form analogous to (8.5.26). Corollary 8.5.VII. The random measure ξ can be predicted forward with predicted intensity at s > t given by ,t (s) ds, E[ξ(ds) | Ht ] = m + X where the conditional expectation is to be understood in the sense of a Hilbertspace projection. Example 8.5(d) A point process with rational spectral density [continued from Example 8.5(c)]. Consider the case where γ(ω) = A2 (α2 + ω 2 )/(β 2 + ω 2 ) .

(8.5.28)

8.5.


355

From the form of g˜(ω) as earlier, it follows that α−β , β − iω c(t) = (α − β)e−βt ,

α−β ˜ , d(ω) = α − iω d(t) = (α − β)e−αt

c˜(ω) =

(t ≥ 0).

Substituting into (8.5.27), we find ht (s − u) = (α − β)e−β(s−u) − (α − β)2 e−β(s−u)

t−u

e−(α−β)v dv

0

= (α − β)e−β(s−t) e−α(t−u) , so that ,t (s) = (α − β)e−β(s−t) X

t

e−α(t−u) ξ 0 (du)

a.s.

(8.5.29)

−∞

Thus, the predictor here is a form of exponential smoothing of the past. How well it performs relative to the full predictor, based on complete information about the past, depends on the particular process that is under consideration. The most instructive and tractable example is again the Hawkes process, which, in order to reproduce the second-order properties above, should have a complete conditional intensity of the special form as in Exercise 7.2.5, t− λ∗ (t) = λ + ν αe−α(t−u) N (du) a.s., ≡ λ + ναY (t), say, (8.5.30) −∞

which leads to (8.5.28) with A2 = λ/2π, β = α(1 − ν) [see equation (8.2.10)]. The full predictor can be found by taking advantage of the special form of the intensity, which implies that the quantity Y (t) as above and in Exercise 7.2.5 ∞ is Markovian. Defining m(t) = E[Y (t)] = 0 y Ft (dy), we find by integrating (7.2.12) that m(t) satisfies the ordinary differential equation dm(t) = −βm(t) + λ dt with solution m(t) =

λ −βt λ + m(0) − e . β β

To apply this result to the nonlinear prediction problem analogous to that ,t (s) in the linear case, we should set m(0) = Y (t) and consider solved by X m(s − t), which gives the solution , ∗ (s) ≡ E[λ∗ (s) | Ht ] = λ + ναE[Y (t + s) | Y (t)] = λ + ναm(s − t) X t λ λ = + να Y (t) − e−β(s−t) . 1−ν β

356


Replacing Y (t) by its representation in terms of the past of the process as in (8.5.30) leads back to (8.5.29). Thus, for a Hawkes process with exponential infectivity function, the best linear predictor of the future intensity equals the best nonlinear predictor of the future intensity. It appears to be an open question whether this result extends to other Hawkes processes or to other stationary point processes. Linear and nonlinear predictors for an example of a renewal process with rational spectral density are discussed in Exercise 8.5.2. Example 8.5(e) Two-point Poisson cluster process [continued from Example 8.5(b)]. While this example does not satisfy the assumptions of the preceding discussion, it is simple enough to handle directly. From the expression for g˜(ω) given earlier, the moving average representation can be written in the form ξ 0 (dt) = (µ/2π)1/2 {V (dt) + V (dt − h)}. The reciprocal has the form 1/˜ g (ω) = (2π/µ)1/2 (1 + eiωh )−1 , which, if we proceed formally, can be regarded as being the sum of an infinite series corresponding to the time-domain representation V (dt) =

2π/µ ξ 0 (dt) − ξ 0 (dt − h) + ξ 0 (dt − 2h) − · · · .

In fact, the sum is a.s. finite and has the effect of retaining in V only those atoms in ξ 0 that are not preceded by a further atom h time units previously; that is, of retaining the atoms at cluster centres but rejecting their cluster companions. From this, it is clear that the process V (·) is just a scaled version of the zero-mean version of the original Poisson process of cluster centres, and the moving average representation is simply a statement of how the clusters are formed. It is now easy to form linear predictors: we have ξ 0 (ds | Ht ) =

0 (s − t > h), (µ/2π)1/2 V (ds − h) (0 ≤ h ≤ s − t),

and on 0 ≤ h ≤ s − t we also have ξˆ0 (ds | Ht ) =

∞

(−1)j ξ 0 (ds − jh).

j=1

The effect of the last formula is to scan the past to see if there is an atom at s − h not preceded by a further atom at s − 2h: the predictor predicts an atom at s when this is the case and nothing otherwise.

8.6.

P.P.D. Measures

357

Exercises and Complements to Section 8.5 8.5.1 Renewal processes with rational spectral density. Show that the Bartlett spectrum for the renewal process considered in Exercise 4.2.4 with interval density µ2 xe−µx has the form µ ω 2 + 2µ2 γ(ω) = . 4π ω 2 + 4µ2 8.5.2 Linear and nonlinear prediction of a renewal process. (a) Show that for any renewal process the best nonlinear predictor E[λ∗ (t+s) | Ht ] for the intensity is the renewal density for the delayed renewal process in which the initial lifetime has d.f. [F (Bt + s) − F (Bt )]/[1 − F (Bt )], where Bt is the backward recurrence time at time t. (b) Find explicitly the best predictor for the process in Exercise 8.5.1. (c) Find the canonical factorization of the spectrum of the renewal process ,t (s), where Bt is in Exercise 8.5.1, and find the best linear predictor B the backward recurrence time at t. When does it coincide with the best nonlinear predictor in (b)? (d) Investigate the expected information gain per event based on the use of the linear and nonlinear predictors outlined above.

8.6. P.P.D. Measures In this section, we briefly develop the properties of p.p.d. measures required for the earlier sections of this chapter. We follow mainly the work of Vere-Jones (1974) and Thornett (1979); related material, in a more abstract setting, is in Berg and Frost (1975). No significant complications arise in developing the theory for Rd rather than for the line, so we follow this practice, although most of the examples are taken from the one-dimensional context. Since the measures we deal with are not totally finite in general, we must first define what is meant by a Fourier transform in this context. As in the theory of generalized functions (see e.g. Schwarz, 1951), we make extensive use of Parseval identities ψ(x) ν(dx) = ψ(ω) µ(dω) (8.6.1) Rd

Rd

to identify the measure ν as the Fourier transform of the measure µ in (8.6.1). Here ψ(ω) = eix·ω ψ(x) dx Rd

is the ordinary (d-dimensional) Fourier transform of ψ(·), but such functions must be suitably restricted. A convenient domain for ψ is the space S of real or complex functions of rapid decay; that is, of infinitely differentiable functions that, together with their derivatives, satisfy inequalities of the form & & & ∂ k ψ(x) & C(k, r) & & & ∂xk1 · · · ∂xkd & ≤ (1 + |x|)r 1 d

358


for some constants C(k, r) < ∞, all positive integers r, and all finite families of nonnegative integers (k1 , . . . , kd ) with k1 + · · · + kd = k. The space S has certain relevant properties, proofs of which are sketched in Exercise 8.6.1: (i) S is invariant under the Fourier transformation taking ψ into ψ. (ii) S is invariant under multiplication or convolution by real- or complexvalued integrable functions g on Rd such that both g and g˜ are zero-free. (iii) Integrals with respect to all functions ψ ∈ S uniquely determine any boundedly finite measure on Rd . The following definitions collect together some properties of boundedly finite measures that are important in the sequel. We use the notation, for complex-valued functions ψ and φ, (ψ ∗ φ)(x) = ψ ∗ (x) = ψ(−x), ψ(y)φ(x − y) dy, Rd

so that ∗

(ψ ∗ ψ )(x) =

Rd

ψ(y)ψ(y − x) dy.

Definition 8.6.I. A boundedly finite signed measure µ(·) on Rd is (i) translation-bounded if for all h > 0 and x ∈ Rd there exists a finite constant Kh such that, for every sphere Sh (x) with centre x ∈ Rd and radius h, & & &µ Sh (x) & ≤ Kh ; (8.6.2) (ii) positive-definite if for all bounded measurable functions ψ of bounded support, (ψ ∗ ψ ∗ )(x) µ(dx) ≥ 0; (8.6.3) Rd

(iii) transformable if there exists a boundedly finite measure ν on Rd such that (8.6.1) holds for all ψ ∈ S; (iv) a p.p.d. measure if it is nonnegative (i.e. a measure rather than a signed measure) and positive-definite. A few comments on these definitions are in order. The concept of translation boundedness appears naturally in this context and is discussed further by Lin (1965), Argabright and de Lamadrid (1974), Thornett (1979), and Robertson and Thornett (1984). If µ is nonnegative, then it is clear that if (8.6.2) holds for some h > 0 it holds for all such h. The notion of positivedefiniteness in (8.6.3) is a direct extension of the same notion for continuous functions; indeed, if µ is absolutely continuous, then it is positive-definite in the sense of (8.6.3) if and only if its density is a positive-definite function in the usual sense. Concerning the Parseval relation in (8.6.1), it is important to note that if the measure µ is transformable, then ν is uniquely determined by µ and conversely. Equation (8.6.1) generalises the relation c(x) = eiω·x F (dω) Rd

8.6.

P.P.D. Measures

359

for the covariance density in terms of the spectral measure F of a mean square continuous process to which it reduces (with the appropriate identifications) when the random measure and associated covariance measure are absolutely continuous. Our main interest is in the class of p.p.d. measures on Rd , denoted below by P + . Some examples may help to indicate the scope and character of P + . Example 8.6(a) Some examples of p.p.d. measures. (1◦ ) A simple counterexample. The measure on Rd with unit mass at each of the two points ±1 is not a p.p.d. measure because its Fourier transform 2 cos ω can take negative values and it thus fails to be positive-definite. On the other hand, the convolution of this measure with itself (i.e. the measure with unit mass at ±2 and mass of two units at 0) is a p.p.d. measure, and its Fourier transform is the boundedly finite (but not totally bounded) measure with density 4 cos ω. This also shows that the convolution square root measure of a p.p.d. measure need not be p.p.d. (2◦ ) Absolutely continuous p.p.d. measures. Every nonnegative positive-definite function defines the density of an absolutely continuous p.p.d. measure. (3◦ ) Counting measure. Let µ have unit mass at every 2πj for j = 0, ±1, . . . . Then, for ψ ∈ S, (8.6.1) reduces to the Poisson summation formula (see Exercise 8.6.4 for details) ∞

ψ(n) =

n=−∞

∞

ψ(2πj);

j=−∞

that is, µ has as its Fourier transform the measure ν with unit mass at each of the integers n = 0, ±1, . . . . It also shows that ν, and thus µ as well, is positive-definite for ψ a function of the form φ∗φ∗ so that the right-hand (take ˜ side becomes |φ(2πj)|2 ≥ 0). ◦ (4 ) Closure under product. Let µ1 , . . . , µd be p.p.d. measures on R with Fourier transforms µ ˜1 , . . . , µ ˜d . Then, the product measure µ1 × · · · × µd is a ˜1 × · · · × µ ˜d . p.p.d. measure on Rd with Fourier transform µ A simple and elegant theory for measures in P + and their Fourier transforms can be developed by the standard device of approximating µ by a smoothed version obtained by convoluting µ with a suitable smoothing function such as the symmetric probability densities t(x) = (1 − |x|)+ eλ (x) =

−λ|x| 1 2 λe

(triangular density), (two-sided exponential density),

and their multivariate extensions d t(x) = (1 − |xi |)+ , i=1

d eλ (x) = 12 λ exp

! −λ

(8.6.4a) d

i=1

" |xi | .

(8.6.4b)

360


Observe that t(x) =

Rd

IUd (x − y) IUd (−y) dy.

(8.6.4a )

We are now in a position to establish the basic properties of P + . Proposition 8.6.II. (a) P + is a closed positive cone in M# (Rd ). (b) Every p.p.d. measure is symmetric and translation-bounded. Proof. In (a), we mean by ‘a positive cone’ a set closed under the formation of positive linear combinations. Then (a) is just the statement that if a sequence of boundedly finite measures in Rd converges vaguely to a limit, and if each measure in the sequence is positive-definite, then so is the limit. This follows directly from the definition of vague convergence and the defining relation (8.6.3). Now let µ be a p.p.d. measure on Rd , and convolve it with t(·) as in (8.6.4a) so that the convolution is well defined. The resultant function c(x) ≡ t(x − y) µ(dy) (8.6.5) Rd

is real-valued, continuous, and for all bounded measurable ψ of bounded support it satisfies, because of (8.6.4a ), Rd

∗

c(u)(ψ ∗ ψ )(u) du =

Rd

(ψ ∗ IUd ) ∗ (ψ ∗ IUd )∗ (y) µ(dy) ≥ 0;

note that (8.6.3) applies because ψ ∗ IUd is measurable and bounded with bounded support whenever ψ is. In other words, the function c(·) is realvalued and positive-definite and hence, from standard properties of such functions, also symmetric and bounded. Since t(·) is symmetric, it is clear that c(·) is symmetric if and only if µ is symmetric, which must therefore hold. Finally, it follows from the positivity of µ and the inequality t(x) ≥ 2−d for x ≤ 14 that if K is a bound for c(·), µ S1/4 (x) ≤ 2d

c(y) dy ≤ 2d K < ∞. S1/4 (x)

Inequality (8.6.2) is thus established for the case h = 14 , and since µ is nonnegative, its validity for any other value of h is now apparent. The Fourier transform properties can be established by similar arguments, though it is now more convenient to work with the double exponential function eλ (·) because its Fourier transform ˜eλ (ω) =

d

λ2 λ2 + ωi2 i=1

8.6.

P.P.D. Measures

361

has no real zeros. The existence of the convolution µ ∗ eλ follows from the translation boundedness just established. The relation eλ (x − y) µ(dy) dλ (x) = Rd

again defines a continuous positive-definite function. By Bochner’s theorem in Rd , it can therefore be represented as the Fourier transform eiω·x Gλ (dω) dλ (x) = Rd

for some totally finite measure Gλ (·). Now let ψ(ω) be an arbitrary element of S, and consider the function κ ˜ (ω) defined by κ ˜ (ω) = (1 + ω 2 )ψ(−ω)/(2π)d . Then κ ˜ ∈ S also, and hence κ ˜ is the Fourier transform of some integrable function κ satisfying ψ(y) = (κ ∗ e1 )(y). From the Fourier representation of d1 , we have κ(x)d1 (x) dx = κ ˜ (ω) G1 (dω) Rd

Rd

for all integrable κ and hence in particular for the function κ just constructed. Substituting for κ, we obtain, for all ψ ∈ S, µ(dy) = (κ ∗ e1 )(y) µ(dy) = κ(x)d1 (x) dx ψ(y) Rd Rd Rd 1 = κ ˜ (ω) G1 (dω) = ψ(ω)(1 + ω 2 ) G1 (−dω). (2π)d Rd Rd We now define the measure ν by ν(dω) = (2π)−d (1 + ω 2 ) G1 (−dω) and observe that ν is boundedly finite and satisfies the equation (8.6.1), which represents ν as the Fourier transform of µ. Thus, we have shown that any p.p.d. measure µ is transformable. Then, interchanging Recall that S is preserved under the mapping ψ → ψ. the roles of ψ and ψ in (8.6.1) shows that every p.p.d. measure is itself a transform and hence that ν is positive-definite as well as positive; that is, it is itself a p.p.d. measure. Since the determining properties of S imply that each of the two measures in (8.6.1) is uniquely determined by the other, we have established the principal result of the following theorem.

362


Theorem 8.6.III. Every p.p.d. measure µ(·) is transformable, and the Parseval equation (8.6.1) establishes a one-to-one mapping of P + onto itself. This mapping can also be represented by the inversion formulae: for bounded ν-continuity sets A, ν(A) = lim (8.6.6) IÃ (ω)˜eλ (ω) µ(dω); λ→∞

Rd

for bounded µ-continuity sets B, 1 µ(B) = lim − I˜B (−x)˜eλ (−x) ν(dx); λ→∞ (2π)d Rd ν({a}) = lim e−iω·a µ(dω); T →∞

Ud T

1 µ({b}) = lim T →∞ (2πT )d

(8.6.7) (8.6.8)

eix·b ν(dx).

(8.6.9)

Ud T

For all Lebesgue integrable φ for which φ˜ is µ-integrable, there holds the extended Parseval relation ˜ φ(x + y) ν(dy) = eiω·x φ(ω) µ(dω) (a.e. x). (8.6.10) Rd

Rd

Proof. It remains to establish the formulae (8.6.6–10), all of which are effectively corollaries of the basic identity (8.6.1). Suppose first that A is a bounded continuity set for ν(·) and hence a fortiori for the smoothed version ν ∗ eλ . Then, for all finite λ, it is a consequence of the Parseval theorem that (ν ∗ eλ )(A) = I˜λ (ω)˜eλ (ω) µ(dω). Rd

Now letting λ → ∞, the left-hand side → ν(A) by standard properties of weak convergence since it is clear that ν ∗ eλ → ν weakly on the closure A¯ of A. This proves (8.6.6), and a dual argument gives (8.6.7). To establish (8.6.8), consider again the convolution with the triangular density t(·). Changing the base of the triangle from (−1, 1) to (−h, h) ensures that the Fourier transform t˜(ω) does not vanish at ω = a for any given a. Now check via the Parseval identity that the totally finite spectral measure corresponding to the continuous function c(x) in (8.6.5) can be identified with t˜(ω)ν(dω). Then, standard properties of continuous positive-definite functions imply 1 ˜ t(a)ν({a}) = lim − e−ia·x c(x) dx. (8.6.11) T →∞ (4πT )d Ud2T

Consider DT ≡ t˜(a)

e Ud 2T

−ia·x

µ(dx) −

Ud 2T

e−ia·x c(x) dx,

8.6.

P.P.D. Measures

363

which on using the definition of c(·) as the convolution t ∗ µ yields T −x1 T −xd −ia·x −ia·y ˜ DT = e µ(dx) t(a)IUd2t (x) − ··· e t(y) dy . Rd

−T −x1

−T −xd

The expression inside the braces vanishes both inside the hypercube with vertices ± (T − h), . . . , ±(T − h) since the second integral then reduces to t˜(a) and outside the hypercube with vertices ± (T + h), . . . , ±(T + h) since both terms are then zero. Because µ is translation-bounded, there is an upper bound, Kh say, on the mass it allots to any hypercube with edge of length 2h. The number of such hypercubes needed to cover the region where the integrand is nonzero is certainly bounded by 2d(2 + T /h)d−1 , within which region the integrand is bounded by M , say. Thus, |DT | 2d ≤ (4πT )d (4π)d

1 2 + h T

d−1

M Kh →0 T

(T → ∞).

Equation (8.6.8) now follows from (8.6.11), and (8.6.9) follows by a dual argument with the roles of µ and ν interchanged. It is already evident by analogy with the argument used in constructing ν(·) that the Parseval relation (8.6.1) holds not only for ψ ∈ S but also for any function of the form (φ ∗ eλ )(x), where φ is integrable. In particular, any function of the form θ(x) = φ(y)ψ(x − y) dy = (φ ∗ ψ)(x) Rd

has this form for ψ ∈ S and φ integrable. Hence, for all ψ ∈ S, ˜ ψ(ω) ψ(x) dx φ(x + y) ν(dy) = φ(ω) µ(dω). Rd

Rd

Rd

If, furthermore, φ˜ is µ-integrable, we can rewrite the right-hand side of this equation in the form ˜ ψ(x) dx eiω·x φ(ω) µ(dω). Rd

Rd

Since equality holds for all ψ ∈ S, the coefficients of ψ(x) in the two integrals must be a.e. equal, which gives (8.6.10). Many variants on the inversion results given above are possible: the essential point is that µ and ν determine each other uniquely through the Parseval relation (8.6.1). A number of further extensions of this relation can be deduced from (8.6.10), including the following important result. Proposition 8.6.IV. For all p.p.d. measures µ with Fourier transform ν as in (8.6.1), and for all bounded functions f of bounded support, (f ∗ f ∗ )(x) ν(dx) = |f˜(ω)|2 µ(dω). (8.6.12) Rd

Rd

364


Proof. Examining (8.6.10), we see that the assumed integrability condition implies that the right-hand side there is continuous in x and consequently that the two sides are equal for any value of x at which the left-hand side is also continuous (note that the a.e. condition cannot be dropped in general because altering φ at a single point will alter the left-hand side whenever ν has atoms while the right-hand side will remain unchanged). Thus, to check (8.6.12), it is enough to establish the continuity of the left-hand side and the integrability of |f˜(ω)|2 with respect to µ on the right-hand side. Appealing to the dominated convergence theorem shows first that Rd f (u)f (x + u) du is a continuous function of x and second, since this function vanishes outside a bounded set within which ν(·) is finite, that the integral Rd

(f ∗ f ∗ )(x + y) ν(dy)

also defines a continuous function of x. To establish that |f˜(ω)|2 is µ-integrable, we use Lemma 8.6.V given shortly (the lemma is also of interest in its own right). Specifically, express the integral on the right-hand side of (8.6.12) as a sum of integrals over regions Bk as in the lemma. For each term, we then have |f˜(ω)|2 µ(dω) ≤ bk µ(Bk ) ≤ Kbk Bk

for some finite constant K using the property of translation boundedness. Finiteness of the integral follows on summing over k and using (8.6.13). Lemma 8.6.V (Lin, 1965). Let A be a bounded set in Rd , h a positive constant, and θ(x) a square integrable function with respect to Lebesgue measure on A. For k = (k1 , . . . , kd ), let Bk be the half-open cube {ki h < xi ≤ ki h + h; i = 1, . . . , d}, and set 2 ˜ bk = sup |θ(ω)| . ω∈Bk

Then, for all such θ(·), there exists a finite constant K(h, A) independent of θ(·) and such that

bk ≤ K(h, A) |θ(x)|2 dx, (8.6.13) A

k

where summation extends over all integers k1 , . . . , kd = 0, ±1, . . . . Proof. For simplicity, we sketch the proof for d = 1, h = 1, A = [−1, 1], leaving it to the reader to supply the details needed to extend the result to the general case. Write 1 αk = 12 eiπkx θ(x) dx −1

8.6.

P.P.D. Measures

365

for the kth Fourier coefficient of θ as a function on the interval (−1, 1). Then, from standard properties of Fourier series, we have 1 ∞

|αj |2 = |θ(x)|2 dx < ∞. (8.6.14) j=−∞

−1

Now let ωk be any point in Bk = (k, k + 1], and consider the Taylor series ˜ expansion of θ(ω) at ωk . Since A is bounded, θ˜ is an entire function, and hence the Taylor series about the point k converges throughout Bk , and we can write & ∞ ∞ &

& ∞ (ωk − k)n (n) &2 ˜ k )|2 = ˜ (k)& & |θ(ω θ & & n! k=−∞ k=−∞ n=0 "! ∞ " ! ∞ ∞

|ωk − k|2n

|θ˜(n) (k)|2 ≤ n! n! n=0 n=0 k=−∞

∞ from the Cauchy inequality. The first series is dominated by n=0 1/n! = e for all choices of ωk ; hence, by analogy with (8.6.14), we obtain ! ∞ " ∞ ∞

1 2 2 (n) ˜ ˜ |θ(ωk )| ≤ e |θ (k)| n! n=0 k=−∞ k=−∞ 1 ∞

1 1 n 2 2 =e |x θ(x)| dx ≤ e |θ(x)|2 dx. n! −1 −1 n=0 ˜ k )|2 and so give bk , (8.6.13) In particular, choosing ωk in Bk to maximize |θ(ω now follows. Another integrability result is noted in Exercise 8.6.8. A simple and characteristic property of a p.p.d. measure is that it remains a p.p.d. measure after addition of an atom of positive mass at the origin. Equally, passing over to the Fourier transforms, it remains a p.p.d. measure after addition of an arbitrary positive multiple of Lebesgue measure. Now suppose that, starting from a given p.p.d. measure µ, we repeatedly subtract multiples of Lebesgue measure in alternation, first from the p.p.d. measure itself and then from its Fourier transform, until one of these measures ceases to be nonnegative. Evidently, certain maximum multiples of Lebesgue measure will be defined by this process, leaving, after subtraction, a p.p.d. measure ν with the additional property that no nonzero multiple of Lebesgue measure can be subtracted from ν or its Fourier transform without destroying the p.p.d. property. Let us call such a measure a minimal p.p.d. measure. This leads us to the following elementary structure theorem. Proposition 8.6.VI. Every p.p.d. measure µ on Rd can be uniquely represented as the sum of a minimal p.p.d. measure, a positive multiple of Lebesgue measure on Rd , and an atom of positive mass at the origin.

366


Very little is known about the structure of minimal p.p.d. measures, even when d = 1. See Exercise 8.6.9. Example 8.6(b). As a simple illustration of (8.6.12), let f (x) be the indicator function of the hyper-rectangle (0, T1 ] × · · · × (0, Td ]. It then follows that 2 d d sin(ωi Ti /2) µ(dω). (Ti − |xi |)+ ν(dx) = ωi /2 Rd i=1 Rd i=1

Exercises and Complements to Section 8.6 8.6.1 The space S. (a) Show that if X = R and ψ: R → R has an integrable kth derivative, then ∞ |ω k ψ(ω)| → 0 as |ω| → ∞, and that, conversely, if −∞ |x|k |ψ(x)| dx < ∞,

is k times differentiable. Deduce that S is invariant under the then ψ(ω) Extend the result to Rd . Fourier mapping taking ψ into ψ. d (b) Let g: R → R be an integrable function with Fourier transform g˜ such that both g and g˜ are zero free on Rd . Show that both the mappings ψ → ψ ∗ g → ψg are one-to-one mappings of S onto itself. In particular, deduce and ψ that this result holds when ψ(·) has the double exponential form eλ (·) of (8.6.4b). (c) Show that if µ, ν are boundedly finite measures on R such that R ψ dµ = ψ dν for all ψ ∈ S, then µ = ν. [Hint: Consider ψ ∈ S of bounded R support and approximate indicator functions.] Extend to Rd .

8.6.2 Let {cn : n = 0, ±1, . . .} denote a doubly infinite sequence of reals. Call {cn } 2π (i) transformable if cn = 0 eiωn ν(dω) for some measure ν on [0, 2π]; and (ii) positive-definite if for all finite families {α1 , . . . , αk } of complex numbers, k k

αi α ¯ j ci−j ≥ 0.

i=1 j=1

Let P + (Z) denote the class of all p.p.d. sequences and P + (0, 2π] the class of all p.p.d. measures on (0, 2π]. Show that every {cn } ∈ P + (Z) is bounded, transformable, and symmetric [i.e. cn = c−n (all n)] and that a one-to-one mapping between P + (Z) and P + (0, 2π] is defined when the Parseval relation k

holds for all a ˜(ω) =

k

j=1

j=1

2π

aj cj =

a ˜(ω) ν(dω) 0

aj eiωj , with a1 , . . . , ak any finite sequence of reals.

8.6.3 Show that not all translation-bounded sequences are transformable. [Hint: Let X = R and exhibit a sequence that is bounded but for which T T −1 j=−T cj does not converge to a limit as T → ∞. Use this to define an atomic measure on R that is not transformable.]

are integrable on R, 8.6.4 Poisson summation formula. Show that if both ψ and ψ then ∞

k=−∞

ψ(2πk + x) =

∞

j=−∞

ψ(j)e−ijx

8.6.

P.P.D. Measures

367

whenever the left-hand side defines a continuous function of x. [Hint: Under the stated conditions, the left-hand side, a(x) say, is a bounded 2π continuous function of x. Denote by an = (2π)−1 0 einx a(x) dx its nth Fourier coefficient, and show by rearrangement that an = ψ(−n). Then, the relation is just the representation of a(·) in terms of its Fourier series. Observe that the conditions hold for ψ ∈ S and that the formula in Example 8.6(a)(3◦ ) is the special case x = 0.] 8.6.5 Show that any p.p.d. measure on R integrates (1 + ω 2 )−α for α > 12 and hence conclude that any p.p.d. measure is a tempered measure in the language of generalized functions. 8.6.6 (a) Let c(x) = |x|−1/2 for (|x| ≤ 1), c(x) = 0 elsewhere, and define g(ω) = ∞ 4 − −∞ eiωx c(x) dx. Show that the measure G with density g is nonnegative and translation-bounded but cannot be made into a p.p.d. measure by adding an atom at the origin. (b) Show that dx ν(A) = (bounded A ∈ B) 2 − sin |x| A defines a measure that is a spectral measure but not a transform (Thornett, 1979). 8.6.7 Show that for 1 < γ < 2 the following functions are densities of p.p.d. measures in R2 , and find their spectral measures: (a) c1 (x, y) = {sin(γπ/2)Γ(γ + 1)/2π}2 |xy|1−γ ; (b) c2 (x, y) = 22(γ−2) π γ−3 (Γ(2 − γ))−1 |x2 + y 2 |1−γ . [Hint: Both spectral measures are absolutely continuous with densities g1 (ω1 , ω2 ) = [ 12 γ(γ − 1)]2 |ω1 ω2 |γ−2 , g2 (ω1 , ω2 ) = π γ−2 /[Γ(γ − 1)|ω12 + ω22 |2 ], respectively. Thornett (1979) has formulae for similar p.p.d. measures in Rd .] 8.6.8 Translation-boundedness characterization. A nonnegative Borel measure µ on B(Rd ) satisfies |IÃ (ω)|2 µ(dω) < ∞

Rd

for all bounded A ∈ B(Rd ), if and only if the measure µ is translation-bounded. [Hint: Establish a converse to Lemma 8.6.V of the form

|f˜(ω)|2 µ(dω) ≤ K 2 sup |f (x)|2 , x∈A

where f , with Fourier transform f˜, is any bounded measurable function vanishing outside the bounded Borel set A and K is an absolute constant that may depend only on µ. See Robertson and Thornett (1984) for further details. Other results and references for such measures, but on locally compact Abelian groups, are given in Bloom (1984).] 8.6.9 Find the minimal p.p.d. measures corresponding to the Hawkes process with Bartlett spectrum (8.1.10).

CHAPTER 9

Basic Theory of Random Measures and Point Processes

9.1 9.2 9.3 9.4 9.5

Definitions and Examples Finite-Dimensional Distributions and the Existence Theorem Sample Path Properties: Atoms and Orderliness Functionals: Definitions and Basic Properties Moment Measures and Expansions of Functionals

2 25 38 52 65

This chapter sets out a framework for developing point process theory as part of a general theory of random measures. This framework was developed during the 1940s and 1950s, and reached a definitive form in the now classic treatments by Moyal (1962) and Harris (1963). It still provides the basic framework for describing point processes both on the line and in higherdimensional spaces, including especially the treatment of finite-dimensional distributions, moment structure, and generating functionals. In the intervening decades, many important alternative approaches have been developed for more specialized classes of processes, particularly those with an evolutionary structure, and we come to some at least of these in later chapters. As far as is convenient, we develop the theory in a dual setting, stating results for general random measures alongside the more specific more clearly the features that are peculiar to point processes. Thus, for results that hold in this unified context, proofs are usually given only in the former, more general, setting. Furthermore, the setting for point processes also handles many of the topics of this chapter for marked point processes (MPPs): an MPP in state space X with mark space K can be regarded as a point process on the product space X × K so far as fidi distributions, generating functionals, and moment measures are concerned. It is only when we consider particular cases, such as Poisson and compound Poisson processes or purely atomic random measures, that distinctions begin to emerge, and become more apparent as we move to 1

2

9. Basic Theory of Random Measures and Point Processes

discuss stationary processes in Chapter 12 and Palm theory and martingale properties in Chapters 13 and 14. The other major approach to point process theory is through random sequences of points. We note that this is equivalent to our approach through random measures, at least in our setting that includes point processes in finite-dimensional Euclidean space Rd . Section 9.1 sets out some basic definitions and illustrates them with a variety of examples. The second section introduces the finite-dimensional (fidi) distributions and establishes both basic existence theorems and a version of Rényi’s theorem that simple point processes are completely characterized by the behaviour of the avoidance function (vacuity function, empty space function), viz. the probability P0 (A) ≡ P{N (A) = 0} over a suitably rich class of Borel sets A. Section 9.3 is concerned with the sample path properties of random measures and point processes, and includes a detailed discussion of simplicity (orderliness) for point processes. The final two sections treat generating functionals and moment properties, extending the treatment for finite point processes given in Chapter 5.

9.1. Definitions and Examples Let X be an arbitrary complete separable metric space (c.s.m.s.) and BX = B(X ) the σ-field of its Borel sets. Except for case (v) of Definition 9.1.II, all the measures that we consider on (X , BX ) are required to satisfy the boundedness condition set out in Definition 9.1.I. It extends to general measures the property required of counting measures in Volume I, that bounded sets have finite counting measure and hence, as point sets, they contain only finitely many points and therefore have no finite accumulation points. Definition 9.1.I. A Borel measure µ on the c.s.m.s. X is boundedly finite if µ(A) < ∞ for every bounded Borel set A. This constraint is incorporated into the definitions below of the spaces which form the main arena for the analysis in this volume. They incorporate the basic metric properties of spaces of measures summarized in Appendix A2 of Volume I. In particular we use from that appendix the following. (1) The concept of weak convergence of totally finite measures on X , namely that µn → µ weakly if and only if f dµn → f dµ for all bounded continuous f on X (see Section A2.3). (2) The extension of weak convergence of totally finite measures to w#(weakhash) convergence of boundedly finite measures defined by f dµn → f dµ for all bounded continuous f on X vanishing outside a bounded set (Section A2.6). (3) The fact that both weak and weak-hash convergence are equivalent to forms of metric convergence, namely convergence in the Prohorov metric

9.1.

Definitions and Examples

3

at equation (A2.5.1) and its extension to the boundedly finite case given by equation (A2.6.1), respectively. Exercise 9.1.1 shows that for sequences of totally finite measures, weak and weak-hash convergence are not equivalent. Many of our results are concerned with one or other of the first two spaces defined below. Both are closed in the sense of the w# -topology referred to above, and in fact are c.s.m.s.s in their own right (Proposition 9.1.IV). At the same time it is convenient to introduce four further families of measures which play an important role in the sequel. Definition 9.1.II. (i) M# X is the space of all boundedly finite measures on BX . # (ii) NX is the space of all boundedly finite integer-valued measures N ∈ M# X, called counting measures for short. (iii) NX#∗ is the family of all simple counting measures, consisting of all those elements of NX# for which N {x} ≡ N ({x}) = 0 or 1

(all x ∈ X ).

(9.1.1)

(iv) NX#g ×K is the family of all boundedly finite counting measures defined on the product space B(X × K), where K is a c.s.m.s. of marks, subject to the additional requirement that the ground measure Ng defined by Ng (A) ≡ N (A × K)

(all A ∈ BX )

(9.1.2)

is a boundedly finite simple counting measure, i.e. Ng ∈ NX#∗ . # (v) M# X ,a is the family of boundedly finite purely atomic measures ξ ∈ MX . (vi) MX (respectively, NX ) is the family of all totally finite (integer-valued) measures on BX . We introduce the family NX#g ×K to accommodate our Definition 9.1.VI(iv) of a marked point process (MPP) (as a process on X with marks in K). In it we require the ground process Ng to be both simple and boundedly finite. Note that in general a simple boundedly finite counting measure on BX ×K need not be an element of this family NX#g ×K . For example, taking X = K = R, realizations of a homogeneous Poisson process on the plane would have ground process elements failing to be members of NR# . See also Exercises 9.1.3 and 9.1.6. Note also that although a purely atomic boundedly finite measure can have at most countably many atoms, these atoms may have accumulation points, so representing such measures as a countable set {(xi , κi )} of pairs of locations and sizes of the atoms can give a counting measure on X × R+ that need not # be in either NX#g ×R+ nor even NX ×R+ [cf. Proposition 9.1.III(v) below]. # In investigating the closure properties of M# X and NX (Lemma 9.1.V below), we use Dirac measures (see Section A1.6) defined for every x ∈ X by 1 if x ∈ Borel set A, (9.1.3) δx (A) = 0 otherwise.

4


Proposition 9.1.III. Let X be a c.s.m.s., and µ a boundedly finite measure on BX (i.e., µ ∈ M# X ). (i) The measure µ is uniquely decomposable as µ = µa + µd , where µa =

i

(9.1.4)

κi δxi

(9.1.5)

is a purely atomic measure, expressed in terms of the uniquely determined countable set {(xi , κi )} ⊂ X × R+ 0 , and µd is a diffuse measure (i.e., it has no atoms). (ii) A boundedly finite measure N on BX is a counting measure (i.e., N ∈ # M# X belongs to NX ), if and only if in part (i) its diffuse component is null, and in (9.1.5) all κi are positive integers, κi = ki say, and {xi } is a countable set with at most finitely many xi in any bounded Borel set; that is, ki δxi . (9.1.6) N= i

(iii) For any N ∈

NX# ,

N∗ =

i

δxi

(9.1.7)

defines the support counting measure N ∗ . Then N ∗ ∈ NX#∗ ; N belongs to NX#∗ if and only if at (9.1.6) ki = 1 (all i); equivalently, N coincides with its support counting measure. (iv) Any counting measure N ∈ NX# may be represented as a counting mea ∈ N #g sure N X ×Z+ with representation {(xi , ki )}, in which the ground measure Ng is equal to the support counting measure N ∗ and the positive integer-valued marks ki represent the multiplicities of the atoms of N. (v) There exists a one-to-one, both ways measurable, correspondence between purely atomic boundedly finite measures µ on BX and counting measures Nµ (A × K) = i δ(xi ,κi ) (A × K) on the Borel sets of X × R+ 0 satisfying the additional requirement that, for all bounded A ∈ BX , κ Nµ (dx × dκ) = κi < ∞; (9.1.8) i:xi ∈A

A×R+

the correspondence is given for bounded A ∈ BX and any K ∈ B(R+ 0) that is bounded away from 0 by κ Nµ (dx × dκ) (9.1.9a) µ(A) = A×R+ 0

and Nµ (A × K) = lim

n→∞

j

IK [µ(Anj )],

(9.1.9b)

where {Anj } is a dissecting system of measurable subsets of A.

9.1.


5

Remarks. The measure Nµ in (v) above, in terms of the representation as δ , may have an accumulation point (x, 0) ∈ X × R+ and therefore (x ,κ ) i i i fail to be boundedly finite, even though for such a measure Nµ we do have κ < ∞. In this case, we have, for some bounded set A, Ng (A) = ∞ i i whereas for > 0, #{(xi , κi ): xi ∈ A, κi > } < ∞. See Definition 9.1.VI(vi). Proof. Part (i) is a standard property of σ-finite measures on BX : see the definition of atomic and diffuse measures in Appendix A1.6. In part (ii), it is clear from (i) that if the diffuse component is null and the κi are positive integers, then the measure is a counting measure. Conversely, if N is integer-valued, any atom of N must have positive integral mass; and because N is boundedly finite there can be at most a finite number of such atoms within any bounded set, and at most countably many in all because we can cover X by a countable number of bounded sets. Hence, to complete the proof, it is enough to show that N has no nonatomic component. Let y be an arbitrary point of X , and { j : j = 1, 2, . . .} a monotonic sequence of positive reals decreasing to zero, so that the spheres Sj (y) ↓ {y} as j → ∞. Then by the continuity lemma for measures (Proposition A1.3.II), N {y} ≡ N ({y}) = lim N Sj (y) . j→∞

Each term on the right-hand side is nonnegative integer-valued; the same therefore applies to N {y}. Thus, if y is not an atom of N , it must be the limit of a sequence of open spheres for which N (Sj (y)) = 0, hence, in particular, the centre of an open sphere with this property. This shows that the support of N (the complement of the largest open set with zero measure) consists exclusively of the atoms of N , or equivalently, that N is purely atomic. Equation (9.1.6) now follows from (9.1.5). The properties of N ∗ in (iii) follow from the representations in (i) and (ii). Part (iv) follows from Definition 9.1.II(iv) and part (ii) because δ(xi ,ki ) can be identified with an atom in the product space. For part (v), (9.1.9a) is a restatement of (9.1.5). The condition at (9.1.8) is a restatement of the requirement that the measure µ be boundedly finite. The representation in (9.1.9b) mimics the construction using decreasing spheres to prove (ii) but with decreasing sets from a sequence of partitions from the dissecting system. Because K is bounded away from 0, there are at most a finite number of atoms with locations in A and values in K. As the sets in the dissecting system shrink, each of these atoms will ultimately be isolated in one of the subsets, leading to the representation (9.1.9b). Notice that this proof makes essential use of the topological structure of X ; Moyal (1962) discusses some of the difficulties that arise in extending it to more general contexts. Basic properties of M# X are set out in Section A2.6 of Appendix 2, from which the key points for our purposes are set out below, together with their counterparts for NX# .

6


Proposition 9.1.IV. (i) Under the w# -topology, M# X is a c.s.m.s. in its own right. (ii) The corresponding Borel σ-algebra, B(M# X ) say, is the smallest σ-algebra on M# with respect to which the mappings µ → µ(A) are measurable X for all A ∈ BX . (iii) Under the w# -topology, NX# is a c.s.m.s. in its own right, and its Borel sets coincide with the Borel sets of NX# as a subset of M# X. # (iv) B(NX ) is the smallest σ-algebra with respect to which the mappings N → N (A) are measurable for each A ∈ BX . Statements (i) and (ii) form Theorem A2.6.III. The extensions to counting measures follow from the next lemma. Lemma 9.1.V. NX# is a closed subset of M# X. Proof. Let {Nk } be a sequence of counting measures converging to some limit measure N in the w# -topology in M# X . As in the proof of Proposition 9.1.III, let y be an arbitrary point of X , and Sj (y) a sequence of spheres, contracting to {y}, with the additional property that N ∂Sj (y) = 0

(j = 1, 2, . . .)

[this is always possible because N S (y) , as a function of , has jumps for at most countably many values of , and thus, the complementary set of values of

, being dense, the j can be chosen in the complementary set]. For each such sphere it follows from the properties of w# -convergence (Proposition A2.6.II) that Nk Sj (y) → N Sj (y) . Once again the terms on the left-hand side are all nonnegative integers, so the same is true for the term on the right-hand side. As in the previous proof, it then follows that N is purely atomic. This argument shows that # NX# is sequentially closed in M# X , and hence closed because MX is separable (Theorem A2.6.III). Exercise 9.1.2 shows that the spaces in Definitions 9.1.II(iii)–(iv), although # # measurable subsets of M# X , are not closed in either MX or NX . # topoloSimilarly, the space M# X ,a is not closed in the weak or weak gies: the sequence of purely atomic measures with atoms of mass 1/n at {1/n, 2/n, . . . , 1} converges weakly to Lebesgue measure on (0, 1). Properties (ii) and (iv) of Proposition 9.1.IV open the way to defining random measures and point processes as measurable mappings involving the spaces of Definition 9.1.II, and lead to simple characterizations of random measures and point processes.

9.1.


7

Definition 9.1.VI. (i) A random measure ξ with phase or state space X , is a measurable map # ping from a probability space (Ω, E, P) into M# X , B(MX ) . (ii) A point process N on state space X is a measurable mapping from a probability space (Ω, E, P) into NX# , B(NX# ) . (iii) A point process N is simple when P{N ∈ NX#∗ } = 1.

(9.1.10)

(iv) A marked point process on X with marks in K is a point process N on BX ×K for which (9.1.11) P{N ∈ NX#g ×K } = 1; its ground process is given by Ng (·) ≡ N (· × K). (v) A purely atomic random measure ξ is a measurable mapping from a # probability space (Ω, E, P) into M# X ,a , B(MX ,a ) . (vi) An extended MPP with positive marks is a point process on BX ×R+ which is finite-valued on all sets of the form A × K for bounded A ∈ BX and Borel sets K ⊂ ( , 1/ ) for some > 0. The notation of Definition 9.1.VI(i) is intended to imply that with every sample point ω ∈ Ω, we associate a particular realization that is a boundedly finite Borel measure on X ; we denote it by ξ(·, ω) or just ξ(·) (or even ξ) when we have no need to draw attention to the underlying spaces. Similar statements can be made for counting measures and point processes N (·, ω), N (·), and so on. A consequence of Definitions 9.1.VI(i)–(ii) above and Definition 9.1.II(ii) for NX# , is that a random measure is a point process if and only if its realizations are a.s. integer-valued. Observe that by choosing the state space X appropriately, Definitions 9.1.VI(i)–(ii) can be made to include not only a number of important special cases but also a number of apparent generalizations. In the case X = R, discussion of one-dimensional random measures is essentially equivalent, as we note in Example 9.1(c), to the discussion of processes with nonnegative increments. The cases X = Rd , d ≥ 2, correspond to multidimensional random measures. If X has the product form Y × K, where K is a finite set, {1, . . . , d} say, and we define distance in Y × K by (for example) d (x, i), (y, j) = ρ(x, y) + |i − j|, the resulting process is a multivariate random measure; each of its d components is a random measure on Y. This itself is a special case of a point process defined on a product space; when both components are metric spaces, any one of a number of combinations of the two individual metrics—the additive form above is one convenient choice—will make the product space into a metric space and so allow the basic machinery to be applied. The assumption that

8


such a choice can and has been made underlies the introduction of MPPs in Section 6.4 and Definition 9.1.VI(iv) which, as already noted, is equivalent when coupled with Definition 9.1.II(iv) to the requirement that for the ground process, Ng (A) ≡ N (A × K) < ∞ a.s. for all bounded A ∈ BX . Exercise 9.1.4 indicates that for a marked point process N and any Borel set K ∈ BK , the ‘K-marginal’ process NK (·) = N (· × K) is a well-defined simple point process. That Ng is boundedly finite and simple follows from Definition 9.1.II(iv). Exercise 9.1.5 sets out in greater detail the extension described in Definition 9.1.VI(vi). The motivation behind this definition is the construction of the counting measure Nµ in Proposition 9.1.III(v) for a purely atomic measure µ. Using this definition and the measurability assertion in Proposition 9.1.III(v) leads to the following equivalence for purely atomic random measures. Lemma 9.1.VII. Equations (9.1.9a, b) establish a one-to-one correspondence between purely atomic random measures ξ and extended MPPs with positive marks, Nξ say, satisfying the condition (9.1.8). A realization of a random measure ξ has the value ξ(A, ω) [or we may write just ξ(A)] on the Borel set A ∈ BX [and, similarly, N (A) for a point process N ]. For each fixed A, ξA ≡ ξ(A, ·) is a function mapping Ω into R+ , and thus it is a candidate for a nonnegative random variable; that it is indeed such is shown in the following proposition. Proposition 9.1.VIII. Let ξ (respectively, N ) be a mapping from a proba# bility space into M# X (NX ) and A a semiring of bounded Borel sets generating ξ is a random measure (N is a point process) if and only if ξA BX . Then N (A) is a random variable for each A ∈ A. Proof. Let U be the σ-algebra of subsets of M# X whose inverse images under ξ are events, and let ΦA denote the mapping taking a measure µ ∈ M# X into µ(A) [hence, in particular, ΦA : ξ(·, ω) → ξ(A, ω) ]. Because ξA (ω) = ξ(A, ω) = ΦA ξ(·, ω) as in Figure 9.1, we have for any B ∈ BR+ −1 ξ −1 Φ−1 (B). A (B) = (ξA ) When ξA is a random variable, (ξA )−1 (B) ∈ E, and then by definition we # have Φ−1 A (B) ∈ U. It now follows from Theorem A2.6.III that B(MX ) ⊆ U and hence that ξ is a random measure. # −1 Conversely, by definition of B(M# X ), ΦA (B) ∈ B(MX ), and when ξ is a −1 −1 random measure, ξ (ΦA (B)) ∈ E, so then ξA is a random variable. Taking for A the semiring of all bounded sets in BX we obtain the following corollary. (respectively, N : Corollary 9.1.IX. ξ: Ω → M# X is a random measure Ω → NX# is a point process) if and only if ξ(A) N (A) is a random variable for each bounded A ∈ BX .

9.1.


9

ξ: ω → ξ(·, ω) Ω ξA : ω → ξ(A, ω)

M# X ΦA : µ → µ(A)

IR+

Figure 9.1 One useful consequence of Proposition 9.1.VIII is that we may justifiably use ξ(A) to denote the random variable ξA as well as the value ξ(A, ω) of the realization of the random measure ξ. Definitions 9.1.VI on their own do not lend themselves easily to the construction of particular random measures or point processes: for this the most powerful tool is the existence Theorem 9.2.VII below. Nevertheless, using Proposition 9.1.VIII or its corollary, we can handle some simple special cases as below in Examples 9.1(a)–(e) and Exercises 9.1.7–8. Example 9.1(a) Uniform random measure. Let X be the real line, or more generally any Euclidean space Rd , and define ξ(A) = Θ(A), where (·) denotes Lebesgue measure on X and Θ is a random multiplier that is nonnegative. To set this up formally, take Ω to be a half-line [0, ∞), E the Borel σ-algebra on Ω, and P any probability measure on Ω, for example, the measure with gamma density xα e−x /Γ(α). This serves as the distribution of Θ. For each particular value of Θ, the corresponding realization of the random measure is the Θ multiple of Lebesgue measure. (Note that this process is random only in a rather artificial sense. Given only one realization of the process, we would have no means of knowing whether it is random. Randomness would only appear if we were to observe many realizations. In the language of Chapter 12, the process is stationary but not ergodic.) # We are left with one task, to verify that the mapping in M# X , B(MX ) is indeed measurable. By Proposition 9.1.VIII, it is sufficient to verify that the mappings ξ(A) are random variables for each fixed A ∈ BX . In our case, ξ(A) is a multiple of the random variable Θ, so the verification is trivial. Thus, we have an example of a random measure. Example 9.1(b) Quadratic random measures: measures with χ2 density [see Example 6.1(c) and Exercise 6.1.3]. In Volume I we sketched a class of random measures constructed as follows. Take X = R, and choose any Gaussian process Z(t, ω) with a.s. continuous trajectories. [Sufficient conditions for this can be expressed in terms of the covariance function: for example, it is enough for Z(·) to be stationary with covariance function c(u) that is continuous at

10


u = 0; Cramér and Leadbetter (1967) give further results of this kind.] Then set Z 2 (t) dt for continuous Z(·, ω), A ξ(A) = 0 otherwise. Let us prove more formally that this construction defines a random measure. Because Z 2 (t) ≥ 0, it is clear that ξ(A) ≥ 0, and countable additivity is a standard property of indefinite integrals. Moreover, because Z 2 (·) is a.s. continuous, it is bounded on bounded sets, and so ξ(·) is boundedly finite. For almost all ω, therefore, ξ(·, ω) is a boundedly finite Borel measure. To complete the proof that ξ(·) is a random measure we check that the condition of Proposition 9.1.VIII is met. Let A be any finite half-open interval (left-open right-closed for definiteness), and let Tn = {Ani : i = 1, . . . , kn } be a sequence of partitions of A into subintervals with lengths 1/n or less. If tni is a representative point from Ani , it follows from standard properties of the Riemann integral that as n → ∞, ξn (A) ≡

kn i=1

Z 2 (tni )(Ani ) →

Z 2 (t) dt = ξ(A)

a.s.

A

Each Z(t) is a random variable by assumption, and therefore so too is ξn (A) (as a linear combination of random variables) and ξ(A) (as the limit of a sequence of random variables). It is then clear that ξ(A) is a random variable for every set A in the semiring of finite unions of left-open right-closed intervals. It now follows from Proposition 9.1.VIII that ξ(·) is a random measure. The distributions of ξ(A) are nearly but not quite of gamma form: each particular value Z 2 (t) has a gamma distribution and is proportional to a χ2 random variable with one degree of freedom, so that the integral defining ξ(A) behaves as a linear combination of gamma variables. Its characteristic function can be obtained in the form of an infinite product of rational factors each associated with a characteristic root of the integral operator with kernel c2 (u − t) on A × A. Exercise 6.1.3 asserts that when Z(·) is stationary, so too is ξ(·), and its moments have been given there [see also Example 9.5(a)], whereas Example 9.3(a) discusses sample-path properties. Because a quadratic random measure has a gamma process as its density, it is a candidate for the directing measure of the class of Cox processes called negative binomial processes in Barndorff-Nielsen and Yeo (1969). More generally, sums of independent quadratic random measures have gamma process densities, and also therefore meet Barndorff-Nielsen and Yeo’s definition, but, as they noted, although these point processes have computable moment properties, their distributional properties are not so readily accessible other than in degenerate cases (cf. Exercise 9.1.9). These negative binomial processes differ from those of Example 6.4(b), where the distributions are exactly of

9.1.


11

negative binomial form, but, in contrast to this example, the earlier processes are either non-orderly or non-ergodic. Example 9.1(c) Processes with nonnegative increments. Let X = R. It seems obvious that any stochastic process X(t), defined for t ∈ R and possessing a.s. finite-valued monotonic increasing trajectories, should define a random measure through the relation ξ(a, b] = X(b) − X(a).

(9.1.12)

We show that this is the case at least when X(t) is also a.s. right-continuous. In any case, (9.1.12) certainly induces, for each realization, a finitely additive set function on the ring of finite unions of half-open intervals. Rightcontinuity enters as the condition required to secure countable additivity on this ring (compare the conditions in Proposition A2.2.VI and Corollary A2.2.VII). Then the set function defined by (9.1.12) can be extended a.s. to a boundedly finite measure, which we may continue to denote by ξ on BR . ξ now represents a mapping from the probability space into M# R . Because X(t) is a stochastic process, ξ(a, b] is a random variable for each half-open interval (a, b]. Proposition 9.1.VIII now implies that ξ is a random measure. The condition of right continuity can always be assumed when X(t) is stochastically continuous, that is, whenever for each > 0, Pr{|X(t + h) − X(t)| > } → 0

as h → 0,

(9.1.13)

because we may then define a new process by setting X ∗ (t) = X(t + 0) and it is easy to verify that X ∗ (t) is a version or copy of X(t), in the sense that it has the same fidi distributions. This condition is satisfied in particular by processes with stationary independent increments, giving rise to stationary random measures with the completely random property of Section 2.2. The next example illustrates the type of behaviour to be expected; Section 10.1 gives a more complete discussion of completely random measures. Example 9.1(d) Gamma random measures—stationary case. We indicated in Example 6.1(b) (see also Exercise 6.1.1) that a stationary random measure is defined by r.v.s ξ(Ai ) that are mutually independent for disjoint Borel sets Ai in Rd and have Laplace–Stieltjes transforms ψ(Ai ; s) = (1 + λs)−α(Ai )

(λ > 0, α > 0, Re(s) ≥ 0);

these transforms show that the ξ(·) are gamma distributed. The convergence ψ(Ai ) → 1 as (Ai ) → 0 shows that the process is stochastically continuous, so we can assume right-continuity of the sample paths. The discussion around Proposition 9.1.VIII then implies that the resulting family of random variables can be extended to a random measure.

12


Despite the condition of stochastic continuity, the measures ξ(·) here are not absolutely continuous, but on the contrary have a purely atomic character. This follows from the Lévy representation theorem which asserts that a process with independent increments can be represented as the sum of a shift, a Gaussian component, and an integral of Poisson components indexed according to the heights of the jumps with which they are associated [see e.g. Feller (1966, Section XVII.2), Bertoin (1996), or Theorem 10.1.III below]. The existence of a Gaussian component is ruled out by the monotonic character of the realizations, which also implies that the jumps are all positive. Thus, the random measure can be represented as a weighted superposition of Poisson processes, in the same kind of way as the compound Poisson process of Section 2.2. In the present case, however, the random measure has a countable rather than a finite number of atoms in any finite interval, but most atoms are so small that the total mass in such an interval is still a.s. finite. See also Example 9.1(g) and the discussion preceding it. Example 9.1(e) Random probability distributions and Dirichlet processes. Random probability distributions play an important role in the theory of statistical inference, in particular, as prior distributions in nonparametric inference. Here we outline one method that has been proposed for constructing such distributions. Further constructions are in Exercises 9.1.10 and 9.3.4. Suppose given a random measure ξ on the c.s.m.s. X , with ξ a.s. totally finite and nonzero, and define ζ(A) = ξ(A)/ξ(X )

(A ∈ BX )

(9.1.14)

[in full, ζ(A, ω) = ξ(A, ω)/ξ(X , ω)]. Proposition 9.1.VIII shows that ζ is a random measure; because ζ(X ) = 1 a.s., it is a random probability measure. The Dirichlet process Dα is the random measure ζ defined by the ratio at (9.1.14) when ξ is a gamma random measure as in the previous example and Exercise 6.1.1. Straightforward algebra shows that ζ(A) has a beta distribution and, more generally, that the fidi distributions of ζ(Ai ) over disjoint sets Ai , i = 1, . . . , r, are multivariate beta distributions. Exercise 9.5.1 gives moment measures of the process. For a Dirichlet process on X = R, write Fζ (·) for the random d.f. associated with the random probability distribution ζ. Then the random variables Zi = −∞ = x0 < x1 < · · · < xr−1 < xr = Fζ (xi ) − Fζ (xi−1 ), i = 1, . . . , r, where ∞ are such that, if each αi = α (xi−1 , xi ] > 0, their joint distribution is singular with respect to r-dimensional Lebesgue measure but absolutely continuous with respect to (r − 1)-dimensional Lebesgue measure on the simplex {(z1 , . . . , zr−1 ): z1 + · · · + zr−1 = 1 − zr ≤ 1}, where it has the density function r r αi −1 zi . αi f (z1 , . . . , zr−1 | α1 , . . . , αr ) = Γ Γ(αi ) i=1 i=1 Ferguson (1973) supposes that an (unobserved) realization ζ of Dα governs independent observations X1 , . . . , Xn for which the parameter α specifies the prior distribution of ζ. He shows that, conditional on (X1 , . . . , Xn ) =

9.1.


13

(x1 , . . . , xn ), the posterior ndistribution of ζ is again that of a Dirichlet process but has parameter α + i=1 δxi . Our later discussion of completely random measures around (10.1.4) implies that both ζ and ξ have realizations that are purely atomic, and hence that the possible d.f.s in such a random distribution are a.s. purely discrete. This is actually an advantage in the above discussion, as it is the feature that allows prior and posterior to have the same distributional form. See also Exercise 9.1.11, where the gamma random measure and the Dirichlet distribution are used to define a prior distribution for an inhomogeneous Poisson process. Concerning the existence of point processes, two basic approaches are widely used in the literature. A point process is defined sometimes as an integer-valued random measure N (·, ω), as above, and sometimes as a sequence of random variables {yi }. When are these approaches equivalent? In one direction the argument is straightforward and covered in the next result where we start from certain finite or countably infinite sequences {yi : i = 1, 2, . . .} of X -valued random elements. Proposition 9.1.X. Let {yi } be a sequence of X -valued random elements defined on a probability space (Ω, E, P), and suppose that there exists an / E0 implies that for any bounded event E0 ∈ E such that P(E0 ) = 0 and ω ∈ set A ∈ BX , only a finite number of the elements of {yi (ω)} lie within A. Define N (·) to be the zero measure on E0 and otherwise set N (A) = #{yi ∈ A} =

i

δyi (A)

(A ∈ BX ).

(9.1.15)

Then N (·) is a point process. Proof. The set function N (A) is clearly a.s. integer-valued and finitely additive on Borel sets. Given a sequence {yi }, choose y ∈ / {yi } and let {Aj } be a sequence of bounded Borel sets decreasing to {y}. Because any Aj is bounded, N (Aj ) < ∞ a.s. Then for each yi ∈ Aj , there exists finite ji such / Aj for j > ji . Therefore, for some finite j , N (Ak ) = 0 for all that yi ∈ k > j , and hence ∞ N A j=1 j = 0 a.s.; that is, N (Aj ) → 0 a.s. Thus, N (·) must be not just finitely but a.s. countably additive. This is enough to show that the sequence {yi } induces a counting measure on X , and so sets up a mapping from its probability space into NX# . To show that N (·) is a point process, the critical step is to show that this mapping is measurable. From Proposition 9.1.VIII it is enough to show that for each Borel set A, N (A) is a random variable. To this end, for each k = 1, 2, . . . and ω ∈ / E0 , write Nk (A, ω) =

k i=1

δyi (ω) (A).

(9.1.16)

14


Because yi is an X -valued random variable, and A is a Borel subset of X , each δyi (A) is a random variable for i = 1, . . . , k and therefore so too is Nk (A). By monotonicity, N (A) = limk→∞ Nk (A) is well defined and therefore a random variable as required. The main problem with this approach to point processes arises from the need to express, in terms of the distributions of the yi , the condition that with probability 1 the counting measures are boundedly finite. Suppose, for example, that the yi are generated as the successive states in a Markov chain with state space X and that the transition function for the chain satisfies an irreducibility condition sufficient to imply the usual classification into recurrent and transient chains. Then the local finiteness condition is satisfied if and only if the chain is transient. Exercises 9.1.12–13 illustrate this point. We now turn to the more difficult question of constructing a sequence of X -valued r.v.s from a given point process. Recall from Proposition 9.1.III that the realizations of the random counting measure N (·) determine a.s. a countable set of atoms without any finite accumulation point, but this ignores the problem of finding a meaningful ordering of the atoms, without which the interpretation of the locations of the atoms as random variables is unresolved. The idea of fixing some enumeration of the points, in a measurable way, prompts the following definition. Definition 9.1.XI. Let N be a point process on a c.s.m.s. X as in Definition 9.1.VI(ii). A measurable enumeration of N is a sequence of X -valued r.v.s {yi (N ) ≡ y i (N (·, ω)): i = 1, 2, . . .} such that for every bounded Borel set A, ∞ N (A, ω) = i=1 δyi (N (·,ω)) (A) a.s. As a first illustration of how a sequence of random variables {yi } can be extracted from the counting measure N (·), we formalize the discussion at the end of Section 3.1 concerning the relation between counting and interval properties of a simple point process N ∈ NR#∗ . Retaining the notation of that section, recall the definitions ⎧ (t > 0), ⎪ ⎨ N ((0, t]) N (t) = 0 (t = 0), (9.1.17) ⎪ ⎩ −N ((t, 0]) (t < 0), and for i = 0, ±1, . . . , ti (N ) = inf{t: N (t) ≥ i}, τi (N ) = ti (N ) − ti−1 (N )

(9.1.18a) (9.1.18b)

(Figure 9.2 illustrates the relationship between {ti } and {τi }). Let S + denote the space of all sequences {τ0 , τ±1 , τ±2 , . . . ; x} of positive numbers τi satisfying 0 ≤ x < τ0 and ∞ ∞ τi = τ−i = +∞. (9.1.19) i=1

i=1

9.1.


15

··· τ0 τ1 ··· τn τn+1 −−−−−−−−|−−−−−−−−|−−−−− | −−−−−−−−−−−− | −−−−−−−−−− | −−−−−−|−−→ time t1 ··· tn−1 tn tn+1 · · · t−1 [0] t0 Figure 9.2 Intervals τ1 , . . . , τn , . . . between successive points t0 , t1 , . . . , tn−1 , tn , . . . . When applicable, 0 satisfies t−1 < 0 ≤ t0 , so that the interval of length τ0 contains 0.

Then adding the relation x(N ) = −t0 (N ) to τi at (9.1.18) defines a mapping R: NR#∗ → S + . The inverse mapping R−1 is defined for s+ ∈ S + by t0 (s+ ) = −x0 (s+ ),

ti (s+ ) =

ti−1 (s+ ) + τi ti+1 (s+ ) − τi+1

(i ≥ 1), (i < 0).

(9.1.20)

We give S + the Borel σ-algebra B(S + ) obtained in the usual way as the product of σ-algebras on each copy of R+ . Proposition 9.1.XII. The mapping at (9.1.18), R say, provides a one-to-one both ways measurable mapping of NR#∗ into S + . In particular, (i) the quantities τi (N ) and x(N ) are well-defined random variables when N is a simple point process; and (ii) there is a one-to-one correspondence between the probability distributions P ∗ of simple point processes on NR#∗ and probability distributions on the space S + . Proof. The relations (9.1.18a) define the ti (N ) as stopping times for the increasing, right-continuous process N (·) (see Definition A3.3.II and Lemma A3.3.III). Hence the ti (N ), and also therefore the τi (N ), are random variables whenever N is a simple point process. To establish the converse, observe that the N can be defined in terms of the sequence of random variables ti , noting that the requirement (9.1.19) implies that the resulting point process is boundedly finite. Then it follows from Proposition 9.1.X that N is a well-defined, simple point process. Note, in particular, the intervention of the initial interval (0, t1 ], the length x of which must be given separately: there is not a one-to-one correspondence between intervals and simple counting measures (Exercise 9.1.14 provides a further example of this type and a counterexample involving the subset N0 ⊂ NR#∗ with an atom at 0 and the subset S0+ for which t0 = x = 0). Sigman’s (1995) Appendix D discusses these questions also. There is no analogous simple representation for point processes in Rd , d = 2, 3, . . . , although Exercise 9.1.15 sketches a possible construction based on distances of points from an origin (see also below Lemma 13.3.III). A general construction that exploits the separability property of the c.s.m.s. X was suggested by Nguyen and Zessin (1976) and incorporated into Theorem 1.11.5 of MKM (1982). It is based on an ordered system of tilings of the c.s.m.s. X , meaning a countable family T = {Tn } of ‘infinite’ dissecting systems of

16


Tn : Tn+1 :

An1 An+1,1

··· ···

x An,in An+1,in+1

··· ···

y An,jn An+1,jn+1

··· ···

Figure 9.3 Tilings Tn = {Ani }, Tn+1 = {An+1,i } with sets containing x, y.

X : each tiling Tn = {Ani : i = 1, 2, . . .} consists of countably many disjoint bounded sets Ani satisfying (i) (Partition and tiling properties) Ani ∩Anj = ∅ for i = j, and i Ani = X ; (ii) (Nesting property) A n−1,i ∩ Anj = Anj or ∅, and any An−1,i ∈ Tn−1 is expressible An−1,i = j∈Jn−1,i Anj for some finite set of indices Jn−1,i ; (iii) (Point-separating property) given distinct x, y ∈ X , there exists an inte/ Ani ; and ger n(x, y) such that for n ≥ n(x, y), x ∈ Ani implies y ∈ (iv) (Enumeration consistency property) given distinct x, y ∈ X , when Tm is such that x ∈ Ami y, for all n ≥ m sets An,in x, An+1,in+1 x and An,jn y, An+1,jn+1 y are such that jn − in and jn+1 − in+1 have the same sign. The first three properties above are analogues of the properties of a dissecting system (Definition A1.6.I). We know from Proposition A2.1.IV that on any bounded Borel set in BX there exists a dissecting system. We also know that a c.s.m.s. is covered by the union of a countable increasing family of boundedly finite sets Si say; then {A1i } = {Si+1 \Si } is a tiling. Now introduce dissecting systems on each A1i , and enumerate their members in one sequence {A2i } so as to satisfy property (iv); then it is a tiling of X . Given an integer-valued measure N on BX , N (A1i ) is a finite integer for each i, and for each i = 1, 2, . . . we can enumerate the atomic support of N within those A1i for which N (A1i ) ≥ 1 via the tilings. Moreover, if x, y ∈ X are such that N ({x}) ≥ 1 and N ({y}) ≥ 1, then x and y will be enumerated (with appropriate multiplicity if either inequality is strict) in a finite number of operations starting from some Si that contains both x and y. The determination of such x or y proceeds via a sequence ofnested sets An,in say for ∞ which N (An,in ) ≥ N (An+1,in+1 ) ≥ 1 for all n with n=1 An,in = {x} say. It follows that an enumeration of the points of N is thereby determined, for a sequence y1 , y2 , . . . , within a finite number of steps for each yr even before its precise location is known. Now write yr = lim j Aj,ij,r for some monotonic decreasing sequence of sets Aj,ij,r for which N (Aj,ij,r ) = 1 for all sufficiently large j. For any given finite enumeration of points, the position in the enumeration is found from a finite number of elements of dissecting systems, with all the associated counting measures N (·) measurable. The limit is therefore measurable; that is, the enumeration is measurable as required. This outline argument leads to the following assertion. Lemma 9.1.XIII. Given a point process N on a c.s.m.s. X as in Definition 9.1.VI(ii), there exists a measurable enumeration of X -valued random

9.1.


17

elements {yi } satisfying (9.1.16). This enumeration is uniquely determined by a given family of bounded sets {Si } with Si ↑ X and of dissecting systems for each Si \ Si−1 . A random measure may be regarded as a family of random variables indexed by the Borel sets of X , but it is considerably more than this. The additivity and continuity properties of measures require at least the truth of ξ(A ∪ B) = ξ(A) + ξ(B) a.s.

(9.1.21)

for all pairs of disjoint Borel sets A, B in X , and ξ(An ) → 0

a.s.

(9.1.22)

for all sequences of bounded Borel sets An such that An ↓ ∅. It is not quite trivial to prove, but fundamental for the resulting theory, that these conditions are in fact sufficient for the family to form a random measure. The difficulty is associated with the exceptional sets of measure zero: because there is an uncountable family of relations (9.1.21) and (9.1.22), it is not clear that the exceptional sets can be combined to form a single set that is still of measure zero. The next lemma indicates one way around the difficulty. Lemma 9.1.XIV. Let A be a countable ring of bounded Borel sets with the self-approximating property of Definition A2.2.VIII, and ξA (ω) a family of nonnegative random variables indexed by the sets A ∈ A. In order that, with probability 1, the ξA (ω) should admit an extension to a measure on σ(A), it is necessary and sufficient that (9.1.21) hold for all disjoint pairs (A, B) of sets in A and that (9.1.22) hold for all sequences {An } of sets in A with An ↓ ∅. Proof. The number of sets in A being countable, it follows immediately that (9.1.21) implies that the ξA (ω) are a.s. finitely additive there. To establish countable additivity we use the covering property of Lemma A2.2.IX, from which it follows that it is enough to know that n F (A; 1/k) = ξ(A) (9.1.23) lim ξ i i=1 n→∞

simultaneously for all sets A ∈ A and integers k < ∞. Because each such relation holds a.s. from (9.1.22), and because the number of sets A ∈ A and integers k < ∞ is countable, this requirement is satisfied almost surely. Then Lemma A2.2.IX implies that the ξA can be a.s. extended to a measure on σ(A). The necessity of both conditions follows directly from the additivity and continuity properties of a measure. As an immediate corollary we obtain the following theorem, in which the point process analogues of (9.1.21) and (9.1.22), for A, B, and {An } as above, are N (A ∪ B) = N (A) + N (B) a.s., (9.1.24a) N (An ) → 0 a.s. (9.1.24b)

18


Theorem 9.1.XV. Let {ξA (ω)} [respectively, NA (ω)] be a family of nonnegative random variables indexed by the sets of BX and a.s. finite-valued (finite integer-valued) on bounded Borel sets. In order that there exist a random measure ξ ∗ (A, ω) (point process N ) such that, for all A ∈ BX , ξ ∗ (A, ω) = ξA (ω) a.s.,

[N (A) = NA

a.s.],

(9.1.25)

it is necessary and sufficient that (9.1.21) [(9.1.24a)] hold for all pairs A, B of disjoint bounded Borel sets and that (9.1.22) [(9.1.24b)] hold for all sequences {An } of bounded Borel sets with An ↓ ∅. Proof. Let A be any countable generating ring of bounded Borel sets with the self-approximating property of Definition A2.2.VIII, as, for example, the ring C following Lemma A2.2.IX. If (9.1.21) and (9.1.22) hold for Borel sets in general, they certainly hold for sets in A. Thus, the conditions of Lemma 9.1.XIV are satisfied, and we can assert that with probability 1 the ξA (ω), initially defined for A ∈ A, can be extended to measures ξ ∗ (A, ω) defined for all A ∈ σ(A) = BX . For ω in the P-null set, U say, where the measures cannot be so extended, set ξ ∗ (A, ω) = 0. Then ξ ∗ (A, ω) is a random measure which coincides a.s. with the original random variables ξ(A, ω) at least on A. It is not immediately obvious, nor indeed is it necessarily true, that the / extensions ξ ∗ (A, ω) coincide with the original random variables ξA (ω) for A ∈ A, even outside the exceptional set U of probability zero where the extension may fail. The best we can do is to show that they are a.s. equal for each particular Borel set A. The exceptional sets may be different for different A, and we do not claim that they can be combined into a single exceptional set of measure zero. Consider the class of sets on which ξ ∗ and ξ coincide a.s. This class includes A, and from the relations (9.1.22) it is closed under monotone limits. By the monotone class theorem it therefore includes σ(A), which by assumption is BX . This proves (9.1.25), and hence also the sufficiency part of the theorem. Necessity is an easy corollary of the additivity and continuity properties of a measure. The arguments can be applied equally to the case that the ξ is a.s. a counting measure, leading to the analogous result for a point process. As a sample application of Theorem 9.1.XV, we outline an approach to the definition of what, loosely speaking, might be termed a conditional random measure. It can be used to provide alternative proofs for the existence of the doubly stochastic and cluster processes introduced in Chapter 6. Proposition 9.1.XVI. Let ξ be a random measure defined on the probability space (Ω, E, P) with some c.s.m.s. X as state space, and let F be a sub-σ-algebra there exists a version of the conditional expectation of E. Then η(A, ω) = E ξ(A) | F (ω) such that (i) for each A ∈ BX , η(A, ·) is an F-measurable r.v.; and (ii) η is a random measure with state space X .

9.1.


19

Proof. It is easy to see from standard properties of conditional expectations that the additivity and consistency relations (9.1.21) and (9.1.22) are both satisfied for the conditional expectations η(A, ω). Furthermore, we may take the probability space here to be (Ω, F, PF ) rather than (Ω, F, P), where PF denotes the restriction of P to sets of F, because by definition the conditional expectations are all F-measurable. It now follows directly from Theorem 9.1.XV that there exists an F-measurable random measure η ∗ such that η ∗ (A) = η(A) a.s. An almost identical argument leads to the classical result on the existence of regular conditional distributions given a σ-algebra (see Exercise 9.1.16 for variants on this theme). To conclude this section, we would again emphasize the essential role played by the assumptions in the definitions. For example, the truth of Proposition 9.1.VIII and Theorem 9.1.XV depends in an essential manner on the assumption of nonnegativity. Corresponding statements for random signed measures are false in general: this is shown by the next example which, superficially, would be regarded as a random measure. Example9.1(f) Wiener’s homogeneous chaos. For A ∈ BX , let ξ(A) have a normal N 0, µ(A) distribution, where µ(A) is a fixed, boundedly finite, Borel measure on X , and suppose that the ξ(A) are independent for disjoint sets. These two requirements immediately allow the joint distributions of finite families ξ(A1 ), . . . , ξ(Ak ) to be written down, and it is easy to check that these joint distributions satisfy the consistency requirements of the Kolmogorov theorem. Thus, there does exist a probability space Ω on which the ξ(A) may be simultaneously defined as random variables. Now consider the random variable W = ξ(A1 ∪ A2 ) − ξ(A1 ) − ξ(A2 ), where A1 , A2 are disjoint bounded Borel sets. It is readily checked that E(W ) = 0 and var W = µ(A1 ∪ A2 ) + µ(A1 ) + µ(A2 ) − 2µ(A1 ) − 2µ(A2 ) = 0, so W = 0 a.s.Next consider the sequence {An } of disjoint bounded Borel ∞ sets with A = j=1 Aj , where A is also bounded, and set Wn ≡ ξ

n j=1

n Aj = ξ(Aj ) a.s., j=1

where the last equality follows by induction from the previous result. Then n n var Wn = µ = A µ(Aj ), j=1 j j=1

20


and if W = ξ

∞ j=1

Aj , we must have var(Wn − W ) → 0. This shows that n

ξ(Aj ) → ξ(A)

j=1

in quadratic mean, and because the ξ(Aj ) are independent, the partial sums converge to ξ(A) almost surely as well [see, e.g., Moran (1968, Theorem 8.24)]. We have shown that the family {ξ(Aj )} satisfies both (9.1.21) and (9.1.22). On the other hand it is not true that for almost all ω the realizations are signed measures. To see this, let {A1 , . . . , An } be a finite partition of A and set n |ξ(Aj )|. Yn = j=1

If the realizations ξ(·) were signed measures, the Yn would remain uniformly bounded a.s. over all possible partitions. But 1/2 n n 1/2 2 µ(Aj ) E |ξ(Aj )| = , E(Yn ) = π j=1 j=1 and

n

µ(Aj )

1/2

j=1

n

≥

µ(Aj ) µ(A) 1/2 = 1/2 , max1≤j≤n µ(Aj ) max1≤j≤n µ(Aj ) j=1

so E(Yn ) can be made arbitrarily large by choosing a partition for which max1≤j≤n µ(Aj ) is sufficiently small. Because var Yn ≤ µ(A) for every partition, an application of Chebyshev’s inequality shows that for any given finite y, a partition can be found for which Pr{Yn ≥ y} can be made arbitrarily close to 1. This is impossible if the Yn are a.s. bounded. Other examples may fail to be a random measure or point process because they fail to satisfy the bounded finiteness condition, as occurs for example with the jump points of many Lévy processes and for certain point sets for which Mandelbrot (1982, p. 78) proposed the term dust. A dust is a point set with infinitely many points in some bounded set and which has topological dimension D = 0 (Mandelbrot, 1982, pp. 15, 409–412). [Stoyan and Stoyan (1994, p. 4) describe a dust as an uncountable point set containing no piece of any curve; the uncountability assumption seems unnecessarily restrictive.] For example, the rationals on [0, 1] constitute an everywhere dense countable dust, whereas the Cantor set (or, Cantor dust) on [0, 1] [see, e.g., Halmos (1950, Exercise 15.5)] is an uncountable nowhere dense dust. Example 9.1(g) Lévy dust. Mandelbrot (1982, p. 240) includes the set of zeroes of Brownian motion B(·) as an example of a Lévy dust. Here we use the term here to mean the class of dusts defined via subordinators of a Brownian motion process [and, by a subordinator η(·) we mean a nonnegative

9.1.


21

3 2

B(η(j/n))

1 0 −1 −2 −3

0

2

4

6

8 η(j/n)

10

12

14

16

Figure 9.4 A space–time skeleton at {t = j/n: j = 1, . . . , 20n} of points of Lévy dust {(xi , B(xi ))} for standard Brownian motion B(·) at jump points {ti } of gamma random measure subordinator η(·), with xi = η(ti −). [n = 50, but η(j/n) − η([j − 1]/n) < 10−7 for over half the skeletal points.]

Lévy process with zero drift coefficient as in, e.g., Bertoin (1996, Chapter III and p. 16)], so that a Lévy dust consists of the set of values of the process y(t) = B[η(t)], where B(·) is a one- or two-dimensional Brownian motion. A subordinator η(·), being nondecreasing and a pure jump process with independent increments on R+ , has a countably infinite number of jump points, {ti } say, on any bounded interval of positive length. When a Markov process is used as a subordinand, the resultant process is again Markovian, and its range remains a countable set of points which, in the case of Brownian motion B(·), is just the countable set {yi } = {B(η(ti ))}. Indeed, the process B(η(·)) is again a Lévy process (Bertoin’s Exercise III.6.1). Figure 9.4 depicts a space–time skeleton of a sample of these points by plotting {(xj , yj )} = {(η(j/n), B(xj )): j = 1, . . . , 20n} in the case of a stationary gamma random measure η(t) = ξ(0, t] of Example 9.1(d) with α = 1. For Lévy dusts on R2 , when B(t) = [Z1 (t), Z2 (t)] and Zj (t) (j = 1, 2) are standard independent one-dimensional Brownian motions, a useful approximation for simulation and illustrative purposes is to ignore the ultra-fine structure (infinitely many exceedingly small increments) and treat the process as a random walk {Xn } in R2 whose steps Yn = Xn+1 − Xn have an isotropic distribution in R2 . For example, when η(·) is a nonnegative stable process, the step lengths follow approximately the Pareto form Pr{|Yn | > r | |Yn | > δ} = (δ/r)α

(r > δ)

[see, e.g., Ogata and Katsura (1991) and the more extended discussion in Mart´ınez and Saar (2002) of astrophysical applications, where the approximation is also known as Rayleigh–Lévy dust or Rayleigh–Lévy flights]. Two illustrations of such approximating point sets are shown in Figure 9.5.

22


1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

0.5

0

1

0

0.5

1

Figure 9.5 Two realizations of 1000 points of ‘Rayleigh–Lévy dust’, that is, a random walk with isotropic steps with d.f. tail (rmin /r)α on r > rmin = 0.001, α = 1.05 and 0.9 (left- and right-hand figures respectively), locations reduced modulo 1 to the unit square.

Exercises and Complements to Section 9.1 9.1.1 By definition, a sequence of totally finite measures {µk } that converges weakly to a totally finite measure µ converges in the w# sense to the same limit. Check that the converse need not be true by taking µk to be Lebesgue measure on [0, k]. [Hint: As k → ∞, µk does not converge weakly but does converge w# to Lebesgue measure on [0, ∞).] 9.1.2 (a) The sequence of measures {Nk } on BR defined by Nk = δ0 + δ1/k converges weakly to the measure N = 2δ0 ; each Nk ∈ NR#∗ but not the limit measure. (b) For k = 1, 2, . . . , let the measure ηk on X × K, with X = R and K = Z+ , have unit atoms at all the points {(i + j/2k , 2k ): i = 1, 2, . . . ; j = 0, 1, . . . , 2k − 1} so ηk has boundedly finite support in R × Z+ , and let Nr = rk=1 ηk . Show that each Nr is an element of NX#∗ ×K but that their limit is not.

9.1.3 Show that an MPP can be simple even if its ground process is not simple. [Hint: Simplicity of the MPP implies only that no single location has two identical marks.] 9.1.4 Let N be a simple point process on BX ×K , and K a fixed bounded Borel set in K. Show that NK (A) = N (A × K) (bounded A ∈ BX ) defines a simple point process. Deduce that the ground process Ng of Definition 9.1.V(iv) is well defined. 9.1.5 (a) Show that an extended MPP in the sense of Definition 9.1.6(vi) may fail to satisfy the requirements either of an MPP with marks in R+ or of a point process on R × R+ . [Hint: Consider as a counterexample the Poisson process arising in the Lévy representation of the gamma random measure of Example 9.1(d). The problem lies in satisfying the bounded finiteness properties as we have defined them.] (b) Show that the mapping (xi , κi ) → (xi , log κi ) defines a one-to-one mapping between the realizations of an extended MPP with marks in R+ and the

9.1.


23

space NX#×R . Use this mapping to define and explore the properties of a form of weak convergence for sequences of extended MPPs. 9.1.6 Show that, except for a set of P-measure zero, a realization of a marked point process (Section 6.4) can be regarded as a simple point process on the product space X ×K∪ , where K∪ = K(1) ∪K(2) ∪· · · , each K(k) consisting of all ordered sets of k-tuples of elements of the k-fold product set of K with itself, and the measure on each A × K(k) is symmetric in the subsets of K(k) . Conclude that any MPP is equivalent to another MPP whose ground process is simple. 9.1.7 Let {X(t): t ∈ R} be a measurable nonnegative stochastic process on (Ω, E, P). Show that, when the integrals concerned are finite, the relation X(t, ω) dt

ξ(A, ω) =

(bounded A ∈ BR )

A

defines a random measure ξ: Ω → M# R. [Hint: Start by considering X(t, ω) of the form j cj IAj (t)IEj (ω).] 9.1.8 Let N be a well-defined point process on X = R2 . With each point yi in a realization of N associate a geometric object in one of the following ways. (a) Construct a disk Sr (yi ) with centre yi and radius r, and let ξ(A) =

i

(A ∩ Sr (yi ))

(bounded A ∈ B(R2 ))

represent the total area of disks intersecting any Borel set A. Use Proposition 9.1.VIII to verify that ξ is a well-defined random measure on R2 . (b) If the radius of each disk is also a random variable, leading to SRi (xi ) say, a conditioning argument as in Example 6.4(e), coupled with some condition (A∩SRi (xi )), is needed. ensuring the a.s. finiteness of the defining sum (c) Instead of disks, construct from yi as endpoint, a finite line segment Li of length d and random orientation θi say, for some random variables {θi } that are i.i.d. on (0, 2π]. For any bounded Borel set A ⊂ R2 let (A ∩ L) now denote the Lebesgue measure (in R1 ) of the intersect of a line L with A. Again use a conditioning argument to show that ξL (A) ≡

i

(A ∩ Li )

(bounded A ∈ B(R2 ))

is a well-defined random measure. 9.1.9 Let N (A) denote the number of points in A ∈ BX of the negative binomial process of Example 9.1(b), involving a Cox process directed by the random measure ξ(A) = A η(u) du for η(·) a gamma process. Show that E(z N (A) ) = E( exp[−(1 − z)ξ(A)]). Derive a negative binomial approximation for suitably small sets A, and relate the first two moments of N (·) to those of ξ(·). 9.1.10 Random probability distributions. (a) Let ξ be a boundedly finite but not totally finite measure on R+ . Use the distribution function Fη (x) ≡ 1 − exp(−ξ[0, x]) to define a measure η on R+ , with η(R+ ) = 1. Show that when ξ is a random measure, η is a random probability measure on R+ . (b) When ξ is completely random (see Section 10.1), η is a ‘neutral process’ in the terminology of Doksum (1974). Show that when ξ has no deterministic component (see Theorem 10.1.III), the distribution η is a.s. purely atomoic.

24


9.1.11 Prior and posterior distributions for an inhomogeneous Poisson process. Suppose it is desired to fit an inhomogeneous but totally finite Poisson process to one or more sets of observations over X . Instead of assuming a specific parametric form for the intensity measure, suppose it is a gamma random measure Λ governed by a constant λ and some totally finite measure α(·) (see Exercise 6.1.1). Given a realization (x1 , . . . , xn ), show that the posterior distribution for Λ is again a gamma random measure, governed by the constant λ + 1 and the totally finite measure α + n i=1 δxi . Equivalently, we may take Λ = C.F where the prior distribution for the constant C is Γ(α(X ), λ), and the prior distribution for the probability distribution F has the Dirichlet form Dα .

9.1.12 Let {Xn } be a stationary ergodic real-valued Markov chain whose first absolute moment is finite, and define Yn = X1 +· · ·+Xn . If EXn = 0, then by the ergodic theorem {Yn } obeys the strong law of large numbers and therefore satisfies the conditions of Proposition 9.1.X. 9.1.13 Let {Yn : n = 0, 1, . . .} be a random walk in Rd ; that is, Y0 = 0 and the Rd valued r.v.s Xn ≡ Yn − Yn−1 , n = 1, 2, . . . are i.i.d. Show that the conditions of Proposition 9.1.X are satisfied if either d ≥ 3 or else d = 1 or 2 and E|Xn | < ∞, EXn = 0. [Hint: Under the stated conditions, a random walk in Rd is transient.] Note that a renewal process is the special case d = 1 and Xn ≥ 0 a.s. (and Xn = 0 a.s.), so that for some d.f. F on R+ , with 0 = F (0−) ≤ F (0+) < 1 = limx→∞ F (x), and any positive integer r, r

P({Xi ∈ (xi , xi + dxi ], i = 1, . . . , r}) =

[F (xi + dxi ] − F (xi )]. i=1

9.1.14 Given a nonnull counting measure N ∈ NR# , define {Yn : n = 0, ±1, . . .} or a subset of this doubly infinite sequence by N (0, Yn ) < n ≤ N (0, Yn ] N (Yn , 0] < −n + 1 ≤ N [Yn , 0]

(n = 1, 2, . . .), (n = 0, −1, . . .).

Show that if N is a point process then {Yn : −N (−∞, 0] + 1 ≤ n ≤ N (0, ∞)} is a set of well-defined r.v.s. Now let N0 be the subspace of NR#∗ consisting of simple counting measures on R with a point at the origin, so N ∈ N0 is boundedly finite, simple, and N {0} = 1. Show that if the atoms of such N yield the ordered set {. . . , t−1 , t0 = 0, t1 , . . .} and τi = ti − ti−1 , then the mapping Θ: N0 → S0+ which takes the counting measure N into the space S0+ of doubly infinite positive sequences {. . . , τ−1 , τ0 , τ1 , . . .} is one-to-one and both ways measurable with respect to the usual σ-fields in N0 and S0+ . Hence, probability measures on N0 and S0+ are in one-to-one correspondence. 9.1.15 To establish a measurable enumeration of the points of a point process on X ⊆ Rd , first locate an initial point in the sequence as the point closest to some spatial origin (recall that the point process is on X ⊆ Rd ), with the proviso that in the event of there being two or more points equidistant

9.2.

Finite-Dimensional Distributions and the Existence Theorem

25

from 0 [i.e., for some sphere Sr (0) and integer k ≥ 2, N (Sr (0)) = k and N (Sr− (0)) = 0 for every > 0], these k points are ordered lexicographically in terms of a coordinate system for X , yielding y1 , . . . , yk say. The remaining points can be found on a sequence of progressively larger spheres centred on 0, using a similar tie-breaking rule as needed. Show that such a sequence of points is a sequence of X -valued r.v.s as required in Definition 9.1.XI. Compare this construction with the discussion of finite point processes summarized in Proposition 5.3.II. 9.1.16 (a) Mimic Theorem 9.1.XV to establish the existence of regular conditional probabilities on a product space X × Y, where (X , E) is an arbitrary measurable space and Y is a c.s.m.s. (cf. Proposition A1.5.III). [Hint: Let π be a probability measure on the product space and πX the marginal distribution on (X , E). For fixed disjoint A, B ∈ BY show the existence of Radon–Nikodym derivatives Q(A | x), Q(B | x), Q(A ∪ B | x) such that Q(A ∪ B | x) = Q(A | x) + Q(B | x)

(πX -a.e. x).

Now identify Q(A | x) with ξA (ω) of the theorem and verify the continuity condition (9.1.22). For alternative approaches see Ash (1972, Section 6.6) and Feller (1966, Section V.10).] (b) Extend the above argument to the case of µ, a boundedly finite measure on X × Y, where X , Y are c.s.m.s.s and there exists a boundedly finite measure λ on BX such that µ(· × B) is absolutely continuous with respect to λ for bounded sets B ∈ BY ; that is, establish the existence of a family of measures µ(· | x) on BY for all x ∈ X such that µ(B | ·) is measurable for each bounded B ∈ BX , and for bounded sets A ∈ BX , B ∈ BY , µ(A × B) =

µ(B | x) λ(dx). A

[Hint: Normalize λ so that it is a probability measure on A.]

9.2. Finite-Dimensional Distributions and the Existence Theorem Only statements about the distributions of a process are amenable, via frequency counts and the like, to direct comparison with observations. This is some justification for the view that the theory of random measures and point to the study of the measures they induce on can be reduced # processes # # ) and N , B(N MX , B(M# X X X ) respectively. The deeper reason, however, is the unity and clarity that this point of view brings to questions concerning the existence of random measures and point processes. Using the characterization of such distributions through their finite-dimensional (fidi) distributions, as set out in Proposition 9.2.III below, we have an unequivocal answer to the problem of how to establish the existence of a particular class of point processes: can we write down for the class

26


a unique and consistent family of fidi distributions? This is the underlying reason for the importance of the studies by Moyal (1962) and Harris (1963), who were the first to set up a systematic theory of point processes in these terms. It also opens up the way for extensions to point processes on general types of spaces. On the other hand, the fidi distributions do not always provide the most convenient framework for examining the structure of particular models. For finite processes, the Janossy densities introduced in Chapter 5 are usually the most effective tool; likewise, for evolutionary processes, the conditional intensities introduced in Chapter 7 may prove extremely useful. But in all such cases, basic questions of existence can be referred back to the possibility of constructing a consistent family of fidi distributions. Definition 9.2.I. The distribution of a random measure or point process # , B(M ) or NX# , B(NX# ) , is the probability measure it induces on M# X X respectively. Definition 9.2.II. The finite-dimensional distributions (fidi distributions for short) of a random measure ξ are the joint distributions, for all finite families of bounded Borel sets A1 , . . . , Ak of the random variables ξ(A1 ), . . . , ξ(Ak ), that is, the family of proper distribution functions Fk (A1 , . . . , Ak ; x1 , . . . , xk ) = P{ξ(Ai ) ≤ xi (i = 1, . . . , k)}.

(9.2.1)

Let us say that the distribution of a random measure is completely determined by some quantities ψ if, whenever two random measures give the same values for ψ, their distributions coincide. Analogously to Theorem A2.6.III and Proposition 9.1.VIII, we have the following result. Proposition 9.2.III. The distribution of a random measure is completely determined by the fidi distributions (9.2.1) for all finite families (A1 , . . . , Ak ) of disjoint sets from a semiring A of bounded sets generating BX . Proof. Let R denote the ring generated by A. Then any element A of R k can be represented as the finite union of disjoint sets from A, A = i=1 Ai say, and thus, because k ξ(A) = ξ(Ai ), (9.2.2) i=1

the distribution of ξ(A) can be written down in terms of (9.2.1) for disjoint Ai . A similar result holds for the joint distributions of the ξ(A) for any finite family of sets Ai in R. Now consider the class of subsets of M# X of the form of cylinder sets, {ξ: ξ(Ai ) ∈ Bi (i = 1, . . . , k)},

(9.2.3)

where the Ai are chosen from R, and the Bi are Borel sets of the real line R. These cylinder sets form a ring, and it follows from Theorem A2.5.III

9.2.


27

that this ring generates B(M# X ). But the probabilities of all such sets can be determined from the joint distributions (9.2.1). Thus, the distribution of ξ is known on a ring generating B(M# X ) and it follows from Proposition A1.3.I(b) that it is determined uniquely. In the terminology of Billingsley (1968, p. 15), Proposition 9.2.III asserts that finite families of disjoint sets from a semiring A generating BX form a determining class for random measures on (X , BX ). Of course we also have the following corollary. Corollary 9.2.IV. The distribution of a random measure is completely determined by its fidi distributions. For a point process, that is, an integer-valued random measure, it is simplest to specify the fidi distributions in the notation of (5.3.9), namely, for bounded Borel sets A1 , A2 , . . . and nonnegative integers n1 , n2 , . . . , Pk (A1 , . . . , Ak ; n1 , . . . , nk ) = P{N (Ai ) = ni (i = 1, . . . , k)}.

(9.2.4)

As in Proposition 9.1.VIII, the distribution of a point process, meaning the measure induced on NX# , B(NX# ) , is completely specified by the fidi distributions of N (A) for A in a countable ring generating the Borel sets. We turn now to the main problem of this section, to find necessary and sufficient conditions on a set of fidi distributions (9.2.1) that will ensure that they are the fidi distributions of a random measure. The conditions fall into two groups: first the consistency requirements of the Kolmogorov existence theorem, and then the supplementary requirements of additivity and continuity needed to ensure that the realizations are measures. Conditions 9.2.V (Kolmogorov Consistency Conditions). (a) Invariance under index permutations. For all integers k > 0 and all permutations i1 , . . . , ik of the integers 1, . . . , k, Fk (A1 , . . . , Ak ; x1 , . . . , xk ) = Fk (Ai1 , . . . , Aik ; xi1 , . . . , xik ). (b) Consistency of marginals. For all k ≥ 1, Fk+1 (A1 , . . . , Ak , Ak+1 ; x1 , . . . , xk , ∞) = Fk (A1 , . . . , Ak ; x1 , . . . , xk ). The first of these conditions is a notational requirement: it reflects the fact that the quantity Fk (A1 , . . . , Ak ; x1 , . . . , xk ) measures the probability of an event {ω: ξ(Ai ) ≤ xi (i = 1, . . . , k)}, that is independent of the order in which the random variables are written down. The second embodies an essential requirement: it must be satisfied if there is to exist a single probability space Ω on which the random variables can be jointly defined.

28


The other group of conditions captures in distribution function terms the conditions (9.1.21) and (9.1.22), which express the fact that the random variables so produced must fit together as measures. Conditions 9.2.VI (Measure Requirements). (a) Additivity. For every pair A1 , A2 of disjoint Borel sets from BX , the distribution F3 (A1 , A2 , A1 ∪A2 ; x1 , x2 , x3 ) is concentrated on the diagonal x1 + x2 = x3 . (b) Continuity. For every sequence {An : n ≥ 1} of bounded Borel sets decreasing to ∅, and all > 0, 1 − F1 (An ; ) → 0

(n → ∞).

(9.2.5)

Conditions 9.2.V imply the existence of a probability space on which the random variables ξ(A), A ∈ BX , can be jointly defined. Then Condition 9.2.VI(a) implies (9.2.6) P{ξ(A1 ) + ξ(A2 ) = ξ(A1 ∪ A2 )} = 1. It follows by induction that a similar relation holds for the members of any finite family of Borel sets. For any given sequence of sets, Condition 9.2.VI(a) implies a.s. finite additivity, and then Condition 9.2.VI(b) allows this finite additivity to be extended to countable additivity. This leads us to the existence theorem itself: it asserts that, in the case of nonnegative realizations, the Conditions 9.2.V and 9.2.VI are not only necessary but also sufficient to ensure that the fidi distributions can be associated with a random measure. Note that Example 9.1(f) implies that without nonnegativity, the sufficiency argument breaks down. It appears to be an open problem to find necessary and sufficient conditions on the fidi distributions that ensure that they belong to a random signed measure. See Exercise 9.2.4. Theorem 9.2.VII. Let Fk (· ; ·) be a family of distributions satisfying the Consistency Conditions 9.2.V. In order that the Fk (·) be the fidi distributions of a random measure, it is necessary and sufficient that (i) the distributions Fk (·) be supported by the nonnegative half-line; and (ii) the Fk (·) satisfy the Measure Conditions 9.2.VI. Proof. Necessity is clear from the necessity part of Theorem 9.1.XIV, so we proceed to sufficiency. Because the Fk (·) satisfy the Kolmogorov conditions, there exists a probability space (Ω, E, P) and a family of random variables ξA for bounded A ∈ BX , related to the given fidi distributions by (9.2.1). Condition (i) above implies ξA ≥ 0 a.s., and condition (ii) that the random variables ξA satisfy (9.1.21) for each fixed pair of bounded Borel sets. Now the random variables are a.s. monotonic decreasing, so (9.2.5) implies the truth of (9.1.22) for each fixed sequence of bounded Borel sets An with An ↓ ∅. As in earlier discussions, the whole difficulty of the proof revolves around the fact that in general there is

9.2.


29

an uncountable number of conditions to be checked, so that even though each individual condition is satisfied with probability 1, it cannot be concluded from this that the set on which they are simultaneously satisfied also has probability 1. To overcome this difficulty, we invoke Theorem 9.1.XIV. It is clear from the earlier discussion that both conditions of Theorem 9.1.XIV are satisfied, so that we can deduce the existence of a random measure ξ ∗ such that ξ ∗ (A) and ξA coincide a.s. for every Borel set A. But this implies that ξ ∗ and ξ have the same fidi distributions, and so completes the proof. Corollary 9.2.VIII. There is a one-to-one correspondence between probability measures on B(M# X ) and families of fidi distributions satisfying Conditions 9.2.V and 9.2.VI. In practice, the fidi distributions are given most often for disjoint sets, so that Condition 9.2.VI(a) cannot be verified directly. In this situation it is important to know what conditions on the joint distributions of the ξ(A) for disjoint sets will allow such distributions to be extended to a family satisfying Condition 9.2.VI(a). Lemma 9.2.IX. Let Fk be the family of fidi distributions defined for finite families of disjoint Borel sets and satisfying for such families the Kolmogorov Conditions 9.2.V. In order for there to exist an extension (necessarily unique) to a full set of fidi distributions satisfying Conditions 9.2.VI(a) as well as 9.2.V, it is necessary and sufficient that for all integers k ≥ 2, and finite families of disjoint Borel sets {A1 , A2 , . . . , Ak }, z Fk (A1 , A2 , A3 , . . . , Ak ; dx1 , z − x1 , x3 , . . . , xk ) (9.2.7) 0 = Fk−1 (A1 ∪ A2 , A3 , . . . , Ak ; z, x3 , . . . , xk ). Proof. The condition (9.2.7) is clearly a corollary of Conditions 9.2.VI(a) and therefore necessary. We show that it is also sufficient. Let us first point out how the extension from disjoint to arbitrary families of sets can be made. Let {B1 , . . . , Bn } be any such arbitrary family. Then there exists a minimal family {A1 , . . . , Ak } of disjoint sets (formed from the nonempty intersections of the Bi and Bic ) such that each Bi can be represented as a finite union of some of the Aj . The joint distribution Fk (A1 , . . . , Ak ; x1 , . . . , xk ) will be among those originally specified. Using this distribution, together with the representations of each ξ(Bi ) as a sum of the corresponding ξ(Aj ), we can write down the joint distribution of any combination of the ξ(Bi ) in terms of Fk . It is clear from the construction that the resultant joint distributions will satisfy Condition 9.2.VI(a) and that only this construction will satisfy this requirement. To complete the proof it is necessary to check that the extended family of distributions continues to satisfy Condition 9.2.V(b). We establish this by induction on the index k of the minimal family of disjoint sets generating the given fidi distribution. Suppose first that there are just two sets A1 ,

30


A2 in this family. The new distributions defined by our construction are F2 (A1 , A1 ∪ A2 ), F2 (A2 , A1 ∪ A2 ), and F3 (A1 , A2 , A1 ∪ A2 ). Consistency with the original distributions F2 (A1 , A2 ), F1 (A1 ), and F1 (A2 ) is guaranteed by the construction and by the marginal consistency for distributions of disjoint sets. Only the marginal consistency with F1 (A1 ∪ A2 ) introduces a new element. Noting that by construction we have min(x,y) F2 (A1 , A1 ∪ A2 ; x, y) = F2 (A1 , A2 ; du, y − u), 0

and letting x → ∞, we see that this requirement reduces precisely to (9.2.7) with k = 2. Similarly, for k > 2, marginal consistency reduces to checking points covered by the construction, by preceding steps in the induction, by Condition 9.2.V(b) for disjoint sets, or by (9.2.7). Example 9.2(a) Stationary gamma random measure [see Example 9.1(d)]. The Laplace transform relation ψ1 (A; s) ≡ ψ(A; s) = (1 + λs)−α(A) determines the one-dimensional distributions, and the independent increments property on disjoint sets implies the relation ψk (A1 , . . . , Ak ; s1 , . . . , sk ) =

k

(1 + λsi )−α(Ai ) ,

i=1

which determines their joint distributions. Consistency of marginals here reduces to the requirement ψk−1 (A1 , . . . , Ak−1 ; s1 , . . . , sk−1 ) = ψk (A1 , . . . , Ak ; s1 , . . . , sk−1 , 0), which is trivially satisfied. Also, if An ↓ ∅, (An ) → 0 by continuity of Lebesgue measure, and thus ψ1 (A; s) = (1 + λs)−α(An ) → 1, which is equivalent to Condition 9.2.VI(b). Finally, to check (9.2.6) we should verify that for disjoint A1 and A2 , ψ1 (A1 ∪ A2 ; s) = ψ2 (A1 , A2 ; s, s), which is a simple consequence of additivity of Lebesgue measure. These arguments establish the consistency conditions when the sets occurring in the fidi distributions are disjoint, and it follows from Lemma 9.2.IX that there is a unique consistent extension to arbitrary Borel sets. The basic existence theorem for point processes is somewhat simpler than Theorem 9.2.VII for general random measures, as we now indicate. Theorem 9.2.X (Kolmogorov Existence Theorem for Point Processes). In order that a family Pk (A1 , . . . , Ak ; n1 , . . . , nk ) of discrete fidi distributions defined on bounded Borel sets be the fidi distributions of a point process, it is necessary and sufficient that

9.2.


31

(i) for any permutation i1 , . . . , ik of the indices 1, . . . , k, ∞

Pk (A1 , . . . , Ak ; n1 , . . . , nk ) = Pk (Ai1 , . . . , Aik ; ni1 , . . . , nik );

(ii) r=0 Pk+1 (A1 , . . . , Ak , Ak+1 ; n1 , . . . , nk , r) = Pk (A1 , . . . , Ak ; n1 , . . . , nk ); (iii) for each disjoint pair of bounded Borel sets A1 , A2 , P3 (A1 , A2 , A1 ∪ A2 ; n1 , n2 , n3 ) has zero mass outside the set where n1 + n2 = n3 ; and (iv) for sequences {An } of bounded Borel sets with An ↓ ∅, P1 (An ; 0) → 1. The task of checking the conditions in detail here can be lightened by taking advantage of Lemma 9.2.IX, from which it follows that if the consistency conditions (i) and (ii) are satisfied for disjoint Borel sets, and if for such disjoint sets the equations n

Pk (A1 , A2 , A3 , . . . , Ak ; r, n − r, n3 , . . . , nk )

r=0

= Pk−1 (A1 ∪ A2 , A3 , . . . , Ak ; n, n3 , . . . , nk )

(9.2.8)

hold, then there is a unique consistent extension to a full set of fidi distributions satisfying (iii). Example 9.2(b) The Poisson process with parameter measure µ [see Section 2.4]. Here the fidi distributions for disjoint Borel sets are readily specified by the generating function relations Πk (A1 , . . . , Ak ; z1 , . . . , zk ) =

k

exp[−µ(Ai )(1 − zi )],

(9.2.9)

i=1

where Πk is the generating function associated with the distribution Pk . Condition (ii) is readily checked by setting Zk = 1; then the term 1 − zk vanishes and reduces the product to the appropriate form for Πk−1 . In generating function terms, equation (9.2.6) becomes, for k = 2, Π2 (A1 , A2 ; z, z) = Π1 (A1 ∪ A2 ; z), which expresses the additivity of the Poisson distribution. Finally, to check condition (iv) we require Π1 (An ; 0) → 1, that is, exp[−µ(An )] → 1, which is a corollary of the assumption that µ is a measure, so µ(An ) → 0 as An ↓ ∅. It should be noted that the form (9.2.9) does not hold for arbitrary sets but has to be replaced by such forms as Π2 (A1 , A2 ; z1 , z2 ) = exp[−µ(A1 )(1 − z1 ) − µ(A2 )(1 − z2 ) + µ(A1 ∩ A2 )(1 − z1 )(1 − z2 )] when the sets overlap. The extension to arbitrary families of nondisjoint sets is unique, but laborious, and need not be pursued in detail.

32


Example 9.2(c) Finite point processes. If the distribution of a finite point process is specified in any of the ways described in Proposition 5.3.II, in particular, say, by its Janossy measures [see around (5.3.2)], then the fidi distributions are given by (5.3.13), namely n1 ! . . . nk ! Pk (A1 , . . . , Ak ; n1 , . . . , nk ) ∞ (n ) (n ) Jn+r (A1 1 × · · · × Ak k × C (r) ) , = r! r=0

(9.2.10)

where C is the complement of the union of the disjoint sets A1 , . . . , Ak and n = n1 + · · · + nk . Although we can infer on other grounds that the point process is well defined, and hence that the fidi distributions must be consistent, it is of interest to check the consistency conditions directly. Because (9.2.10) is restricted to disjoint sets, the appropriate conditions are (i), (ii), and (iv) of Theorem 9.2.X together with (9.2.8). The permutation condition (i) follows from the symmetry of the Janossy measures. Also, condition (iv) reduces to ∞ Jr (X \ An )(r) →1 if An ↓ ∅. P1 (An ; 0) = r! r=0 But then X \ An ↑ X , and the result follows from dominated convergence, the fact ∞ that the(r)Jr (·) are themselves measures, and the normalization condition )]/r! = 1 as in (5.3.9). r=0 [Jr (X The additivity requirement (9.2.8) follows from identities of the type

(n1 )

Jn+r (A1

n1 +···+nk =n

(nk )

× · · · × Ak n1 ! . . . nk !

× C (r) )

Jn+r (A1 ∪ · · · ∪ Ak )(n) × C (r) = , n! which are immediate applications of Lemma 5.3.III. Similarly, the marginal condition (ii) reduces to checking the equations (nk−1 ) (n ) (n ) ∞ ∞ Jν+nk +r (A1 1 × · · · × Ak−1 × Ak k × C (r) )

nk ! r!

nk =0 r=0

∞ s 1 (nk−1 ) (n ) (t) Jν+s (A1 1 × · · · × Ak−1 × Ak × C (s−t) ) s! s=0 t=0 (n ) (nk−1 ) ∞ Jν+s A1 1 × · · · × Ak−1 × (Ak ∪ C)(s) , = s! s=0

=

where ν = n1 + · · · + nk−1 , the first equation is a regrouping of terms, and the second equation is a further application of Lemma 5.3.III.

9.2.


33

Underlying Example 9.2(c) is a mapping of X ∪ → NX ; see also Propo∞ ∪∗ = n=0 X (n)∗ , where X (n)∗ is the n-fold sitions 9.1.XI–XII. The space X product of the c.s.m.s. X subject to the constraint that any unordered set x = {x1 , . . . , xn } ∈ X (n)∗ satisfies xi = xj for i = j [cf. (5.3.10)], is a candidate space for describing finite simple point processes [cf. Chapter 5 and Definition 9.1.II(iii)]. Important questions relate to the characterization of subclasses of point processes through their fidi distributions. Characterizing simple point processes leads to the discussion of orderliness taken up in Section 9.3. Marked point processes can be treated as point processes on product spaces, with the defining sets for the fidi distributions restricted to ‘rectangle sets’ Ai × Ki where Ai ∈ BX , Ki ∈ BK . The fidi distributions of a process on the product space correspond to those of a marked point process if the distributions of the ground process, obtained by setting all Ki = K in the product sets, as in Pkg (A1 , . . . , Ak ; n1 , . . . , nk ) = Pk (A1 × K, . . . , Ak × K; n1 , . . . , nk ), are proper, and satisfy the conditions of Proposition 9.2.X. To conclude the present section, we outline an extension of Rényi’s (1967) result, quoted at Theorem 2.3.II, that a simple Poisson process in Rd , whether homogeneous or not, is determined by the values of its avoidance function P0 (A) = P{N (A) = 0}

(9.2.11)

on a suitably rich class A of Borel sets. The essence of this result is that, for a simple point process, the avoidance function alone is enough to determine the full set of fidi distributions. Our aim is to describe an interaction of structural properties of the space X and the function P0 (·) which are enough for P0 (·) to retain this determining character without the strong probabilistic assumptions of the Poisson process. Concerning terminology, Kendall (1974) used the term avoidance function in a more general (stochastic geometry) context, reflecting the fact that P0 (A) gives the probability of the support of a random set function avoiding a prescribed set A; other possible terms include zero function, avoidance probability function, and vacuity function [McMillan (1953)]. Extensions of Rényi’s result are due to M¨ onch (1971), who showed that the Poisson assumption is not needed [see also Kallenberg (1973, 1975)], and a characterization of the avoidance function due to Kurtz (1974). Unpublished work of Karbe (1973) is presumably the basis of some discussion in MKM (1978, Section 1.4). Much of the work, largely couched in algebraic language, was developed by McMillan (1953) who used the term vacuity function in lectures in Berkeley in 1981. If only the state space X of the simple point process N (·) were countable, Rényi’s result would be almost trivial, for with i, j, . . . denoting distinct points

34


of X , we should have for the first few fidi distributions P1 ({i}; 0) = 1 − P1 ({i}; 1) = P0 ({i}), P2 ({i}, {j}; 0, 0) = P0 ({i, j}), P2 ({i}, {j}; 0, 1) = P0 ({j}) − P0 ({i, j}), P2 ({i}, {j}; 1, 1) = 1 − P0 ({i}) − P0 ({j}) + P0 ({i, j}). Continuing in this way, all the fidi distributions could be built up through a sequence of differencing operations applied to P0 (·), and it is clear that the avoidance function would thereby determine the fidi distributions uniquely. Our task here is to extend this argument to a general c.s.m.s. X as state space. Following Kurtz (1974), the equations ∆(A)ψ(B) = ψ(B) − ψ(A ∪ B), ∆(A1 , . . . , Ak , Ak+1 )ψ(B) = ∆(Ak+1 )[∆(A1 , . . . , Ak )ψ(B)]

(9.2.12a) (k = 1, 2, . . .), (9.2.12b)

define a difference operator ∆(A) and its iterates acting on any set function ψ(·) for A, A1 , A2 , . . . , B in a ring of sets on which ψ(·) is defined. This operator is tailored to the needs of (9.2.16) in the lemma below; the sign convention in its definition here is opposite that of Kurtz and Kallenberg. Lemma 9.2.XI. For every integer k ≥ 1 and all Borel sets A1 , A2 , . . . , B, ∆(A1 , . . . , Ak )P0 (B) = P{N (Ai ) > 0 (i = 1, . . . , k), N (B) = 0}.

(9.2.13)

Proof. For k = 1 we have P{N (A1 ) > 0, N (B) = 0} = P0 (B) − P0 (A1 ∪ B) = ∆(A1 )P0 (B). The general form follows by an induction argument (see Exercise 9.2.5). As a special case of (9.2.13) with B the null set, ∆(A1 , . . . , Ak )P0 (∅) = P{N (Ai ) > 0 (i = 1, . . . , k)}.

(9.2.14)

The nonnegativity of (9.2.11) appears later in Theorem 9.2.XV in a characterization of the avoidance function. In the meantime, Lemma 9.2.XI provides a useful notational convention and serves as a reminder that the probability of the complex event on the right-hand side of (9.2.13) can be expressed immediately in terms of the avoidance function. The basic idea motivating the introduction of the operator ∆ at (9.2.12) is that it leads to a succinct description of the fidi distributions of a point process when, for suitable sets Ai , N (Ai ) is ‘small’ in the sense of having P{N (Ai ) = 0 or 1 (all i)} ≈ 1. Such an approximation can be realized only if N is simple and the class of sets on which the values of the avoidance function are known contains a dissecting system for X (see Definition A1.6.I). Fortunately, on a c.s.m.s. X the Borel sets BX necessarily contain a dissecting system and hence a dissecting ring (Definition A2.1.V).

9.2.


35

Theorem 9.2.XII (Rényi, 1967; M¨ onch, 1971). The distribution of a simple point process N on a c.s.m.s. X is determined by the values of the avoidance function P0 on the bounded sets of a dissecting ring A for X . Proof. It is enough to show that the fidi distributions of a point process as at (9.2.4), involving only bounded subsets of X , are determined by the avoidance function. We use an indicator function Z(B), B ∈ A, and a dissecting system T as in and below (9.3.12), for which the r.v.s ζn (A) =

kn

Z(Ani )

(n = 1, 2, . . .),

(9.2.15)

i=1

count the numbers of sets in Tn containing points of N (·). Because every Ani is the union of elements in Tn+1 , and the r.v.s Z(·) are subadditive set functions, it follows that {ζn (A)} is a nondecreasing sequence. Moreover, because N is simple and {Tn } is a dissecting system, the limit N (A) ≡ lim ζn (A)

(9.2.16)

n→∞

exists a.s. Now the joint distribution of the Z(Ani ), and hence of ζn (A), and (more generally) of {ζn (Ai ) (i = 1, . . . , k)}, is expressible directly in terms of the avoidance function: for example, r (9.2.17) ∆(Ani1 , . . . , Anir ) P0 A \ Anij , P{ζn (A) = r} = {i1 ,...,ir }

j=1

where the sum is taken over all krn distinct combinations of r sets from the kn (≥ r) sets in the partition Tn of A. Rather more cumbersome formulae give the joint distributions of ζn (Ai ). Because the convergence of {ζn } to its limit is monotonic, the sequence of events {ζn (Ai ) ≤ ni (i = 1, . . . , k)} is also monotone decreasing in n, and thus P{ζn (Ai ) ≤ ni (i = 1, . . . , k)} → P{N (Ai ) ≤ ni (i = 1, . . . , k)}. Thus, P0 determines the fidi distributions as asserted. Corollary 9.2.XIII. Let N1 , N2 be two point processes on X whose avoidance functions coincide on the bounded sets of a dissecting ring for X . Then their support point processes N1∗ and N2∗ are equivalent. Versions of these results that apply to random measures can be given (see Exercise 9.2.7): the avoidance functions are replaced by the Laplace transforms E(e−sξ(A) ) for fixed s > 0. We turn finally to a characterization problem. Definition 9.2.XIV. A set function ψ defined on a ring R of sets is completely monotone on R if for every sequence {A, A1 , A2 , . . .} of members of R, (every n = 1, 2, . . .). ∆(A1 , . . . , An ) ψ(A) ≥ 0

36


Note that this definition of complete monotonicity differs from conventional usage by the omission of a factor (−1)n on the left-hand side of the inequality [see also the definition of ∆ at (9.2.12)]. Using Definition 9.2.XIV, Lemma 9.2.XI asserts that the avoidance function of a point process is completely monotone on BX . Complete monotonicity of a set function is not sufficient on its own to characterize an avoidance function. Theorem 9.2.XV (Kurtz, 1974). Let ψ be a set function defined on the members of a dissecting ring R covering the c.s.m.s. X . In order that there exist a point process on X with avoidance function ψ, it is necessary and sufficient that (i) ψ be completely monotone; (ii) ψ(∅) = 1; (iii) ψ(An ) → 1 for any bounded sequence {An } in R for which An → ∅ (n → ∞); and (iv) for every bounded A ∈ R, r k = 1, ∆(Ani1 , . . . , Anir ) ψ A \ Anij lim lim ψ(A) + r→∞ n→∞

k=1 {i1 ,...,ir }

j=1

where {Tn } = {Ani : i = 1, . . . , kn } is a dissecting system for A, {Tn } ⊆ R, and the inner summation is over all distinct combinations of k sets from the kn sets in the partition Tn for A. Proof. The necessity of (i) has been noted in Lemma 9.2.XI, condition (ii) is self-evident, and condition (iii) here is the same as (iv) of Theorem 9.2.X. Condition (iv) here follows most readily from (9.2.17) when written in the form r P{ζn (A) = k} = lim P{N (A) ≤ r} = 1 lim lim r→∞ n→∞

k=0

r→∞

and expresses the fact that a point process N is boundedly finite. For the sufficiency, it is clear from (i) and (ii) that we can construct an indicator process Z on bounded A ∈ R with fidi distributions (for any finite number k of disjoint bounded A1 , . . . , Ak ∈ R) Pr{Z (A1 ) = 0} = 1 − Pr{Z (A1 ) = 1} = ψ(A1 ) = ∆(A1 ) ψ(∅), (9.2.18a) ⎫ Pr{Z (Ai ) = 1 (i = 1, . . . , k} = ∆(A1 , . . . , Ak ) ψ(∅), ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ Pr{Z (Aj ) = 0, Z (Ai ) = 1 (all i = j)} (9.2.18b) = ∆(A1 , . . . , Aj−1 , Aj+1 , . . . Ak ) ψ(Aj ), ⎪ ⎪ ⎪ ⎪ ⎪ k ⎭ Pr{Z (Ai ) = 0 (all i)} = ψ i=1 Ai ); nonnegativity is ensured by (i), summation to unity by (ii), and marginal consistency reduces to ∆(A1 , . . . , Ak+1 )ψ(B) + ∆(A1 , . . . , Ak )ψ(B ∪ Ak+1 ) = ∆(A1 , . . . , Ak )ψ(B).

9.2.


37

In other words, we have a family of fidi distributions that, being consistent in the sense of the Kolmogorov existence theorem [e.g., Parthasarathy (1967, Chapter V)], enable us to assert the existence of a probability space (Z, E, P ) on which are jointly defined {0, 1}-valued r.v.s {Z (A): bounded A ∈ R}, and P is related to ψ via relations such as (9.2.18) (with Pr replaced by P ). We now introduce r.v.s ζn (A) (bounded A ∈ R) much as at (9.2.18) and observe that the subadditivity of Z implies that ζn (A) are a.s. monotone nondecreasing under refinement as before, so lim ζn (A) ≡ N (A)

n→∞

(9.2.19)

exists a.s. and, being the limit of an integer-valued sequence, is itself integervalued or infinite. From the last relation at (9.2.18b), we have P {ζn (A) = 0} = ψ(A) for all n, so P {N (A) = 0} = ψ(A)

(all bounded A ∈ R).

(9.2.20)

The a.s. finiteness condition that N must satisfy on bounded A ∈ R is equivalent to demanding that lim lim P {ζn (A) ≤ y} = 1,

y→∞ n→∞

which, expressed in terms of the functions ψ via (9.2.21) and relations such as (9.2.18) (with P and ψ replacing P and P0 ), reduces to condition (iii). For bounded disjoint A, B ∈ R, we find by using a dissecting system for A ∪ B containing dissecting systems for A and B separately that N (A ∪ B) = lim ζn (A ∪ B) = lim [ζn (A) + ζn (B)] = N (A) + N (B) a.s., n→∞

n→∞

and thus N is finitely additive on R. Let {Ai } be any disjoint sequence in R with bounded union ∞ Ai ∈ R; A≡ i=1

∞ ∞ we i=1 N (Ai ). Let Br = i=r+1 Ai = A \ r seek to show that N (A) = A , so B is bounded, ∈ R, ↓ ∅ (r → ∞), and thus P {N (Br ) = 0} = r i=1 i ψ(Br ) ↑ 1; that is, N (Br ) → 0 a.s. Define events Cr ∈ E for r = 0, 1, . . . by C0 = {N : N (A) = 0}

Cr = {N : N (Br ) = 0 < N (Br−1 )}. r Then P (C0 ∪ C1 ∪ · · ·) = 1, and N (A) = i=1 N (Ai ) + N (Br ) on Cr . Also, 0 = N (Br ) = limn→∞ ζn (Br ) that N (Ai ) = 0 for on Cr , it follows from ∞ ∞ i ≥ r + 1 and hence i=r+1 N (Ai ) = 0 on Cr . Because P r=0 Cr = 1, it now follows that N is countably additive on R. Then by the usual extension theorem for measures, N can be extended a.s. to a countably additive boundedly finite nonnegative integer-valued measure on BX . This extension, with the appropriate modification on the P -null set where the extension may fail, provides the required example of a point process with avoidance function P {N (A) = 0} = ψ(A) (A ∈ R) satisfying conditions (i)–(iv). and

38


Exercises and Complements to Section 9.2 9.2.1 Give an example of a family of fidi distributions satisfying (9.2.6) for disjoint sets and all other requirements of Theorem 9.2.VII apart from Condition 9.2.VI(a), which is not satisfied. [Hint: Let X be a two-point space, {x, y} say, and construct a r.v. Z and a random set function ξ for which Z has the distribution of ξ({x, y}) but Z = ξ({x}) + ξ({y}).] 9.2.2 Give an example of a family of fidi distributions that satisfy (9.2.6) for k = 2 but not for some k ≥ 3, and hence do not satisfy the consistency Condition 9.2.V(b). [Hint: Modify the previous example.] 9.2.3 Show that the joint distributions of the Dirichlet process of Example 9.1(e) are consistent. 9.2.4 Let Ψ = ξ1 − ξ2 be the difference of two random measures. For each ω, let Ψ = Ψ+ −Ψ− be the Jordan–Hahn decomposition of Ψ(ω) (Theorem A1.3.IV). Determine conditions under which the mappings Ψ+ and Ψ− are measurable and hence define random measures. Investigate the extent to which such conditions can be extended to more general settings. 9.2.5 Let A1 , . . . , An be disjoint, A = n i=1 Ai , and ψ(∅) = 1. Verify that the operator ∆ at (9.2.12) satisfies (a) and (b) below, and complete the induction proof of (9.2.13).

n

(a) ∆(A)ψ(B) =

n

(1 − ψ(Ai )) =

9.2.6 Let A1 , A2 , . . . be disjoint, A =

∞

1 − ψ(A) =

1−ψ

k j=1

k ∆(Ai1 , . . . , Aik ) ψ A \

Aij

k j=1

,

Aij .

k=1 1≤i1 ξ(An,j(n) ). Because ξ is a random measure, we then have (n → ∞), ξ(An,j(n) ) ↓ ξ(A∞ ) and therefore P{ξ(A∞ ) ≥ } ≥ η. Because ξ(∅) = 0 a.s. and > 0, we must have A∞ = {x0 }, and thus ξ has at least one fixed atom. Conversely, if ξ has a fixed atom x0 ∈ A, there exists > 0 for which 0 < P{ξ{x0 } > } ≡ η . Then, given any dissecting system T for A, there exists a set An,j(n) in each Tn such that x0 ∈ An,j(n) , and P{ξ(An,j(n) ) > } ≥ P{ξ{x0 } > } = η

(all n),

so (9.3.2) fails for any dissecting system T for A. Once the fixed atoms have been identified, one would anticipate representing the random measure as the superposition of two components, the first containing all fixed atoms and the second free from fixed atoms. It is not absolutely clear, however, that this procedure corresponds to a measurable operation on the original process. The following establishes this fact. Lemma 9.3.III. The set D of fixed atoms of a random measure is countably infinite at most. Proof. Suppose on the contrary that D is uncountable. Because X can be covered by the union of at most countably many bounded Borel sets, there exists a bounded set, A say, containing uncountably many fixed atoms. Define the subset D of D ∩ A by D = {x: P{ξ{x} > } > },

(9.3.4)

observing by monotonicity that D ∩ A = lim↓0 D . If D is finite for every

> 0, then by a familiar construction we can deduce that D ∩ A is countable, so for some positive which we fix for the remainder of the proof, D is infinite. We can extract from D an infinite sequence of distinct points {x1 , x2 , . . .} for which the events En ≡ {ξ: ξ{xn } > } have P(En ) > . Because ξ is boundedly finite, 0 = P{ξ(A) = ∞} ≥ P{ξ{x} > for infinitely many x ∈ D } ≥ P{infinitely many En occur} ∞ ∞ ∞ Ek = limn→∞ P Ek =P n=1 k=n

≥ > 0, thereby yielding a contradiction.

k=n

9.3.

Sample Path Properties: Atoms and Orderliness

41

It is convenient to represent the countable set D by {xk } and to write Uk ≡ Uk (ω) for the random variable ξ{xk }. Using Dirac measure δx as in (9.1.3), the set function ξc defined for bounded Borel sets A by ξc (A, ω) = ξ(A, ω) − Uk (ω) δxk (A) xk ∈D

is positive and countably additive in A, and for every such A it defines a random variable. Thus, it defines a new random measure that is clearly free from fixed atoms, and we have proved the following extension of Proposition 9.1.III(i) of properties of a fixed measure µ to those of a random measure ξ. Proposition 9.3.IV. Every random measure ξ can be written in the form ξ(·, ω) = ξc (·, ω) +

∞

Uk (ω) δxk (·),

k=1

where ξc is a random measure without fixed atoms, the sequence {xk : k = 1, 2, . . .} constitutes the set D of all fixed atoms of ξ, and {Uk : k = 1, 2, . . .} is a sequence of nonnegative random variables. Consider next the more general question of finding conditions for the trajectories to be a.s. nonatomic. As before, let A be a bounded Borel set and T = {Tn : n = 1, 2, . . .} a dissecting system for A. For any given > 0, we can ‘trap’ any atoms of ξ with mass or greater by the following construction. For each n, set (9.3.5) N(n) (A) = #{i: Ani ∈ Tn , ξ(Ani ) ≥ }. (n)

Then each N (A) is a.s. finite, being bounded uniformly in n by ξ(A)/ . (n) Moreover, as n → ∞, N (A) converges a.s. to a limit r.v., N (A) say, which is independent of the particular dissecting system T and which represents the number of atoms in A with mass or greater (see Exercise 9.3.2 for a more formal treatment of these assertions). Consequently, ξ is a.s. nonatomic on (n) A if and only if for each > 0, N (A) = 0 a.s. Because N (A) converges a.s. to N (A) irrespective of the value of the latter, a necessary and sufficient (n) condition for N = 0 a.s. is that N → 0 in probability. This leads to the following criterion. Lemma 9.3.V. The random measure ξ is a.s. nonatomic on bounded A ∈ BX if and only if for every > 0 and for some (and then every) dissecting system T for A,

(n → ∞). (9.3.6) P #{i : ξ(Ani ) ≥ } > 0 → 0 Corollary 9.3.VI. A sufficient condition for ξ to be a.s. nonatomic on bounded A ∈ BX is that for some dissecting system for A and every > 0, kn

P{ξ(Ani ) ≥ } → 0

(n → ∞).

(9.3.7)

i=1

If ξ is a completely random measure then this condition is also necessary.

42


Proof. Equation (9.3.7) is sufficient for (9.3.6) to hold because P

k n

kn {ξ(Ani ) ≥ } ≤ P{ξ(Ani ) ≥ }.

i=1

i=1

When ξ is completely random (Definition 10.1.I), the r.v.s ξ(Ani ) are mutually independent and hence k k n n {ξ(Ani ) ≥ } = P {ξ(Ani ) < } 1−P i=1

=

kn

i=1

P{ξ(Ani ) < } =

i=1

kn

1 − P{ξ(Ani ) ≥ } .

(9.3.8)

i=1

If now ξ is a.s. nonatomic, then by (9.3.6) the left-hand side of (9.3.8) converges to 1 as n → ∞. Finally, the convergence to 1 of the product of the right-hand side of (9.3.8) implies (9.3.7). Exercise 9.3.3 shows that (9.3.7) is not necessary for a random measure to be absolutely continuous. Indeed, it appears to be an open problem to find simple sufficient conditions, analogous to Corollary 9.3.VI, for the realizations of a random measure to be a.s. absolutely continuous with respect to a given measure (see also Exercise 9.1.7). Example 9.3(a) Quadratic random measures [see also Example 9.1(b)]. Take A to be the unit interval (0, 1], and for n = 1, 2, . . . divide this into kn = 2n subintervals each of length 2−n to obtain suitable partitions for a dissecting system T . Each ξ(Ani ) can be represented in the form ξni ≡ ξ(Ani ) =

(i+1)/kn

Z 2 (t) dt ≈ (1/kn )Z 2 (i/kn ).

i/kn

Because Z 2 (i/kn ) has a χ2 distribution on one degree of freedom, we deduce (as may be shown by a more careful analysis) that Pr{ξni > } = Pr{Z 2 (i/kn ) > kn } 1 + O(1) ≤ C exp(−kn ) for some finite constant C. Then kn

Pr{ξni > } ≤ Ckn e−kn → 0

(n → ∞).

i=1

Thus, ξ being an integral with a.s. continuous integrand its trajectories are a.s. nonatomic. We turn now to point processes; ultimately we generalize the results of Section 3.3. Simplicity is again a sample-path property; in terms of Propo-

9.3.


43

sition 9.1.III, it occurs when N coincides with its support counting measure N ∗ , although this is not the only way it can be described. For MPPs, we note as in Exercise 9.1.6 that any MPP that is not simple can be redefined as a simple MPP with marks in the compound space K∪ . In particular, a nonsimple point process on X can be regarded as a simple point process on X × {1}∪ = X × Z+ [recall also Proposition 9.1.III(iv)]. In practice, it is arguably more useful to have analytic conditions for a point process to be simple. The main approach in developing such conditions is again via dissecting systems. We start from the representation (9.1.6) of a counting measure N , writing it now in the form ∞ kNk∗ , (9.3.9) N= k=1

where

Nk∗ (A) = #{xi ∈ A : N {xi } = k}

(k = 1, 2, . . .).

(9.3.10)

∗

Then for the support counting measure N of N we can write (cf. (9.1.7)) N∗ =

∞

Nk∗ .

(9.3.11)

k=1

We would like to regard (9.3.9) and (9.3.11) as statements concerning point processes as well as statements about individual realizations. To this end we use Proposition 9.1.VIII. It is clear from the construction that N ∗ and each Nk∗ are elements of NX#∗ ; the essential point is to show that for any bounded Borel set A, N ∗ (A) and Nk∗ (A) are r.v.s. We establish this by a construction which plays an important role in later arguments. Suppose then that N is a point process, and for bounded B ∈ BX define the indicator functions 1 if 1 ≤ N (B) ≤ k, (9.3.12a) Zk (B) = I{1≤N (B)≤k} = 0 otherwise, 1 if N (B) ≥ 1, (9.3.12b) Z(B) = I{N (B)≥1} = 0 otherwise, which are r.v.s because the N (B) are r.v.s. All of the Zk , as well as Z, are subadditive set functions, meaning, for example, that for A, B ∈ BX , Zk (A ∪ B) ≤ Zk (A) + Zk (B). Such subadditive set functions have the important properties that P{1 ≤ N (A ∪ B) ≤ k} = E[Zk (A ∪ B)] ≤ P{1 ≤ N (A) ≤ k} + P{1 ≤ N (B) ≤ k}, (9.3.13) with similar inequalities for P{N (B) ≥ 1}.

44


As around (9.3.9), let T = {Tn } = {Ani } be a dissecting system for any given bounded A ∈ BX . Then (n)

ζk (A) =

Zk (Ani ),

ζ (n) (A) =

i:Ani ∈Tn

Z(Ani )

i:Ani ∈Tn

are further r.v.s. Here, for example, ζ (n) counts the number of subsets in the (n) nth partition containing at least one point of N . The ζk are nondecreasing in n, as follows from (9.3.13) and the fact that Tn+1 partitions each Ani ∈ Tn . So for fixed k and A, (n)

(n+1)

ζk (A) ≤ ζk

(A) ≤ N (A) < ∞

(n = 1, 2, . . .);

hence (n)

ζk (A) = lim ζk (A) n→∞

and

ζ(A) = lim ζ (n) (A), n→∞

(9.3.14)

being monotone limits of bounded sequences, exist a.s. and again define r.v.s. Although spheres are used in the derivation of (9.1.5), we could equally use elements of T , thereby showing explicitly that ζ1 (A) = N1∗ (A) and that limk→∞ ζk (A), which exists because ζk (A) ≤ ζk+1 (A) ≤ N (A) (k = 1, 2, . . .), equals N ∗ (A). Thus N1∗ (A), N ∗ (A), and Nk∗ (A) = ζk (A) − ζk−1 (A)

(k = 2, 3, . . .)

are all random variables. It follows from Proposition 9.1.VIII that, whenever N is a point process, the random set functions Nk∗ (·) and N ∗ (·) are also point processes. We summarize this discussion in the following proposition. Proposition 9.3.VII. For any point process N, the constructions at (9.1.7) and (9.3.10) applied to realizations of N define simple point processes N ∗ and Nk∗ , and the relations (9.3.9) and (9.3.11) hold as relations between point processes. In particular, N is simple if and only if Nk∗ = 0 a.s. (k = 2, 3, . . .). The next result generalizes Proposition 3.3.I and identifies a general form of the intensity with the first moment measure of N ∗ . Definition 9.3.VIII. The intensity measure of a point process N is the set function P{N (Ani ) ≥ 1} (A ∈ BX ). λ(A) ≡ sup Tn ∈T (A) i:A

ni ∈Tn

9.3.


45

Proposition 9.3.IX (Khinchin’s Existence Theorem). Whether finite or infinite, (9.3.15) λ(A) = EN ∗ (A) ≡ M ∗ (A); it is independent of the choice of dissecting system T , and defines a measure when it is boundedly finite. Proof. Using the monotonicity of ζ (n) and the interchange of limits that monotone convergence permits, we find that for any given dissecting system for A, M ∗ (A) ≡ E N ∗ (A) = E lim ζ (n) (A) n→∞ (n) = lim E ζ (A) n→∞ = lim P{N (Ani ) ≥ 1} n→∞

= sup Tn

i:Ani ∈Tn

P{N (Ani ) ≥ 1} = λ(A).

i:Ani ∈Tn

The relation N ∗ (A) = limn→∞ ζ (n) (A), and hence also λ(A) itself, is independent of the particular choice of dissecting system. The rest of the proposition follows easily from the fact that the first moment measure M (·) of a point process is indeed a measure [see the discussion around (5.4.1)]. A consequence of the equality of λ(·) with M ∗ (·) is that it is a well-defined measure, possibly ‘extended’ in the sense that we may have λ(A) = ∞ for some bounded A ∈ BX . Notice also that Definition 9.3.VIII and the subsequent proposition form a particular case of a subadditive set function yielding a measure by addition under refinement (see Exercise 9.3.5). In the next result the direct part is now trivial; it generalizes Proposition 3.3.IV and so may be called Korolyuk’s theorem. The converse may be referred to as Dobrushin’s lemma (cf. Proposition 3.3.V), because it extends naturally a result first referenced in Volkonski (1960) for stationary point processes. Proposition 9.3.X. For a simple point process N , λ(A) = M (A)

(all A ∈ BX ).

(9.3.16)

Conversely, if (9.3.16) holds and λ(A) < ∞ for all bounded A, then N is simple. Proof. When N is not simple, there is some bounded A ∈ BX for which ∆ ≡ P{N (A) = N ∗ (A)} > 0. Then M (A) = EN (A) ≥ ∆ + EN ∗ (A) = ∆ + λ(A) and (9.3.16) cannot hold when λ(A) < ∞.

46


Proposition 9.3.VII asserts that each Nk∗ is a simple point process and thus has an intensity measure λ∗k = ENk∗ . From (9.3.9) we may therefore deduce the generalized Korolyuk equation M (A) = EN (A) =

∞

kλ∗k (A) = λ(A) +

k=1

∞

[λ(A) − λk (A)],

(9.3.17)

k=1

where λk = λ∗1 + · · · + λ∗k is the intensity measure of the simple point process ζk . Exercise 9.3.6 notes a version of (9.3.17) applicable to atomic random measures. Further analytic conditions for simplicity involve infinitesimals directly, and usually bear the name orderly or ordinary, the latter deriving from transliteration rather than translation of Khinchin’s original terminology. Definition 9.3.XI. A point process N is ordinary when, given any bounded

A ∈ BX , there is a dissecting system T = {Tn } = {Ani : i = 1, . . . , kn } for A such that kn P{N (Ani ) ≥ 2} = 0. (9.3.18) inf Tn

i=1

Thus, for an ordinary point process, given > 0, there exists a dissecting system for bounded A and some n such that for n ≥ n , k kn n

> P{N (Ani ) ≥ 2} ≥ P {N (Ani ) ≥ 2} ≡ P(Bn ). i=1

i=1

Now the sequence of sets {Bn } is monotone decreasing, so for any > 0,

> P limn→∞ Bn = P N {x} ≥ 2 for some x ∈ A , which establishes the direct part of the following result. The proof of the converse is left as Exercise 9.3.7. Proposition 9.3.XII. An ordinary point process is necessarily simple. Conversely, a simple point process of finite intensity is ordinary, but one of infinite intensity need not be ordinary. The following two analytic conditions have a pointwise character, and hence may be simpler to check in practice than the property of being ordinary. To describe them, take any bounded set A containing a given point x, let T be a dissecting system for A, and for each n = 1, 2, . . . , let An (x) denote the member of Tn = {Ani } that contains x. Definition 9.3.XIII. (a) A point process on the c.s.m.s. X is µ-orderly at x when µ is a boundedly finite measure and fn (x) ≡

P{N (An (x)) ≥ 2} →0 µ An (x)

(n → ∞),

(9.3.19)

where if µ An (x) = 0 then fn (x) = 0 or ∞ according as P{N (An (x)) ≥ 2} = or > 0.

9.3.


47

(b) The process is Khinchin orderly at x if gn (x) ≡ P{N (An (x)) ≥ 2 | N (An (x)) ≥ 1} → 0

(9.3.20)

as n → ∞, where gn (x) = 0 if P{N (An (x)) ≥ 1} = 0. In many situations the state space X is a locally compact group with a boundedly finite invariant measure ν. If a point process is ν-orderly for a dissecting system based on spheres, we speak of the process as being orderly. Such usage is consistent with Khinchin’s (1955) original use of the term for stationary point processes on R; a point process on R uniformly analytically orderly in Daley’s (1974) terminology is orderly in the present sense. Proposition 9.3.XIV. Suppose that for bounded A ∈ BX the point process N is µ-orderly at x for x ∈ A, and satisfies sup sup fn (x) < ∞,

(9.3.21)

n x∈A

where fn (·) is defined by (9.3.19). Then N is simple on A. Proof. From the definition at (9.3.19), kn

P{N (Ani ) ≥ 2} =

fn (x) µ(dx). A

i=1

Here, fn (x) → 0 pointwise, and by using (9.3.21) to justify appealing to the dominated convergence theorem, the integral → 0 as n → ∞. The process is thus ordinary, and Proposition 9.3.XII completes the proof. When λ is boundedly finite, the hypotheses here can be weakened by dropping the requirement (9.3.21) and demanding merely that (9.3.19) and (9.3.20) should hold for µ-a.e. x in A. This observation is the key to obtaining a partial converse to the proposition: we give it in the context of Khinchin orderliness. Proposition 9.3.XV. A point process N with boundedly finite intensity measure λ(·) is simple if and only if it is Khinchin orderly for λ-a.e. x on X . Proof. It suffices to restrict attention to a bounded set A. To prove sufficiency, use the fact that P{N (Ani ) ≥ 1} ≤ λ(Ani ) to write kn

P{N (Ani ) ≥ 2} ≤

i=1

gn (x) λ(dx),

→ 0 (n → ∞)

A

because from (9.3.20), 1 ≥ gn (x) → 0 for λ-a.e. x and λ(A) < ∞. To prove the necessity, suppose that N is simple: we first show that hn (x) ≡

P{N (An (x)) ≥ 1} →1 λ(An (x))

(n → ∞) for λ-a.e. x.

(9.3.22)

48


We establish this convergence property via a martingale argument much as in the treatment of Radon–Nikodym derivatives [see Lemma A1.6.III or, e.g., Chung (1974, Section 9.5(VIII))]. Construct a sequence of r.v.s {Xn } ≡ {Xn (ω)} on a probability space (A, BA , IP), where IP(·) = λ(·)/λ(A), by introducing indicator r.v.s Ini (x) = 1 for x ∈ Ani , = 0 otherwise, and setting Xn (x) =

kn

hn (x)Ini (x).

(9.3.23)

i=1

Let Fn denote the σ-field generated by the sets of T1 ∪ · · · ∪ Tn ; because {Tn } is a nested system of partitions, Fn has quite a simple structure (!). Then, {Xn } is a submartingale because {Fn } is an increasing sequence of σ-fields, and on the set An (x) = Ani say, λ(An+1,j ) P{N (An+1,j ) ≥ 1} E Xn+1 | Fn = In+1,j (x) λ(An+1,j ) λ(Ani ) j:An+1,j ⊆Ani

≥

P{N (Ani ) ≥ 1} Ini (x) = Xn λ(Ani )

IP-a.s.,

where the inequality comes from the subadditivity of P{N (·) ≥ 1}, so {Xn } is a submartingale. Now hn (x) ≤ 1 (all x), so Xn (ω) ≤ 1 (all ω), and we can apply the submartingale convergence theorem [Theorem A3.4.III or, e.g., Chung (1974, Theorem 9.4.4)] and conclude that Xn (x) converges IP-a.s. Equivalently, limn→∞ hn (x) exists λ-a.e. on A, so to complete the proof of (9.3.22) it remains to identify the limit. For this, it is enough to show that lim supn→∞ hn (x) = 1 λ-a.e., and this last fact follows from hn (x) ≤ 1 (all x) and the chain of relations hn (x) λ(dx) ≤ lim sup hn (x) λ(dx) ≤ λ(A), λ(A) = lim n→∞

A

A

n→∞

in which we have used the lim sup version of Fatou’s lemma. The same martingale argument can be applied to the function

! P N An (x) = 1 λ An (x) if λ An (x) > 0, (1) hn (x) ≡ 0 otherwise, because the set function P{N (·) = 1} is again subadditive. Now for a simple point process with boundedly finite λ, kn

P{N (Ani ) = 1} → λ∗1 (A) = λ(A)

(n → ∞),

i=1

so it again follows that h(1) n (x) → 1

λ-a.e.

Combining (9.3.22) and (9.3.24), gn (x) = 1 − for λ-a.e. x.

(1) hn (x)/hn (x)

(9.3.24) → 0 as n → ∞

9.3.


49

Exercises 9.3.9–10 show that the λ-a.e. qualification cannot be relaxed. Clearly, from (9.3.22) and (9.3.24), the proposition could equally well be phrased in terms of λ-orderliness rather than Khinchin orderliness. In this form the significance of the result is more readily grasped, namely, that for a simple point process with boundedly finite λ, not only are M (·) and λ(·) interchangeable but also, for suitably ‘small’ sets δA and λ-a.e., we can interpret M (δA) = λ(δA) = P{N (δA) = 1} 1 + o(1) .

(9.3.25)

Note that for any x with λ{x} > 0, λ{x} = P{N {x} = 1} = P{N {x} ≥ 1} because N is simple. Equation (9.3.25) provides a link between the statements in Chapter 3 of conditional probabilities as elementary limits and those in Chapter 13 derived by direct appeal to the Radon–Nikodym theorem. The converse parts of the last few propositions in this section have included the proviso that the intensity measure be boundedly finite; without this proviso, the assertions may be false (see Exercise 9.3.11 and the references there).

Exercises and Complements to Section 9.3 9.3.1 Use Lemma 9.3.II to show that a general gamma random measure process [see Example 9.1(d)] has no fixed atoms if and only if its shape parameter is a nonatomic measure. [See also the remark in Example 9.5(c).] 9.3.2 An elaboration of the argument leading to Lemma 9.3.V is as follows. Suppose given µ ∈ M# X , bounded A ∈ BX , a dissecting system T = {Tn } for A, and

> 0. Define ν(n) (A) = #{i: Ani ∈ Tn , µ(Ani ) ≥ }. Let x1 , . . . , xk be atoms in A whose masses u1 , . . . , uk are at least [and, because µ(A) < ∞, k ≡ k( ) is certainly finite]. Verify the following. (n) (a) k = 0 if and only if ν (A) = 0 for all sufficiently large n (cf. Lemma 9.3.III). (n)

(b) More generally, there exists n < ∞ such that ν (A) = k for all n ≥ n . (n)

(c) Because k is independent of T , so is ν (A) ≡ limn→∞ ν (A). (d) Let A vary over the bounded Borel sets. Then ν (·) is a measure (in fact, ν ∈ NX#∗ of Definition 9.1.II). (e) Given a random measure ξ, so that ξ(·, ω) ∈ M# X , denote by N (·, ω) the counting measure that corresponds to ν defined from µ = ξ(·, ω). Use Proposition 9.1.VIII to verify that N is a random measure [indeed, by (d) and Definition 9.1.V, it is a simple point process]. k() (f) For each > 0 the relation µ (A) = i=1 ui δxi (A) defines a measure. Then lim↓0 µ (A) = µ(A) if and only if µ is purely atomic. Ê 9.3.3 For every A ∈ B([0.1]) let ξ(A, ω) = A f (u, ω) du, where for 0 ≤ u ≤ 1, f (u, ω) = r(r + 1)u(1 − u)r−1 with probability 1/[r(r + 1)] for r = 1, 2, . . . . Note that ξ([0, 1], ω) = 1 a.s., but there is no interval of positive length in the neighbourhood of 0 where ξ is ‘small’ a.s. Investigate whether (9.3.7) is satisfied.

50


9.3.4 Absolutely continuous random distributions [cf. Example 9.1(e)]. Construct a sequence {Fn (·)} of distribution functions on (0, 1] as follows: Fn (0) = 0, Fn (1) = 1 (n = 1, 2, . . . , ), F1 ( 12 ) = Fn ( 12 ) = U11 , and for k = 1, 3, . . . , 2n − 1, Fn (k/2n ) = (1 − Unk )Fn−1 ((k − 1)/2n ) + Unk Fn−1 ((k + 1)/2n ) = Fn+r (k/2n )

(n = 1, 2, . . . ; r = 1, 2, . . .),

where Un ≡ {Unk : k = 1, . . . , 2n − 1} is a family of i.i.d. (0, 1)-valued r.v.s with EUnk = 12 and σn2 = var Unk , the families {Un } are mutually independent, and Fn (x) is obtained by linear interpolation between Fn (jn (x)/2n ) and Fn ((jn (x) + 1)/2n ), where jn (x) = largest integer ≤ 2n x. With Fn (x) so defined on 0 ≤ x ≤ 1, the derivative fn (·) of Fn (·) is well defined except at x = anj ≡ j/2n (j = 0, 1, . . . , 2n ), where we adopt the convention that f (anj ) = 1 (all n, all anj ). (a) Show that there is a d.f. F on 0 ≤ x ≤ 1 such that

Pr{Fn (x) → F (x) (0 ≤ x ≤ 1) as n → ∞} = 1.

(b) Provided the Unj are sufficiently likely to be close to 2 condition ∞ n=1 σn < ∞ is sufficient, show that

1 , 2

for which the

Pr{fn (x) → f (x) for 0 < x < 1} = 1 x

for some density function f (·) for which F (x) = 0 f (u) du a.s. for 0 < x < 1. Thus, the random d.f. F (·) is a.s. absolutely continuous with respect to Lebesgue measure. [Hint: Let the r.v. W be uniformly distributed on [0, 1] and independent of {Un }. Show that {fn (W )} is a martingale, and assuming σn2 < ∞, use the mean square martingale convergence theorem to deduce that fn (W ) converges a.s. and that F is the integral of its limit.] (c) Investigate other conditions such as ∞ n=1 Pr{|2Unk − 1| < 1 − } = ∞ for some > 0 or lim inf n→∞ σn2 < 14 that may be sufficient to imply either that F (·) is continuous on (0, 1) or else that F (·) has jumps. [Remarks: For constructions related to the above by Kraft (1964), see Dubins and Freedman (1967)—the random d.f.s they construct on [0, 1] are a.s. singular continuous—and Métivier (1971) where the construction leads to random d.f.s on [0, 1] that are a.s. absolutely continuous.] 9.3.5 Let the nonnegative set function ψ defined on the Borel subsets of a space X be subadditive under refinement. Define a set function, λψ say, much as in Definition 9.3.VIII. Show that if λψ (A) is finite on bounded A ∈ BX , then λψ is a measure. [Hint: Check that λψ is continuous on the empty set.] 9.3.6 Korolyuk equation for purely atomic random measure. Let N be a marked point process on X × (0, ∞) for which the ground process Ng has boundedly finite intensity measure λ. Denote by {(xi , κi )} the points of a realization κi δxi [see also (6.4.6) and of N , and consider the random measure ξX = (9.1.5)]. Show that for each finite x > 0 the set function λx (A) ≡ sup T

kn

P{0 < ξX (Ani ) ≤ x}

(bounded A ∈ BX ),

i=1

is an intensity measure for those points with marks ≤ x, and that [cf. (9.3.17)] ∞ E[ξX (A)] = [λ(A) − λx (A)] dx, finite or infinite. 0

9.3.


51

9.3.7 Use equation (9.3.16) to show that a simple point process with boundedly finite intensity measure is ordinary. 9.3.8 Show that a Poisson process is simple if and only if it is ordinary. [Hint: Show that being simple is equivalent to the parameter measure being nonatomic.] 9.3.9 Let the point process N on R+ have points located at {n2 U : n = 1, 2, . . .}, where the r.v. U is uniformly distributed on (0, 1). Show that for 0 < h < 1, P{N (0, h] ≥ k} 6/π 2 . = λ((0, h]) k2

P{N (0, h] ≥ k} 1 = 2, P{N (0, h] ≥ 1} k

Conclude that the λ-a.e. constraint in Proposition 9.3.XV cannot be relaxed. 9.3.10 Suppose that the simple point process N on X has a boundedly finite intensity measure λ(·) that is absolutely continuous with respect to a measure µ(·) on X , and that there is a version of the Radon–Nikodym derivative dλ/dµ coinciding µ-a.e. with a continuous function. Use the techniques of the proof of Proposition 9.3.XV to show that, in the notation of (9.3.22), P {N (An (x)) ≥ 1} dλ → dµ µ(An (x))

(n → ∞) for µ-a.e. x.

Deduce in particular that when X = Rd and N is stationary, P {N (An (x)) ≥ 1} → const. (An (x))

-a.e. x,

and then use stationarity both to identify the constant and to eliminate the -a.e. condition (cf. Theorem 13.3.IV). 9.3.11 When N is a stationary mixed Poisson process on R with mixing distribution function F (·), P{N (0, x] = k} =

∞ 0

e−λx

(λx)k dF (λ) k!

(0 < x < ∞, k = 0, 1, . . .).

Prove the following. (i) N is simple [because F (∞−) = 1]. (ii) The intensity of N is finite or infinite with limy→∞ F1 (y), where y

F1 (y) = 0

[1 − F (u)] du.

(iii) Whether the intensity is finite or not, the conditional probabilities at (9.3.20) converge to zero when F1 is slowly varying [i.e., F1 (2y)/F1 (y) → 1 as y → ∞]. (iv) N is orderly when y[1 − F (y)] → 0 as y → ∞. (v) N is ordinary when lim inf y→∞ y[1 − F (y)] = 0. [Hint: It follows from results in Daley (1982b) that these results are in fact necessary and sufficient. It is not difficult to find d.f.s F showing that none of the implications (iii) ⇒ (iv) ⇒ (v) ⇒ (i) can be reversed; for other examples see Exercise 3.3.2, Daley (1974, 1982a), and MKM (1978, pp. 68, 371), but note that MKM use the term orderly for what we have called ordinary.]

52


9.4. Functionals: Definitions and Basic Properties In the study of families of independent random measures and point processes, transforms analogous to Laplace–Stieltjes transforms and probability generating functions of nonnegative and integer-valued r.v.s play a central role. In this section we outline a general setting in the context of random measures and give general characterization results, before extending the discussion in Chapter 5 of the properties of probability generating functionals (p.g.fl.s) of point processes. Then in Section 9.5 we discuss moment measures and some of their connections with functionals of random measures and point processes. Let f be a Borel measurable function, defined on the same c.s.m.s. space as the random measure ξ. No special definitions are needed to introduce the integral f (x) ξ(dx) ≡ f dξ, (9.4.1) ξf = X

for by assumption each realization of ξ is a boundedly finite Borel measure, and the usual theory of the Lebesgue integral applies on a realization-byrealization basis. In particular, if we introduce the space BM(X ) of bounded measurable functions which vanish outside a bounded set in X , then with probability 1 the integral exists and is finite. It is also a random variable; this can be seen by first taking f to be the indicator function of a bounded Borel set and then applying the usual approximation arguments using linear combinations and monotone limits. Now the class of r.v.s is closed under both these operations, and ξf = ξ(A) is certainly an r.v. when f = IA , so it follows that ξf is a (proper) r.v. for any f ∈ BM(X ). The study of such random integrals, which are evidently linear in f , links the theory of random measures with a whole hierarchy of theories of random linear functionals, of which the theory of random distributions is perhaps the most important, and is relevant in discussing second-order properties (see Chapter 8). We pause, therefore, to give a brief introduction to such general theories. Given any linear space U on which the notions of addition and scalar multiplication are defined, the concept of a linear functional, that is, a mapping γ from U into the real line satisfying γ(αu + βv) = αγ(u) + βγ(v)

(α, β ∈ R; u, v ∈ U)

(9.4.2)

makes sense, and we may consider the space of all such linear functionals on a given U. Furthermore, if U has a topology conformable with the linear operations (i.e., one making these continuous), we may consider the smaller space of continuous linear functionals on U. Many different possibilities arise, depending on the choice of U and of the topology on U with respect to which continuity is defined. With any such choice there are several ways in which we may associate a random structure with the given space of linear functionals. Of these we

9.4.

Functionals: Definitions and Basic Properties

53

distinguish two general classes, which we call strict sense and broad sense random linear functionals. A natural σ-algebra in the space CU of continuous linear functionals on U is the smallest σ-algebra with respect to which the mappings γ: u → γ(u) are measurable with respect to each u ∈ U. Endowing CU with this σ-algebra, we may define a strict sense random linear functional on U as a measurable mapping Γ(·) from a probability space into CU . This ensures, as a minimal property, that Γ(u) is a random variable for each u ∈ U. On the other hand. it is often difficult to determine conditions on the distributions of a family of r.v.s {Γu }, indexed by the elements of U, that will allow us to conclude that the family {Γu } can be identified a.s. with a random functional Γ(u) in this strict sense. The same difficulty arises if we attempt to define a random linear functional as a probability distribution on CU . How can we tell, from the fidi distributions or otherwise, whether such a distribution does indeed correspond to such an object? Even in the random measure case this discussion is not trivial, and in many other situations it remains unresolved. The alternative, broad sense, approach is to accept that a random linear functional cannot be treated as anything more than a family of r.v.s indexed by the elements of U and to impose on this family appropriate linearity and continuity requirements. Thus, we might require that Γαu+βv = αΓu + βΓv

a.s.

(9.4.2 )

and, if un → u in the given topology on U, ξun → ξu

a.s.

(9.4.3)

or, at (9.4.3), we could merely use convergence in probability or in quadratic mean. If Γu = Γ(u) for all u ∈ U, where Γ(·) is a strict sense random linear functional, then of course both (9.4.2 ) and (9.4.3) hold a.s. Dudley (1969) reviews some deeper results pertaining to random linear functionals. Example 9.4(a) Generalized random processes (random Schwartz distributions). Take X = R, and let U be the space of all infinitely differentiable functions on R that vanish outside some finite interval; that is, U is a space of test functions on R. Introduce a topology on U by setting un → u if and only if the {un } vanish outside some common finite interval, and for all k ≥ 0, (k) the kth derivatives {un } converge uniformly to u(k) . Then CU , the space of all functionals on U satisfying (i) the linearity condition (9.4.2), and (ii) the continuity condition γ(un ) → γ(u) whenever un → u in U, is identified with the space of generalized functions, or more precisely Schwartz distributions. Any ordinary continuous function g defines such a distribution through the relation ∞ g(x)u(x) dx, γ(u) = −∞

54


the continuity condition (ii) following from the boundedness of g on the finite interval outside which the un vanish and the uniform convergence of the un themselves. Similarly, any bounded finite measure G on R defines a distribution by the relation u(x) G(dx). γ(u) = R

However, many further types of Schwartz distribution are possible, relating, for example, to linear operations on the derivatives of u. The corresponding strict sense theory has been relatively little used, but the broad sense theory plays a central role in the second-order theory of stationary generalized processes, of which the second-order theory of stationary point processes and random measures forms a special case. A similar theory for random generalized fields can be developed by taking test functions on Rd in place of test functions on R. Gelfand and Vilenkin (1964) and Yaglom (1961) give systematic treatments of these broad sense theories. A natural tool for handling any type of random linear functional is the characteristic functional, defined by ΦΓ [g] = E[exp(iΓg )]

(g ∈ U),

(9.4.4)

where Γg is a random linear functional (strict or broad sense) on U. It can be described as the characteristic function E(eisΓg ) of Γg , evaluated at the arbitrary value s = 1 and treated as a function of g rather than s. Example 9.4(b) Gaussian measures on Hilbert space. Random variables taking their values in a Hilbert space H can be placed within the general framework of random linear functionals by taking advantage of the fact that the space of continuous linear functionals on a Hilbert space can be identified with the given Hilbert space itself. In this interpretation Γ(u) is identified with the inner product Γ, u for u ∈ H. When H is finite-dimensional, the characteristic functional reduces to the multivariate characteristic function n = φ(u1 , . . . , un ) = φ(u), Γk u k E[exp(iΓ, u)] = E exp i k=1

where Γk and uk are the coordinates of Γ and u. In this case a Gaussian measure is just the ordinary multivariate normal distribution: setting the mean terms equal to zero for simplicity, the characteristic function has the form φ(u) = exp − 12 u Au , where u Au is the quadratic form associated with the nonnegative definite (positive semidefinite) symmetric matrix A. This suggests the generalization to infinite-dimensional Hilbert space of Φ[u] = E[exp(iΓu )] = exp(− 12 u, Au),

(9.4.5)

9.4.


55

where A is now a positive definite self-adjoint linear operator. The finitedimensional distributions nof Γu1 , . . . , Γun for arbitrary u1 , . . . , un in H can be determined by setting k=1 sk uk in place of u in (9.4.5): they are of multivariate normal form with n × n covariance matrix having elements ui , Auj . From this representation the consistency conditions are readily checked, as well as the linearity requirements (9.4.2). If un → u (i.e., un − u → 0), it follows from the boundedness of A that (un − u), A(un − u) → 0 and hence that Γun → Γu in probability and in quadratic mean. These arguments suffice to show that (9.4.5) defines a broad sense random linear functional on H, but they are not sufficient to imply that (9.4.5) defines a strict sense random linear functional. For this, more stringent requirements are needed; these have their roots in the fact that a probability measure on H must be tight, and hence in a loose sense approximately concentrated on a finite-dimensional subset of H. It is known [see, e.g., Parthasarathy (1967, Chapter 8)] that the necessary and sufficient conditions for (9.4.5) to be the characteristic functional of a strict sense random linear functional on H [so that we can write Γu = Γ(u) = Γ, u], or, equivalently, of a probability measure on H itself, is that the operator A be of Hilbert–Schmidt type. In this case the characteristic functional has the more special form λk (hk , u)2 , Φ[u] = exp − 12 where {hk } is a complete set eigenvectors for A, {λk } is the set of corre of sponding eigenvalues, and λ2k < ∞. Returning to the random measure context, let us first note that random measures can just as easily be characterized by the values of the integrals (9.4.1) as they can by their evaluations on Borel sets; indeed, the latter are just a special case of the former when f is an indicator function. It follows at once thatB(M# X ) is the smallest σ-algebra with respect to which the random integrals f dξ are measurable for each f ∈ BM(X ), and that a mapping ξ f dξ is from a probability space into M# X is a random measure if and only if a random variable for each f ∈ BM(X ) [a smaller class of functions f suffices: Kallenberg (1983a, Exercise 3.1) indicates a stronger version of this result]. A more useful result is the following analogue of Proposition 9.1.VIII, the proof of which is left to Exercise 9.4.1. Proposition 9.4.I. Let {ξf } be a family of random variables, indexed by the elements f of BM(X ). Then there exists a random measure ξ such that ξf = f dξ a.s. if and only if (i) ξαf +βg = αξf + βξg a.s. for all scalars α, β and f , g ∈ BM(X ); and (ii) ξfn → ξf a.s. as n → ∞ for all monotonically converging nonnegative sequences {fn } ⊂ BM(X ) (i.e. fn ≥ 0 and fn ↑ f ).

56


Conditions (i) and (ii) are, of course, just the conditions (9.4.2) and (9.4.3) in a form suitable for random measures; the importance of the proposition is that it implies that the broad and strict sense approaches are equivalent for random measures. From this point it is easy to move to a characterization of the fidi distributions of the integrals f dξ for a random measure; we state this in the form of a characterization theorem for characteristic functionals, which we define by (f ∈ BM(X )), (9.4.6) Φξ [f ] = E exp i f dξ as the appropriate special form of (9.4.4). Theorem 9.4.II. Let the functional Φ[f ] be real- or complex-valued, defined for all f ∈ BM(X ). Then Φ is the characteristic functional of a random measure ξ on X if and only if (i) for every finite family f1 , . . . , fn of functions fk ∈ BM(X ), the function n φn (f1 , . . . , fn ; s1 , . . . , sn ) = Φ sk fk

(9.4.7)

k=1

is the multivariate characteristic function of proper random variables ξf1 , . . . , ξfn , which are nonnegative a.s. when the functions f1 , . . . , fn are nonnegative; (ii) for every sequence {fn } ⊂ BM(X ) with fn ≥ 0 and fn ↑ f pointwise, Φ[fn ] → Φ[f ] ; and

(9.4.8)

(iii) Φ(0) = 1 where 0 here denotes the zero function in BM(X ). Moreover when the conditions are satisfied, the functional Φ uniquely determines the distribution of ξ. Proof. If ξ is a random measure, conditions (i) and (ii) are immediate, and imply that the fidi distributions of ξ, and hence the distribution of ξ itself, are uniquely determined by ξ. If fn → f pointwise, and the fn are either monotonic or bounded by a common element of BM(X ), it follows from the Lebesgue convergence theorems that for each realization, ξfn =

fn dξ →

f dξ = ξf ,

so that ξfn → ξf a.s. Equation (9.4.8) follows from " " application of a further the dominated convergence theorem, using " exp i fn dξ " ≤ 1. Suppose next that conditions (i)–(iii) are satisfied. Condition (i) subsumes both Kolmogorov consistency conditions for the fidi distributions of the r.v.s ξf defined by (9.4.7) for f ∈ BM(X ). For example, in characteristic function

9.4.


57

terms, the requirement of marginal consistency reduces to the trivial verification that n−1 n−1 si fi + 0 · fn = Φ si fi . Φ k=1

k=1

Thus, we may assume the existence of a jointly distributed family of r.v.s {ξf , f ∈ BM(X )}. Condition (i) also implies the linearity property (i) of Proposition 9.4.I, for the condition ξf3 = ξf1 + ξf2 a.s. is equivalent to the identity Φ[s1 f1 + s2 f2 + s3 f3 ] = Φ[(s1 + s3 )f1 + (s2 + s3 )f2 ], which will certainly be valid if f3 = f1 + f2 . A similar argument applies when scalar multipliers α, β are included. Finally, condition (ii) of the theorem implies that the distribution of ξf −fn approaches the distribution degenerate at 0, and hence that ξf −fn converges in probability to zero. From the linearity property of the ξf we deduce that ξfn → ξf

in probability.

(9.4.9)

However, because we assume in condition (ii) that the sequence {fn } is monotonic increasing, it follows from condition (i) that the sequence {ξfn } is a.s. monotonic increasing. Because ξfn ≤ ξf a.s. by similar reasoning, ξfn converges a.s. to a proper limit r.v. X say. But then (9.4.9) implies that X = ξf a.s., so condition (ii) of Proposition 9.4.I is also satisfied. The existence of a random measure with the required properties now follows from Proposition 9.4.I and the part of the theorem already proved. Variants of condition (ii) above are indicated in Exercise 9.4.2. As described above, the characteristic functional emerges naturally from the context of random linear functionals, but in the study of random measures and point processes, which are nonnegative by definition, it is enough to use real variable counterparts. The Laplace functional is defined for f ∈ BM+ (X ), the space of all nonnegative f ∈ BM(X ), by Lξ [f ] = E exp − f dξ

(f ∈ BM+ (X ))

(9.4.10)

[this is the same as (6.1.8)]. An exact counterpart of Theorem 9.4.II holds for Laplace functionals; see Exercise 9.4.3 for some detail. Observe, in particular, that the theorem implies that the distribution of a random measure is completely determined by the Laplace functional. The class BM+ (X ) of functions on which the Laplace functional is defined can be restrictive for some applications, as, for example, in discussing the mixing properties of cluster processes in Section 12.3. We therefore define the extended Laplace functional for use in such contexts much as for L but on the space of functions BM+ (X ) consisting of functions f that are expressible as

58


the monotone limit of an increasing sequence of functions {fn } ⊂ BM+ (X ). Then by the monotone convergence theorem, fn (x) ξ(dx) ↑ f (x) ξ(dx) a.s., X

X

whether the limit is finite or infinite, and then by dominated convergence, Lξ [fn ] → Lξ [f ] ≡ E exp − X f dξ (f ∈ BM+ (X )), (9.4.11) where we use Lξ or, more briefly, L, both for the functional as originally defined and for its extension to BM+ (X ). The extended Laplace functional, as defined over functions f ∈ BM+ (X ), has continuity properties as below, but it need not be continuous for monotone sequences {fn } ⊂ BM+ (X ): take f (x) = (all x ∈ X ), ξ(X ) = ∞ a.s.; then for all > 0, L[f ] = 0 = L[0] = 1. Proposition 9.4.III. The extended Laplace functional L[·] satisfies L[fn ] → L[f ]

(fn , f ∈ BM+ (X ))

whenever either (a) fn (x) ↑ f (x), or (b) fn (x) → f (x) and there exists a nonnegative measurable function ∆(·) such that X ∆(x) ξ(dx) < ∞ a.s. and |fn (x) − f (x)| ≤ ∆(x) for all sufficiently large n. Proof. If fn ↑ f , then it is easy to construct a monotone sequence of functions {fn } ⊂ BM+ (X ) with fn (x) ↑ f (x), and (9.4.11) holds by definition. In the other case, we have for all n ≥ some n0 , fn (x) ≤ f (x) + ∆(x) ≤ fn0 (x) + 2∆(x), so X fn (x) ξ(dx) ≤ X [fn0 (x) + 2∆(x)] ξ(dx) ≤ ∞ a.s., and by dominated convergence applied to the sequence {fn (·)}, fn (x) ξ(dx) → f (x) ξ(dx) < ∞ a.s.

X

X

A second appeal to dominated convergence now implies (9.4.11). Under conditions (b) here, it follows that L[ f ] → 1 = L[0] as → 0, so for such fn and f ∈ BM+ (X ), the extended Laplace functional has all the properties of the ordinary Laplace functional (9.4.10). Because random measures are inherently nonnegative, the Laplace functional is often the most appropriate tool to use in handling random measures, just as the Laplace–Stieltjes transform is generally the most useful tool for handling nonnegative random variables. It is only when the r.v. is integervalued, or, in our context, when the random measure is a point process, that

9.4.


59

there are advantages in moving to the probability generating function (p.g.f.) or its counterpart the probability generating functional (p.g.fl). We have already discussed the p.g.fl. of finite point processes in Chapter 5, defined there on the class U of complex-valued Borel measurable functions ζ satisfying the condition |ζ(x)| ≤ 1 (all x). But just as the Laplace functional is discussed more advantageously as a functional over a space of real-valued functions, so is the p.g.fl. discussed better over a narrower class of functions. We largely follow Westcott’s (1972) general treatment. Definition 9.4.IV. V(X ) denotes the class of all real-valued Borel functions h defined on the c.s.m.s. X with 1 − h vanishing outside some bounded set and satisfying 0 ≤ h(x) ≤ 1 (all x ∈ X ). V0 (X ) is the subset of h ∈ V(X ) satisfying inf x∈X h(x) > 0. V(X ) is the space of functions h expressible as limits of monotone sequences hn ∈ V(X ). Extending Definition 5.5.I, the probability generating functional (p.g.fl.) of a (general) point process N on the c.s.m.s. X is defined by log h(x) N (dx) (h ∈ V(X )). (9.4.12) G[h] ≡ GN [h] = E exp X

Because a point process is a.s. finite on the bounded set where 1 − h does not vanish, the exponential of the integral at (9.4.12) can legitimately be written in the product form GN [h] = E h(xi ) , (9.4.13) i

where the product is taken over the points of each realization of N (recall Proposition 9.1.V), with the understanding that it takes the value zero if h(xi ) = 0 for any xi , and unity if there are no points of N within the support of 1 − h. If h is such that − log h ∈ BM+ (X ), then the equation G[h] = LN [− log h]

(9.4.14)

relates the p.g.fl. to the Laplace functional of the point process [cf. (9.4.10)]. Indeed, − log h ∈ BM+ (X ) implies that the values of h lie within a closed subset of (0, 1], so because the distribution of a random measure is determined by all f ∈ BM+ (X ), the distribution of a point process is determined by all h ∈ V0 (X ). Although results for point processes need to be proved only with this more restricted class of functions, it is only in our discussion of mixing properties of cluster processes (see Proposition 12.3.IX) that we need the constraint, so mostly we use V(X ). Note that if {hn (x)} is a pointwise convergent sequence of functions ∈ V0 (X ) with the support of each 1 − hn contained by some fixed bounded set, then the pointwise limit, h say, has h ∈ V(X ) but not necessarily h ∈ V0 (X ). To this extent, V(X ) is a simpler class with which to work.

60


By putting additional restrictions on the point process, the p.g.fl. can be defined for more general classes of functions h. For example, if the expectation measure M of N exists [cf. around (9.5.1) below], and nonnegative h is so chosen that | log h(x)| M (dx) < ∞, (9.4.15) X

then the integral in (9.4.12) converges a.s. to a finite quantity, and the expectation exists. The p.g.fl. of a point process can be characterized as in Theorem 9.4.V. It is an exact analogue of Theorem 9.4.II, so proof is left to the reader. Theorem 9.4.V. Let the functional G[h] be real-valued, defined for all h ∈ V(X ). Then G is the p.g.fl. of a point process N on X if and only if (i) for every h of the form n (1 − zk )IAk (x), 1 − h(x) = k=1

where the bounded Borel sets A1 , . . . , An are disjoint and |zi | ≤ 1, the p.g.fl. G[h] reduces to the joint p.g.f. Pn (A1 , . . . , An ; z1 , . . . , zn ) of an ndimensional integer-valued random variable; (ii) for every sequence {hn } ⊂ V(X ) with hn ↓ h pointwise, G[hn ] → G[h] whenever 1 − h has bounded support; and (iii) G[1] = 1 where 1 denotes the function identically equal to unity in X . Moreover, when these conditions are satisfied, the functional G uniquely determines the distribution of N . Variants on the continuity condition (ii) are again possible, although more is needed than just pointwise convergence (see Exercise 9.4.5). Indeed, we shall have a need for the extended p.g.fl. G[·] defined by analogy with the extended Laplace functional at (9.4.11) as log h(x) N (dx) (h ∈ V(X)) G[h] ≡ E exp X

[see Definition 9.4.IV for V(X)]. Further details are given in Exercise 9.4.6. Example 9.4(c) Poisson and compound Poisson processes. The form of the p.g.fl. for the Poisson process is already implicit in the form of the p.g.f. obtained in Chapter 2 and its multivariate extension [see (2.4.5) and (9.2.9)] k (1 − zj )µ(Aj ) , (9.4.16) Πk (A1 , . . . , Ak ; z1 , . . . , zk ) = exp − j=1

where A1 , . . . , Ak are disjoint and µ(·) is the parameter measure of the process. k Writing h(x) = 1 − j=1 (1 − zj )IAj (x), so that h(x) = zj on Aj and = 1 outside the union of all the Aj , (9.4.16) is expressible as [1 − h(x)] µ(dx) , (9.4.17) G[h] = exp − X

which is evidently of the required form for the p.g.fl.

9.4.


61

The following heuristic derivation of (9.4.17) also throws light on the character of the p.g.fl. Suppose that the support of 1 − h(x) is partitioned into small subsets ∆Ai in each of which is a ‘representative point’ xi . Then, approximately, log h(x) N (dx) ≈ log h(xi )N (∆Ai ), X

i

where because the r.v.s N (∆Ai ) are independent (by assumption), N (∆Ai ) log h(x) N (dx) ≈E exp log h(xi ) E exp X

=

i

E h(xi )N (∆Ai ) = exp − [1 − h(xi )]µ(∆Ai ) i

i

[1 − h(x)] µ(dx) . ≈ exp − X

The corresponding expression for the p.g.fl. of a compound Poisson process, understood in the narrow sense of Section 2.4 as a nonorderly point process, is [1 − Πh(x) (x)] µ(dx) G[h] = exp − X (9.4.18) πn (x) 1 − [h(x)]n µ(dx) ; = exp − X

n

this reduces to (2.4.5) for the univariate p.g.f. when h(x) = 1 − (1 − z)IA (x). It is not difficult to verify that G satisfies the conditions of Theorem 9.4.V, and therefore it represents the p.g.fl. of a point process which enjoys the complete independence property discussed further in Section 10.1. Indeed, it is not difficult to check from the representation in (9.4.18) that the compound Poisson process can be characterized as a completely random measure which has no drift component, is free of fixed atoms, and has random atoms of positive integral mass. The generalized compound Poisson process described in Section 6.4 corresponds to a marked Poisson process with independent nonnegative marks. As a point process on X × K (with K = R+ ) it has a Laplace functional of the form −f (x,κ) 1−e π(dκ) µ(dx) (f ∈ BM+ (X × K)). L[f ] = exp − X

K

Here the interpretation as a Poisson process on the product space is immediately evident. Furthermore, when K = R+ , the cumulative process ξ(A) = i:xi ∈A κi has the Laplace functional, now defined over f ∈ BM+ (X ), − κi f (xi ) −κf (x) i = exp − 1−e π(dκ) µ(dx) . L[f ] = E e X

K

This exhibits the process as a completely random measure (Definition 10.1.I) with neither any fixed atoms nor a drift component.

62


Example 9.4(d) Mixed Poisson process. Referring to (9.4.16), denote the fidi distributions of a Poisson process by Pk (· | µ) for short. Then N is a mixed Poisson process when for some r.v. Λ and boundedly finite measure µ its fidi distributions Pk (·) are given by Pk (·) = E[Pk (· | Λµ)],

(9.4.19)

the expectation being with respect to Λ. Write L(s) = E(e−sΛ ) [Re(s) ≥ 0] for the Laplace–Stieltjes transform of Λ. It then follows from (9.4.19) and (9.4.17) that the p.g.fl. of a mixed Poisson process is given by [1 − h(x)]Λ µ(dx) =L [1 − h(x)] µ(dx) . G[h] = E exp − X

X

We have already remarked that one of the most important properties of the transforms of r.v.s is the simplification they afford in handling problems involving sums of independent r.v.s. The summation operator for random measures is defined by (ξ1 + ξ2 )(A) = ξ1 (A) + ξ2 (A)

(all A ∈ BX ),

(9.4.20)

and it is both obvious and important that it extends the notion of superposition of point processes. Note that (9.4.20) has the equivalent form (all f ∈ BM+ (X )). (9.4.20 ) f d(ξ1 + ξ2 ) = f dξ1 + f dξ2 Now suppose that {ξi : i = 1, 2, . . .} is an infinite sequence of random measures, each defined on (Ω, E, P), such that ζ(A) ≡

∞

ξi (A)

(9.4.21)

i=1

is a.s. finite on all bounded A ∈ BX . It is well known and easy to check that a countable sum of measures is again a measure. Thus, ζ(·) is a boundedly finite measure, at least on the ω set where the ξi are simultaneously measures, which set has probability 1 by assumption and the fact that only a countable family is involved. Redefining ζ to be zero on the complementary ω set of P-measure zero, and observing that a countable sum of r.v.s is again a r.v., we obtain a mapping from (Ω, E, P) into M# X satisfying the condition of Corollary 9.1.IX and which is therefore a random measure. Thus, we have the following lemma. Lemma 9.4.VI. ζ(·) defined at (9.4.21) is a random measure if and only if the infinite sum at (9.4.21) converges for all bounded A ∈ BX . No new concepts arise in the following definition of independence of two random measures; it extends to the mutual independence of both finite and infinite families of random measures in the usual way.

9.4.


63

Definition 9.4.VII. The random measures ξ1 and ξ2 are independent when they are defined on a common space (Ω, E, P) and are such that P(F1 ∩ F2 ) = P(F1 )P(F2 ) for all finite families Fi of events defined on ξi (i = 1, 2). Let ξi have characteristic functional Φi and Laplace functional Li . By writing ζn = ξ1 + · · · + ξn , the following assertions are simple consequences of the definitions and Lemma 9.4.VI, and can be proved by methods exploited already (see Exercise 9.4.7). Proposition 9.4.VIII. When the random measures ξ1 , ξ2 , . . . are mutually independent, the sum ζn = ξ1 + · · · + ξn has the characteristic functional n

Φi [f ] (all f ∈ BM(X )) (9.4.22a) Φζn [f ] = i=1

and Laplace functional Lζn [f ] =

n

Li [f ]

(all f ∈ BM+ (X )).

(9.4.22b)

i=1

Lζn [f ] converges as n → ∞ to a nonzero limit Lf for each f ∈ BM+ (X ) if and only if the infinite sum at (9.4.21) is finite on bounded A ∈ BX , and then Lf is the Laplace functional of the random measure ζ at (9.4.21). The analogue of this result for the p.g.fl. G of the superposition of the independent point processes N1 , . . . , Nn with p.g.fl.s G1 , . . . , Gn is easily given (see Exercise 9.4.8 for proof). Proposition 9.4.IX. When the point processes N1 , N2 , . . . are mutually independent, the superposition N1 + · · · + Nn has p.g.fl. n

Gi [h] (h ∈ V(X )). G[h] = i=1

This sequence of finite products converges if and only if the infinite sum ∞ Ni (A) (9.4.23) N (A) = i=1

is a.s. finite on bounded A ∈ BX , and the infinite product is then the p.g.fl. of the point process N at (9.4.23).

Exercises and Complements to Section 9.4 9.4.1 Use linear combinations of indicator functions and their limits to prove that conditions (i) and (ii) of Proposition 9.4.I imply the a.s. form of (9.4.1). 9.4.2 Condition (ii) of Theorem 9.4.II requires the continuity of characteristic functionals (9.4.24) Φ[fn ] → Φ[f ] for f , fn ∈ BM(X ) when fn (x) → f (x) (x ∈ X ) pointwise monotonically from below with f , fn in fact ∈ BM+ (X ). Show that (9.4.24) holds without this monotonicity of convergence or nonnegativity of f if either

64

9. Basic Theory of Random Measures and Point Processes (i) P{ξ(X ) < ∞} = 1, bounded measurable f and fn , with sup |f (x) − fn (x)| → 0

(n → ∞); or

(9.4.25)

x∈X

(ii) f and fn ∈ BM(X ), the union of their support is a bounded set, and (9.4.25) holds. Give an example of a random measure ξ and functions f , fn ∈ BM(X ) satisfying (9.4.25) for which (9.4.24) fails. [Hint: Consider a stationary Poisson process on R+ with f (x) = 0 (all x ∈ R+ ), fn (x) = n−1 I[0,n] (x).] 9.4.3 Laplace functional analogues of various results for characteristic functionals are available, subject to modifications reflecting the different domain of definition; below, f ∈ BM+ (X ). (a) [See Theorem 9.4.II.] Show that {L[f ]: all f } uniquely determines the distribution of a random measure ξ. (b) [See Exercise 9.4.2.] For sequences fn , the convergence L[fn ] → L[f ] holds as supx∈X |fn (x) − f (x)| → 0 if (i) ξ is totally bounded; or (ii) the pointwise convergence fn → f is monotonic; or (iii) there is a bounded Borel set containing the support of every fn . Give examples to show that, if otherwise, the convergence L[fn ] → L[f ] may fail. 9.4.4 For a random measure ξ on the c.s.m.s. X with Laplace functional L[·], show that for any bounded A ∈ BX , P{ξ(A) = 0} = lims→∞ L[sIA ], whereas for any A ∈ BX , P{ξ(A) < ∞} = lims↓0 limn→∞ L[sIAn ], where {An } is an increasing sequence of bounded sets in BX for which A = limn→∞ An (the case A = X is of obvious interest). 9.4.5 Suppose that the functions hn ∈ V(X ) and that hn (x) → h(x) (n → ∞) for every x ∈ X . Show that G[hn ] → G[h] (n → ∞) if, in place of the conditions at (ii) of Theorem 9.4.V, either (a) N is a.s. totally finite, or (b) N has a finite first moment measure M and X |hn (x) − h(x)| M (dx) → 0 as n → ∞. Let hn (x) = 1 − n−1 for |x| < n, = 1 for |x| ≥ n, so that hn (x) → 1 (all x) for n → ∞. Show that for a stationary Poisson process at rate λ, G[hn ] = e−2λ = 1 = G[limn→∞ hn (·)] (cf. Exercise 9.4.2). 9.4.6 Let {hn } ⊂ V(X ) have hn (x) → h(x) pointwise as n → ∞. Show that the extended p.g.fl. convergence result G[hn ] → G[h] holds if any one of the following conditions holds. (a) h is the monotone limit of functions hn ∈ V(X ). (b) | log[hn (x)/h(x)]| < (x) (all n) and X (x) N (dx) < ∞ a.s. (c) inf x∈X hn (x) > c for some c > 0 and sufficiently large n, and |hn (x) − h(x)| N (dx) < ∞ a.s. X

[Hint: This is the p.g.fl. analogue of Proposition 9.4.III. For part (b), use the method of proof of Proposition 9.4.VIII. Part (c) follows from (b). See Daley and Vere-Jones (1987).]

9.5.

Moment Measures and Expansions of Functionals

65

9.4.7 For the partial sums ζn of random measures (cf. Proposition 9.4.VIII), show that Lζn [f ] has a nonzero limit for f ∈ BM+ (X ) if and only if the infinite product ∞ i=1 (1 − exp[−ξi (A)]) > 0 a.s., that is, when the infinite series at (9.4.21) converges. Hence, complete the proof of the proposition.

9.4.8 For an infinite sequence N1 , N2 , . . . of independent point processes, show that the necessary and sufficient condition for the infinite superposition ∞ i=1 Ni to be a well-defined point process is the convergence for every bounded A ∈ BX ∞ of the sum i=1 pi (A), where pi (A) = Pr{Ni (A) > 0}. Hence, establish Proposition 9.4.IX.

9.4.9 Let N be a renewal process with lifetime d.f. F , and denote by Gb|a [h] the p.g.fl. of the process on the interval [a, b] conditional on the occurrence of a point of the process at a. Much as in Bol’shakov (1969), show that this conditional p.g.fl. satisfies the integral equation Gb|a [h] = [1 − F (b − a)] +

b

(h(x) + 1)Gb|x [h] dx F (x − a).

a

Similarly, if Ga|b [h] denotes the p.g.fl. conditional on a point at b,

Ga|b [h] = [1 − F (b − a)] +

b

(h(x) + 1)Ga|x [h] |dx F (b − x)|.

a

Find extensions of these equations to the case where the renewal process is replaced by a Wold process.

9.5. Moment Measures and Expansions of Functionals As with ordinary random variables and characteristic functions, the characteristic functional is closely associated with the moment structure of the random measure, which, as in the point process case studied in Chapter 5, is expressed through a family of moment measures. In particular, for any random measure ξ on the c.s.m.s. X and any Borel set A, consider the expectation M (A) = E[ξ(A)]

(finite or infinite).

(9.5.1)

Clearly, M inherits the property of finite additivity from the underlying random measure ξ. Moreover, if the sequence {An } of Borel sets is monotonic increasing to A, then by monotone convergence M (An ) ↑ M (A). Thus, M (·) is continuous from below and therefore a measure. In general, it need not take finite values, even on bounded sets, but when it does, we say that the expectation measure of ξ exists and is given by (9.5.1). The expectation measure M (·) may also be called the first moment measure of ξ. When M exists, the above argument can readily be extended to the random integrals f dξ for f ∈ BM(X). Thus, if f is the indicator function of the bounded Borel set A, E f dξ = M (A). Extending in the usual way through linear combinations and monotone limits it follows that E f dξ = f dM (f ∈ BM(X )). (9.5.2)

66


Equations of the form (9.5.2) have been included under the name Campbell theorem [see, e.g., Matthes (1972), MKM (1978)] after early work by Campbell (1909) on the shot-noise process in thermionic vacuum tubes [see also Moran (1968, pp. 417–423)]. Campbell measures that we discuss in Chapter 13 constitute a significant extension of this simple concept. Consider next the k-fold product of ξ with itself, that is, the measure defined a.s. for Borel rectangles A1 × · · · × Ak by ξ (k) (A1 × · · · × Ak ) =

k

ξ(Ai )

(9.5.3)

i=1

and extended to a measure, necessarily symmetric, on the product Borel σalgebra in X (k) . Now the rectangles form a semiring generating this σ-algebra, and (9.5.3) defines a random variable for every set in this semiring, so it follows from Proposition 9.1.VIII that ξ (k) is a random measure on X (k) . Definition 9.5.I. The kth order moment measure Mk (·) of ξ is the expectation measure of ξ (k) , whenever this expectation measure exists. This identification is illustrated forcefully in the proof below of a result already given at Proposition 5.4.VI and where a key role is played by diag A(k) ≡ {(x1 , . . . , xk ) ∈ X (k) : x1 = · · · = xk = x ∈ A}.

(9.5.4)

Proposition 9.5.II. A point process N with boundedly finite second moment measure has M2 (diag A(2) ) ≥ M (A) for all bounded A ∈ BX ; equality holds if and only if N is simple. Proof. Let Tbe a dissecting system for A, so diag A(2) equals the monotone kn Ani × Ani . Because M2 is a measure and M2 (A(2) ) < ∞, limit limn→∞ i=1 M2 (diag A(2) ) = M2

lim

n→∞

kn

kn (Ani × Ani ) = lim M2 (Ani × Ani ) n→∞

i=1

i=1

kn kn N 2 (Ani ) = M (A) + lim E N (Ani )[N (Ani ) − 1] . = lim E n→∞

i=1

n→∞

i=1

Write the last term as E(Xn ). From the nesting property of the Tn , the r.v.s Xn are a.s. nonincreasing and ≥ 0, so by monotone convergence M2 (diag A(2) ) = M (A) + E lim Xn , n→∞

and limn→∞ Xn = 0 if and only if limn→∞ supi N (Ani ) ≤ 1 a.s.; that is, P{N ({x}) ≤ 1 for all x ∈ A} = 1; equivalently, N = N ∗ a.s. The following kindred property of random measures is proved similarly; see Exercise 9.5.5(b), and Section 9.3 and Exercise 9.5.6 for related results. Proposition 9.5.III. A random measure ξ with boundedly finite second moment measure M2 is a.s. nonatomic if and only if M2 (diag X (2) ) = 0.

9.5.


67

Example 9.5(a) Mixtures of quadratic random measures. A Gaussian r.v. has moments of all orders, from which it follows that the same is true for the stationary quadratic random measure in Example 9.1(b). In particular, its first and second moment measures are defined by the equations Z 2 (x) dx = σ 2 (A), (9.5.5a) M (A) = E[ξ(A)] = E A

2 2 M2 (A × B) = E[ξ(A)ξ(B)] = E Z (x)Z (y) dx dy A B = [σ 4 + 2c2 (x − y)] dx dy. A

(9.5.5b)

B

From these representations it is clear that M and M2 are both absolutely continuous with respect to Lebesgue measure on R and R2 , with derivatives σ 2 and σ 4 + 2c2 (x − y), respectively, where c(·) is the covariance function for Z. Similar representations can be obtained for higher moments. Example 9.5(b) Mixed random measure. Let Λ be a positive r.v., independent of the random measure ξ(·), and set ξΛ (A) = Λξ(A). (k)

Using independence, the kth order moment measures MΛ for ξΛ are related to those of ξ by the equations (k)

MΛ (·) = E(Λk )Mk (·). Thus, if Λ has infinite moments of order k and higher, the same will be true for the moment measures of ξΛ , and conversely, if the kth moment of ξΛ is finite, the kth order moment measure of ξ exists. This particular example is nonergodic [meaning, the values of M (A) cannot be determined from observations on a single realization of the process], but this is not a necessary feature of examples with infinite moment measures: for example, in place of the r.v. Λ, we could multiply ξ by any continuous ergodic process λ(t) with infinite moments and integrate to obtain a random measure with similar moment properties. This procedure of mixing, or randomizing, with respect to a given parameter of a process is a rich source of examples. Example 9.5(c) Moments of completely random measures (see Sections 2.2 and 10.1). If ξ(·) is completely random, the first and second moment measures (assuming these are finite) are given by relations of the type M (A) = E[ξ(A)] = µ(A), M2 (A × B) = E[ξ(A)ξ(B)] = µ(A)µ(B) + var ξ(A ∩ B).

(9.5.6)

Particular interest here centres on the variance term: it vanishes unless the set A ∩ B = ∅, so this term represents a measure concentrated along the

68


diagonal. For the stationary gamma random measure studied in Example 9.1(d), var ξ(A) = λ2 α(A) and so M2 (A × B) = λ2 α(A ∩ B) + λ2 α2 (A)(B).

(9.5.7)

2 2

Thus, M2 has a constant areal density λ α off the diagonal and a concentration with linear density λ2 α along it. Such concentrations are associated with the a.s. atomic character of the random measure (see Proposition 9.5.III) and should be contrasted with the absolutely continuous moment measures of Example 9.5(a) in which the realizations themselves are a.s. absolutely continuous measures. The next lemma summarizes the relation between moment measures and the moments of random integrals, and is useful in discussing expansions of functionals. Note also the identification property at Exercise 9.5.7. Lemma 9.5.IV. Let the kth moment measure Mk of the random measure ξ exist. Then for all f ∈ BM(X ), the random integral f dξ has finite kth moment satisfying k = f (x1 ) . . . f (xk ) Mk (dx1 × · · · × dxk ). (9.5.8) E f dξ X (k)

Proof. Apply (9.5.2) to the product measure ξ (k) , for which Mk is the expectation measure. This gives h(x1 , . . . , xk ) Mk (dx1 × · · · × dxk ) X (k) (9.5.9) ··· h(x1 , . . . , xk ) ξ(dx1 ) . . . ξ(dxk ) =E X

X

for all k-dimensional bounded Borel measurable functions h(x1 , . . . , xk ): #k X (k) → X . Then (9.5.8) is the special case h(x1 , . . . , xk ) = i=1 f (xi ). We now consider the finite Taylor series expansion of the characteristic functional (see Exercise 9.5.8 for an expansion for the Laplace functional). Proposition 9.5.V. Let Φ be the characteristic functional of the random measure ξ, and suppose that the kth moment measure of ξ exists for some k ≥ 1. Then for each fixed f ∈ BM(X ) and real s → 0, Φ[sf ] = 1 +

k (is)r r=1

r!

X (r)

f (x1 ) . . . f (xr ) Mr (dx1 × · · · × dxr ) + o(|s|k ).

(9.5.10) Furthermore, if the (k + 1)th moment exists, the remainder term o(|s|k ) is bounded by |s|k+1 k+1 (k+1) C Mk+1 (Af ), (9.5.11) (k + 1)! f where Cf is a bound for f , and f vanishes outside the bounded Borel set Af .

9.5.


69

Proof. Because Φ[sf ] = φf (s), where φf is the ordinary characteristic function for the r.v. f dξ, both assertions follow from (9.5.8) and the corresponding Taylor series results for ordinary characteristic functions. The bound (9.5.11), with Cf and Af as defined there, is derived from "k+1 k+1 " ≤E |f | dξ ≤ E [Cf ξ(Af )]k+1 . E " f dξ " The analogy with the finite-dimensional situation may be strengthened by noting that the moment measures can be identified with successive Fréchet derivatives of Φ. Specifically, we can write formally Mk (A) = Φ(k) (IA ). The difficulty with such expressions is that they rarely give much information, either theoretical or computational, concerning the analytic form or other characteristics of the moment measures. The corresponding expression for the logarithm of the characteristic functional leads to a new family of measures associated with ξ, the cumulant measures. The first cumulant measure coincides with the expectation measure, whereas the second is the covariance measure defined by (9.5.12) C2 (A × B) = M2 (A × B) − M (A)M (B) = cov ξ(A), ξ(B) . In Example 9.5(a), the covariance measure is absolutely continuous with respect to two-dimensional Lebesgue measure, the covariance density being given by c2 (x, y) = 2c2 (x − y). The covariance density for this random measure is just the ordinary covariance function of the process forming the density of the random measure. Similar relations between the moment and cumulant densities of the random measure, and the moment and cumulant functions of its density, hold whenever the random measure can be represented as the integral of an underlying process. By contrast, in Example 9.5(c), the covariance measure is singular, consisting entirely of the concentration along the diagonal y = x with linear density λ2 α. The relation can be expressed conveniently using the Dirac delta function as cξ (x, y) = λ2 α δ(x − y). The general relation between the moment measures and the cumulant measures is formally identical to the relation between the factorial moment measures and factorial cumulant measures studied in Chapter 5, inasmuch as both are derived by taking logarithms of an expression of the type (9.5.9). Finally we consider analogues of Proposition 9.5.V for point processes that no longer need be finite as in the discussion in Section 5.4. The advantages of working with factorial moment measures M[k] remain: the same definition

70


at (5.4.2) holds when we require the sets Ai there to be bounded. Then M[k] exists (i.e., is boundedly finite) if and only if Mk exists, and the definitions (5.4.3) and (5.4.4) continue, now with bounded sets Ai . In the proof below N [k] (·) denotes the k-fold factorial product measure as described below (5.4.4) (the notation recalls the factorial power used around Definition 5.2.I). It has expectation measure M[k] (·). For simple point processes, integration with respect to N [k] (dx1 × · · · × dxk ) is the same as integration with respect to N (dx1 ) . . . N (dxk ) if we add the restriction that x1 , . . . , xk must all be distinct. Thus, the integral at (7.1.13) could be written without the coincidence annihilating function I(·) there if instead the product measure is replaced by the factorial product measure. Proposition 9.5.VI. Let G be the p.g.fl. of a point process whose kth order moment measure exists for some positive integer k. Then for 1 − η ∈ V(X ) and 0 < ρ < 1, G[1 − ρη] = 1 +

k (−ρ)j j=1

j!

X (j)

η(x1 ) . . . η(xj ) M[j] (dx1 × · · · × dxj ) + o(ρk ). (9.5.13)

Proof. Some care is needed in evaluating the difference between G[1 − ρη] and the finite sum on the right-hand side of (9.5.13). For fixed η and a given realization {yi } of the point process, consider the expressions Sm (ρ) = 1 +

m (−ρ)j j=1

j!

X (j)

η(x1 ) . . . η(xj ) N [j] (dx1 × · · · × dxj ),

where N [j] is the modified product counting measure formed by taking all possible ordered j-tuples of different points of the realizations of N (with the convention that if {yi } has multiple points these should be treated as different points with the same state space coordinates, that is, as if they represented distinct particles). Each integral then reduces to a sum Qj = η(yi1 ) . . . η(yij ) over all such j-tuples. Effectively, each sum is a.s. finite, because with probability 1 only a finite number of points of the process will fall within the support of 1 − η. Moreover, the sum vanishes whenever j is larger than the number of points in this support. Because N [j] includes all possible orderings of a given j-tuple, each distinct term in the sum occurs j! times, so that we can write j! qj , where the sum on qj extends over all distinct combinations of Qj = j points from {yi } (with the same convention as before regarding multiple points). In this notation we have Sm (ρ) = 1 +

m j=1

(−ρ)j qj ,

(9.5.14)

9.5.


71

and it is not difficult to verify (e.g., by induction) that for all m and η,

S2m+1 (ρ) ≤ [1 − ρη(yi )] ≡ Π(ρ) ≤ S2m (ρ), (9.5.15) i

the product being taken over {yi }; Exercise 9.5.9 interprets (9.5.15) in terms of Bonferroni inequalities. Equation (9.5.14) implies that |Sk − Sk−1 | ≤ ρk qk , which with (9.5.15) and its implication that the sums Sk (ρ) are alternately above and below Π(ρ) implies both |Sk (ρ) − Π(ρ)| ≤ ρk qk

and

|Sk (ρ) − Π(ρ)| ≤ ρk+1 qk+1 .

(9.5.16)

Now suppose that Mk , and hence M[k] , exist. The first inequality at (9.5.16) implies that [Sk (ρ) − Π(ρ)]/ρk is bounded by a random variable with finite expectation 1 η(x1 ) . . . η(xk ) M[k] (dx1 × · · · × dxk ), E(qk ) = k! X (k) because M[k] is just the expectation of N [k] . The second inequality at (9.5.16) implies that [Sk (ρ) − Π(ρ)]/ρk → 0 a.s. as ρ → 0. The limit behaviour of the remainder term in (9.5.13) now follows by dominated convergence. Uniqueness of the expansion (9.5.13) follows from the uniqueness of the coefficients in a power series expansion and the fact that, as symmetric measures, the moment measures are uniquely specified by integrals of the type appearing in the expansion (see Exercise 9.5.7). Taking expectations in (9.5.16) yields the following corollary. Corollary 9.5.VII. When Mk+1 exists, the remainder term in (9.5.13) is bounded by ρk+1 η(x1 ) . . . η(xk+1 ) M[k+1] (dx1 × · · · × dxk+1 ). (k + 1)! X (k+1) On taking logarithms of the expression (9.5.13) and using the expansion k log(1 − y) = − j=1 y j /j + o(|y|j ) (y → 0), equation (9.5.17) below follows. Corollary 9.5.VIII. Under the conditions of Proposition 9.5.VI, the p.g.fl. can be expressed in terms of the factorial cumulant measures C[j] , for ρ → 0, as k (−ρ)j η(x1 ) . . . η(xj ) C[j] (dx1 × · · · × dxj ) + o(ρk ). log G[1 − ρη] = j! (j) X j=1 (9.5.17) Equation (9.5.17) serves to define the cumulant measures, which can be expressed explicitly in terms of the measures M[k] as in Chapter 5. Unfortunately it does not seem possible to provide a simple bound for the remainder term in (9.5.17) analogous to that of Corollary 9.5.VII (but, see Exercise 9.5.10).

72


Example 9.5(d) Moment measures of Poisson and compound and mixed Poisson processes [continued from Examples 9.4(c)–(d)]. Expanding the p.g.fl. at (9.4.17) formally, in terms of η with 1 − η ∈ V(X ), k ∞ 1 η(x) µ(dx) G[1 + η] = 1 + k! X k=1 ∞ 1 =1+ ··· η(x1 ) . . . η(xk ) µ(dx1 ) . . . µ(dxk ). k! X X k=1

Thus, for the Poisson process with parameter measure µ, the kth order factorial moment measure M[k] is the k-fold product measure of µ with itself. The situation with the cumulant measures is even simpler: here, log G[1 + η] = η(x) µ(dx) so that for a Poisson process the second and all higher factorial X cumulant measures vanish. This last result is in marked contrast with the situation for compound Poisson processes for which log G[1 + η] equals ∞ ∞ [η(x)]k [k] [1 + η(x)]n − 1 πn (x) µ(dx) = n πn (x) µ(dx) k! X n k=1 X n=k ∞ = [η(x)]k m[k] (x) µ(dx), k=1

X

where m[k] (x) is the kth factorial moment of the batch-size distribution {πn (x)} at the point x, assuming the moment exists. This representation implies that C[k] is concentrated on the diagonal elements (x, . . . , x) where it reduces to a measure with density m[k] (x) with respect to µ(·). For the mixed Poisson process, suppose that Λ has a finite kth moment. Then in a neighbourhood Re(s) ≥ 0 of s = 0, the Laplace–Stieltjes transform L(s) = 1 +

k (−s)j E(Λj ) j=1

j!

+ o(|s|k ).

Then for η with 1 − η ∈ V(X ), G[1 − ρη] equals 1+

k (−ρ)j E(Λj ) j=1

j!

X

···

X

η(x1 ) . . . η(xj ) µ(dx1 ) . . . µ(dxj ) + o(ρk ),

and thus the factorial moment measures of the process are given by M[j] (dx1 × · · · × dxj ) = E(Λj ) µ(dx1 ) . . . µ(dxj )

(j ≤ k).

(9.5.18)

If in particular, X = Rd and µ(·) is Lebesgue measure on Rd , then M[j] has a density m[j] with respect to such Lebesgue measure given by m[j] (x1 , . . . , xj ) = E(Λj )

(j ≤ k).

9.5.


73

Thus, the factorial moment measures for the mixed Poisson process retain the product form of the Poisson case but are multiplied by the scalar factors E(Λj ). McFadden (1965a) and Davidson (1974c) established the following converse to this result. Let {M[j] (·)} be a sequence of product measures of the form (9.5.18) with {E(Λj )} replaced by a sequence {γj }. Then the M[j] (·) are the factorial moment measures of a point process if and only if {γj } is the moment sequence of some nonnegative r.v. Λ0 . A sufficient condition for −1/j < ∞, in which the resulting process to be uniquely defined is that γj case it is necessarily the mixed Poisson process with parameter measure Λ0 µ, where Λ0 has a uniquely defined distribution with moments {γj }. Proposition 5.4.VII implies a weaker version of this result (see Exercise 9.5.11). For completeness here recall that in Example 6.4(b) we discussed negative binomial processes, meaning, point processes N for which N (A) has a negative binomial distribution [but see also Example 9.1(b)]. In particular, we noted two mechanisms leading to such processes, one starting from compound Poisson processes and the other from mixed Poisson processes. Not infrequently our concern with a point process may be with its structure only on some bounded region A of the state space. Within A (assumed to be Borel), N is a.s. finite-valued by assumption, and its probabilistic structure must be expressible in terms of some family of local probability distributions or Janossy measures as in Definition 5.4.IV and, for the p.g.fl., Example 5.5(b). However, because such point processes are in general a.s. infinite on the whole of X , no such measures exist for the process as a whole. We illustrate in the next example how local characteristics can be described: we take the case of the negative binomial process in the setting of its local Janossy measure. Example 9.5(e) Local properties of the negative binomial process. Recall from Example 5.5(b) that, given the p.g.fl. G[·] of a point process and a bounded Borel set A, the p.g.fl. GA [·] of the local process on A is given by GA [h] = G[1 − IA + h∗ ] ∗

∗

[h ∈ V(A)],

where h (x) = h(x)IA (x) so that h ∈ V(X ). Example 6.4(b)(i) gives the p.g.fl. log [1 − ρh(x)]/(1 − ρ) µ(dx) G[h] = exp log(1 − ρ) X for a negative binomial process coming from a Poisson cluster process with clusters degenerate at a point and size following a negative binomial distribution. Then because the integral over Ac vanishes, we deduce that 1 1 − ρh µ(dx) . log GA [1 − IA + h∗ ] = exp log(1 − ρ) A 1−ρ Thus, the localized process is still a negative binomial process. The local Janossy measures can be found from the expansion ∞ ρn (n) 1 − ρh = − log(1 − ρ) + h , log 1−ρ n n=1

74


from which we deduce that p0 (A) = exp[−µ(A)] and J1 (dx | A) = ρp0 (A) µ(dx), J2 (dx1 × dx2 | A) = ρ2 p0 (A)[µ(dx1 )µ(dx2 ) + δ(x1 , x2 )µ(dx1 )], where the two terms in J2 represent contributions from two single-point clusters at x1 and x2 (x1 = x2 ) and a two-point cluster at x1 = x2 .

Exercises and Complements to Section 9.5 9.5.1 Moment measures of Dirichlet process. Let ξ be a random probability measure on X . Show that for every k, the kth moment measure exists and defines a probability measure on X (k) . Find these measures for the Dirichlet process ζ of Example 9.1(e), showing in particular that Eζ(A) =

α(A) , α(X )

var ζ(A) =

α(A) 1 − α(A)/α(X ) · . α(X ) α(X ) + 1

9.5.2 For the random measure induced by the limit random d.f. of Exercise 9.3.4, show that the first moment measure is Lebesgue measure on [0, 1]. 9.5.3 Let ξ be a random measure on X = Rd , and for g ∈ BM+ (X ) define G(A) = g(x) (dx), where denotes Lebesgue measure on Rd . Define η on BX by A G(A − x) ξ(dx). η(A) = X

(a) Show that η(A) is an a.s. finite-valued r.v. for bounded A ∈ BX , that it is a.s. countably additive on BX , and hence invoke Proposition 9.1.VIII to conclude that η is a well-defined random measure. (b) Show that if ξ has moment measures up to order k, so does η, and find the relation between them. Verify that the kth moment measure of η is absolutely continuous with respect to Lebesgue measure on (Rd )(k) . (c) Denoting the characteristic functionals of ξ and η by Φξ [·] and Φη [·], show that for f ∈ BM+ (X ), h(x) = X

f (y)g(y − x) dy

is also in BM+ (X ), and Φη [f ] = Φξ [h]. 9.5.4 (Continuation). By its very definition, η is a.s. absolutely continuous with respect to Lebesgue measure and its density g(t − x) ξ(dx), Y (t) ≡ X

when ξ is completely random, is called a linear process. Find the characteristic functional of Y when ξ is a stationary gamma random measure. [Remark: Examples of linear processes are provided by equation (9.5.2) in connection with the original Campbell theorem and by the shot-noise process as in Examples 6.1(d) and 6.2(a). See Exercise 10.1.3(b) for the case that ξ is completely random; for other references see, e.g., Westcott (1970).]

9.5.


75

9.5.5 (a) For the random probability distribution on R+ defined as in Exercise 9.1.4 by Fη (x) = 1 − exp(−ξ([0, x])), show that ξ is a.s. nonatomic if and only if the second moment measure M2 of η has M2 (diag(R(2) )) = 0. (b) Prove the more general Proposition 9.5.III. 9.5.6 (a) Use the intensity measure λ∗k of the simple point process Nk∗ in the decomposition at (9.3.9) [see also (9.3.17) and Proposition 9.5.II] to show that M2 (diag A(2) ) = M (A) +

∞

k(k − 1)λ∗k (A) = M (A) + M[2] (diag A(2) ).

k=2

r ∗ Conclude more generally that Mr (diag A(r) ) = ∞ k=1 k λk (A). (b) When Mr (A(r) ) < ∞, deduce that M[r] (diag A(r) ) = 0 if and only if P{N ({x}) ≤ r − 1 for all x ∈ A} = 1.

9.5.7 Let M(k) be a symmetric measure on (X (k) , B(k) ). By starting from indicator functions, show that M(k) is uniquely determined by integrals of the form

η(x1 ) . . . η(xk ) M(k) (dx1 × · · · × dxk )

X (k)

(1 − η ∈ V(X )).

9.5.8 Expand E(e−X−Y ) for nonnegative r.v.s X and Y to deduce that if a random measure ξ has a finite kth order moment measure, then for > 0 and for functions f , g ∈ BM+ (X ), using ξf as at (9.5.1), L[f + g] = L[f ] − E[ξg exp(−ξf )] + · · · + 9.5.9 Let QK =

K i=1 (1

(− )k E[(ξg )k exp(−ξf )] + o( k ). k!

− αi ), where 0 < αi < 1 for i = 1, . . . , K, and write qk =

···

1≤i1 0. Use (5.4.8) to show that local Janossy measures are defined and that they determine a point process whenever {γj } is a moment sequence.

CHAPTER 10

Special Classes of Processes

10.1 10.2 10.3 10.4

Completely Random Measures Infinitely Divisible Point Processes Point Processes Defined by Markov Chains Markov Point Processes

77 87 95 118

We have already discussed in Volume I a variety of particular models for point processes and random measures, and described many of their properties. With the added benefit of the basic theory in Chapter 9, we return here to the study of four important classes of models: completely random measures; infinitely divisible point processes; point processes generated by Markov chains; and Markov point processes in space. Each class has interest in its own right, and contains models which are widely used in applications. Although it is the intrinsic interest of the models that motivates the discourse, our immediate aims are to use the theory of the last chapter to establish structure theorems for these classes, to show that they are well-defined mathematical objects, and to establish some of their general properties. A key feature of the first two classes is their close link to the Poisson process. Indeed, they form natural extensions of the compound Poisson processes discussed in Chapters 2 and 9. Because of this feature, many of their properties can be handled compactly by p.g.fl. techniques, and we make extensive use of this approach. It should be borne in mind, however, that the main advantage of this approach lies precisely in its compactness: it quickly summarizes information that can still be derived quite readily without it and that in less tractable examples may not be so easily expressible in p.g.fl. form. The other two sections illustrate both the power and limitations of the ideas of Markov chains that so pervade applied probability. When the process has a temporal ingredient, it is natural to include this in any probabilistic description so that it is the ‘future’ that is predicted (stochastically) on the basis of sufficient conditions described by the ‘present’ so that any other knowledge from the past is superfluous in terms of making any prediction better informed. In reality, this amounts to a factorization of the stochastic structure 76

10.1.

Completely Random Measures

77

of the evolution of the process as a product of probabilistic terms (densities) ‘chained’ through adjacent epochs in time. It is exactly a product probabilistic structure that underlies the class of so-called Markov point processes in space. The class of models that can be described in this way leads to representations of the type now known under the umbrella of Hammersley–Clifford theorems. This Section 10.4 serves as a taste of further results in Chapter 15.

10.1. Completely Random Measures This section represents both an illustration of the ideas of the sample-path properties expounded in Section 9.3 and an extension of the discussion of Section 2.4 on the general Poisson process. The principal result is the Representation Theorem 10.1.III, which comes from Kingman (1967); it is based on a careful study of sample-path structures to which we proceed immediately. Although completely random measures have been referred to at several points in the text already, we state the following for the record. Definition 10.1.I. A random measure ξ on the c.s.m.s. X is completely random if for all finite families of disjoint, bounded Borel sets {A1 , . . . , Ak }, the random variables {ξ(A1 ), . . . , ξ(Ak )} are mutually independent. Of course, the Poisson process discussed extensively in Chapter 2 and elsewhere is the prime example. A compound Poisson process with marks in the c.s.m.s. K is a completely random process on X × K with the additional requirement that the ground process Ng on X be well defined (and then a Poisson process in its own right). In one dimension, a random measure ξ is completely random if and only if the corresponding cumulative process t η(t) = 0 ξ(dx) has independent increments. Recall from Corollary 9.3.VI that a completely random measure ξ on X is nonatomic if and only if for some dissecting system {Tn } and every > 0, kn i=1 P{ξ(Ani ) ≥ } → 0 (n → ∞). A substantial step in the proof of the main representation result, equation (10.1.4) in Theorem 10.1.III, is the following result, which is of interest in its own right. Proposition 10.1.II. If the completely random measure ξ is a.s. nonatomic, then there is a fixed nonatomic measure ν such that ξ(·) = ν(·)

a.s.

(10.1.1)

Proof. Let T = {Tn } = {Ani : i = 1, . . . , kn } be a dissecting system for any given bounded Borel set A; define the transforms ψni (s) = E exp[−sξ(Ani )]

(Re(s) ≥ 0).

78

10. Special Classes of Processes

Because ξ is completely random and ξ(A) =

kn i=1

ξ(Ani ), we have

kn ψA (s) ≡ E exp[−sξ(A)] = ψni (s)

(n = 1, 2, . . .),

i=1

and 1 − ψni (s) = E 1 − exp[−sξ(Ani )] =

∞

s e−sy P{ξ(Ani ) > y} dy.

0

Appealing to Corollary 9.3.VI and the dominated convergence theorem, ξ being nonatomic implies that max [1 − ψni (s)] → 0

(n → ∞)

1≤i≤kn

for every fixed real s ≥ 0. Using this result in an expansion of the logarithmic term below, it now follows that − log ψA (s) = −

kn

log ψni (s) = − lim

n→∞

i=1 kn

= lim

n→∞

log ψni (s)

i=1

[1 − ψni (s)]

i=1 ∞

= lim

n→∞

kn

(10.1.2)

(1 − e−sy ) Gn (dy),

0

where Gn (·) is the sum of the kn individual probability measures of ξ(Ani ). Again from (9.3.7), ∞ Gn (dy) → 0 (n → ∞) (10.1.3)

for every fixed > 0, and from the limit relation (10.1.2) it follows that y Gn (dy) remains bounded as n → ∞. Thus, the sequence of measures 0 Hn (dy) ≡ min(1, y) Gn (dy) is not merely bounded in total mass but, from (10.1.3), is uniformly tight. Again, using (10.1.3), it follows that the only possible limit for {Hn } is a degenerate measure with its mass concentrated at the origin. Uniform tightness implies the existence of a convergent subsequence, thus there must exist a constant ν ≡ ν(A) and a sequence {nk } for which Hnk → ν(A)δ0 weakly, and therefore ∞

− log ψA (s) = lim

nk →∞

0

= lim

nk →∞

0

∞

(1 − e−sy ) Gnk (dy)

1 − e−sy Hnk (dy) = sν(A). min(y, 1)

10.1.


79

This result is equivalent to ξ(A) = ν(A) a.s. for the given bounded Borel set A. Because such a relation holds for any bounded Borel set A, the family ν(·) must be a finitely bounded Borel measure and ξ = ν must hold for almost all realizations. Finally, ξ being a.s. free of atoms, the same must be true of ν also. The major goal now is the following representation theorem. Theorem 10.1.III (Kingman, 1967). Any completely random measure ξ on the c.s.m.s. X can be uniquely represented in the form ∞ ∞ Uk δxk (A) + ν(A) + y N (A × dy), (10.1.4) ξ(A) = 0

k=1

where the sequence {xk } enumerates a countable set of fixed atoms of ξ, {Uk } is a sequence of mutually independent nonnegative random variables determining (when positive) the masses at these atoms, ν(·) is a fixed nonatomic boundedly finite measure on X , and N (·) is a Poisson process on X × (0, ∞), independent of {Uk }, the parameter measure µ of which may be unbounded on sets of the form A × (0, ) but satisfies µ(A × dy) < ∞, (10.1.5a) y> y µ(A × dy) < ∞ (10.1.5b) 0 > 0, the finiteness of (10.1.5a) is assured. For (10.1.5b), we observe moreover that for every > 0, (A × dy) ≤ ξ(A) < ∞ a.s., yN (10.1.10) 0

because the integral is simply a sum (possibly an infinite series) of the contribution of the random atoms of ξ to its total mass ξ(A) on A. To see that the convergence of the integral at (10.1.10) implies the convergence of its expectation as at (10.1.5b), partition the interval (0, ) into the sequence of subintervals {[ /(r + 1), /r) : r = 1, 2, . . .}. Then ∞

y µ(A × dy) ≤ yr µ(Ar ), where yr = and Ar = A × yr+1 , yr . r 0 r=1 Let {ζr } be a sequence of independent Poisson r.v.s with parameters µ(Ar ). Then for the Laplace transform of yr ζr we have ∞ ∞

− log E exp − s = − log yr ζr exp − µ(Ar )[1 − exp(−syr )] r=1

r=1

=

∞

µ(Ar )[1 − exp(−syr )] ≥

1 2s

r=1

∞

yr µ(Ar )

r=1

for 0 < s < −1 , because then0 < syr ≤ 1. ∞ Thus, the convergence of r=1 yr µ(Ar ) is implied by the a.s. convergence ∞ of r=1 yr ζr , concerning which we have ∞ ∞ ∞ ∞

(A × dy) ≥ ζr ≥ ζr ≥ 12 yr ζr = yN yr ζr . r r+1 0 r=1 r=1 r=1 r=1 The asserted finiteness at (10.1.5b) is now established. Finally observe that the measure ∞ (A × dy) ν˜(A) ≡ ξ(A) − Uk δxk (A) − yN 0

is a.s. nonatomic (by construction) and completely random inasmuch as ξ is by assumption and the other two terms have been demonstrated to have the property. Then by Proposition 10.1.I, ν˜ = ν a.s. for some fixed nonatomic measure ν. The theorem is proved on noting that of the three terms in (10.1.4), the first consists of fixed atoms, the second is a constant measure, and the third is purely atomic, so uniqueness is assured. As a simple special case we obtain the Lévy-type representation for a process with nonnegative independent increments.

82


Example 10.1(a) Nonnegative Lévy processes. Here we use ‘Lévy process’ to mean a process X(t) on the real line with independent increments; nonnegativity ensures that the corresponding set process is a measure and not a signed measure. Thus, this example excludes the Brownian motion process and its fractional derivatives. The form of the representation (10.1.4) is unchanged; all that is required is the identification of X with R. In applications, it is common to require the process to have increments that are both stationary and independent. Stationarity then rules out the existence of fixed atoms, the fixed measure ν reduces to a multiple of Lebesgue measure, and the compound Poisson process in the representation (10.1.4) inherits the stationarity property from X(t). Thus the representation takes the simpler form, for any finite interval (a, b], X(b) − X(a) = ν(b − a) +

∞

y N (a, b] × dy ,

0

where ν is a nonnegative real constant, called the drift coefficient in, for example, Bertoin (1996, p. 16), and N is an extended stationary compound Poisson process, meaning that the intensity measure of the corresponding Poisson process on R × R+ has the form µ = × Ψ where Ψ, although not necessarily totally finite, does have finite total mass beyond any > 0 [corresponding to (10.1.5a)], and integrates y at the origin [corresponding to (10.1.5b)]. It is often convenient to describe the representation of Theorem 10.1.III in terms of Laplace functionals, as in the proposition below [cf. Kingman (1967)]. Exercise 10.1.2 summarizes the corresponding representations of the Laplace– Stieltjes transforms for the process increments in the real line case; these are standard representations for the transforms of nonnegative infinitely divisible distributions. Proposition 10.1.IV. In order that the family {ψA (·), A ∈ BX } denote the Laplace transforms of the one-dimensional distributions of a completely random measure on X , it is necessary and sufficient that ψA (·) have a representation of the form, for Re(s) ≥ 0, log ψA (s) = −

∞ k=1

θk (s)δxk (A)−

∞

(1−e−sy ) µ(A×dy)−sν(A),

(10.1.11)

0

where {xk } is a fixed sequence of points, each θk (·) is the logarithm of the Laplace transform of a positive random variable, and the measures ν, µ have the same properties as in Theorem 10.1.III. Conversely, given any such family {xk , θk (·), ν, µ}, there exists a completely random measure with one-dimensional Laplace transforms given by (10.1.11). Proof. The representation (10.1.11) follows immediately on substituting for ξ(A) from (10.1.4) in the expectation ψA (s) = E(e−sξ(A) ).

10.1.


83

To prove the converse it is sufficient to show that the form (10.1.11), together with the definition of joint distributions through the completely random property, yields a consistent family of finite-dimensional distributions. The details of the verification are left as Exercise 10.1.1. Example 10.1(b) Stable random measures; nonnegative stable processes. A special case of interest is the class of stable random measures for which the measure µ of Theorem 10.1.III takes the form µ(dx × dy) = κ(dx) y −(1+1/α) dy for 1 < α < ∞ and some boundedly finite measure κ(·) on BX . For such random measures the Laplace–Stieltjes transform of the one-dimensional distributions take the form ∞

1 − e−sy −ξ(A)s ] = exp −κ(A) dy = exp −Cα κ(A)s1/α , ψA (s) = E[e 1+1/α y 0 (10.1.12) where Cα = αΓ([α − 1]/α) [see, e.g., Bertoin (1996, p. 73)]. An alternative representation for the Laplace–Stieltjes transforms of nonnegative stable processes, due to Kendall (1963), is given in Exercises 10.1.2–3. When X = Rd and κ reduces to Lebesgue measure, the process is both selfsimilar and stationary, with index of similarity α (we also discuss self-similar random measures in Section 12.8). For more detail concerning stable random measures and related processes, see Samorodnitsky and Taqqu (1994). The case when the fidi distributions are gamma distributions has already been discussed in Example 9.1(d). A related but more extended family of completely random measures is described below, following Brix (1999) whose exposition covers earlier material. Example 10.1(c) G-random measures. Here the one-dimensional distributions have Laplace transforms (10.1.11) of the form

ψA (s) = exp − κ(A)[(θ + s)ρ − θρ ]/ρ , where ρ and θ are parameters satisfying either ρ ≤ 0 and θ > 0, or 0 < ρ ≤ 1 and θ ≥ 0. The case ρ = 0 can be obtained as a limit for ρ ↓ 0, and gives back a gamma distribution for ψA (s). When ρ < 0, the underlying measure µ of (10.1.11) is a product of Lebesgue measure and a gamma distribution (i.e., in this case the jumps have gamma distributed heights). In the case 0 < ρ < 1 the corresponding ∞ density is improper (its integral diverges) but still satisfies the condition 0 yf (y) dy < ∞, implying that conditions (10.1.5) hold. The Lévy representation has density [cf. (10.1.11)] κ(A) y −ρ−1 e−θy dy. µ(A × dy) = Γ(1 − ρ) Lee and Whitmore (1993) describe the Lévy processes corresponding to the one-dimensional versions of these processes as Hougarde processes.

84


Brix (1999) also describes the use of these G-measures, or smoothed versions thereof, as directing measures for a Cox process. In one dimension, that is, when X = R, the smoothed version can then be made to correspond to a type of shot-noise process; Exercise 10.1.7 gives some details. No essentially new ideas arise in extending the complete randomness property to marked point processes. We say that an MPP on X with marks in K is completely random when the associated point process on X × K is completely random. It is somewhat surprising that when the MPP has a simple ground process Ng , this condition is equivalent to the apparently weaker condition that the random variables N (Ai × Ki ) should be mutually independent whenever the sets Ai are disjoint, irrespective of whether the corresponding sets Ki are disjoint. Because the construction of Exercise 9.1.6 indicates that, by adjusting the mark space if necessary, we can always find an equivalent description of an MPP as an MPP with simple ground process, the lemma below is rather generally applicable. Lemma 10.1.V. An MPP with simple ground process Ng is completely random if and only if for every finite n, bounded Ai ∈ BX and Ki ∈ BK (i = 1, . . . , n), the random variables N (Ai × Ki ) are mutually independent whenever the Ai are mutually disjoint. Proof. It is obvious that if the complete randomness property holds, the asserted independence property holds because sets in a product space with disjoint marginals are disjoint. For the converse, suppose given two product sets in the product space, Aj × Kj say, for j = 1, 2. Consider first the case A1 = A2 = A say but K1 ∩ K2 = ∅. We want to show that under the condition of Ng being simple, the N (A × Kj ) are independent. Let T be a dissecting system for A, and consider for elements Ani of a partition Xni,j = min(1, N (Ani × Kj )). Simplicity of Ng implies that N (A×Kj ) = limn→∞ i Xni,j . We can now imitate that part of the proof of Theorem 10.1.III around (10.1.6–9) to conclude that the N (A × Kj ) are independent. In the general case, with both A12 = A1 ∩ A2 and A1 ∪ A2 nonempty, where Aj = Aj \ A12 for j = 1, 2, the product sets Aj × K are disjoint in their X -components and therefore independent, and N (A12 × Kj ) are independent for disjoint Kj by the case already considered. Independence of N (Vj ) (j = 1, 2) for arbitrary bounded Borel sets Vj in the product space follows from their independence when the Vj are product sets by standard extension arguments. The argument extends to any finite number of sets by induction. We proceed to examine the structure of a completely random MPP. We know from Chapter 2 (or as a special case of Theorem 10.1.III) that a simple

10.1.


85

completely random point process reduces to a Poisson process; in the present case the parameter measure µg of the ground process must satisfy µg (A) = E[N (A × K)] < ∞

(10.1.13)

for all bounded A ∈ BX , inasmuch as Ng is boundedly finite by assumption. Introduce a family of probability measures P (K | x) on the mark space (K, B(K)) by means of the Radon–Nikodym derivatives P (K | x) µg (dx).

E[N (A × K)] ≡ µ(A × K) =

(10.1.14)

A

Then the absolute continuity condition µ(· × K) µg is satisfied, and the property P (K | x) = 1 a.s. follows from the definition of µg . As in the discussion of regular conditional probability (Proposition A1.5.III), we can and do assume that the family {P (B | x): B ∈ B(K), x ∈ X } is so chosen that P (· | x) is a probability measure on B(K) for all x ∈ X . With this understanding we arrive at the next proposition which effectively implies that completely independent MPPs reduce to some general type of compound Poisson process. Proposition 10.1.VI. A completely random MPP with simple ground process is fully specified by the two components: (i) a Poisson process of locations with parameter measure µg ; and (ii) a family of probability distributions P (· | x) giving the distribution of the mark in K with the property that P (B | x) is measurable in x for each fixed B ∈ B(K). Conversely, given such µg and P (· | ·), there exists a completely random MPP having these as components. Proof. For the converse, it suffices to construct a Poisson process on X × K with parameter measure (10.1.14); we leave it to Exercise 10.1.6 to verify that the resultant process is an MPP with the complete randomness property.

Exercises and Complements to Section 10.1 10.1.1 Imitate the discussion of Example 9.2(a) to verify that the fidi distributions of completely random measure, constructed from the one-dimensional distributions as at (10.1.12), satisfy Conditions 9.2.V and 9.2.VI. 10.1.2 Let ξ be a stationary, completely random measure on R, and let ψt (s) denote the Laplace–Stieltjes transform of ξ(0, t]. (a) Deduce from Proposition 10.1.IV that ψt (s) = exp

− stν − t

(0,∞)

[1 − e−sy ] Ψ(dy) ,

(10.1.15)

where ν is a positive constant, and the σ-finite ∞ measure Ψ on (0, ∞) satisfies, for some > 0, 0 y Ψ(dy) < ∞ and Ψ(dy) < ∞.

86

10. Special Classes of Processes (b) Establish also the equivalent representation ψt (s) = exp

− stν − t

(0,∞]

1 − e−sy G(dy) 1 − e−y

(10.1.16)

for some totally finite measure G and some nonnegative finite constant ν [ν here equals ν(0, 1] in the notation of (10.1.11)]. [Remark: This form parallels that given in Kendall (1963); the measure µ of (10.1.5) and (10.1.11) satisfies (0,∞) (1 − e−y ) µ(A × dy) < ∞.] 10.1.3 (Continuation). Using the representation above and the same notation, show that P{ξ(0, t] = 0} > 0 if and only if both (0,1) y −1 G(dy) < ∞ and ν = 0. [Remark: The condition ν = 0 precludes any positive linear trend; the other condition precludes the possibility of an everywhere dense set of atoms.] 10.1.4 For a given random measure ξ let Y denote the family of measures η satisfying P{ξ(A) ≥ η(A) (all A ∈ BX )} = 1. (a) Define νd (A) = supη∈Y η(A); check that it is a measure, and confirm that ξ − νd is a random measure. (b) Extract from ξ − νd the random measure ζa consisting of all the fixed atoms of ξ − νd , leaving ξr = ξ − νd − ζa . (c) The result of (a) and (b) is to effect a decomposition ξ = νd + ζa + ξr of ξ into a deterministic component νd , a component of fixed atoms ζa , and a random component ξr . Give an example showing that there may still be bounded A ∈ BX for which P{ξr (A) ≥ } = 1 for some > 0. [Hint: Let ξr give mass 1 to either U or U + 1, where U is uniformly distributed on (0, 1).] 10.1.5 Proposition 10.1.IV coupled with the independence property for disjoint sets A implies that the Laplace functional (9.4.10) of a completely random measure is expressible for f ∈ BM+ (X ) as

− log E exp

−

f (x) ξ(dx)

X

=

f (x) α(dx) + X

X ×R+

(1 − e−yf (x) ) µ(dx × dy),

# −y ) γ(B × dy) < ∞ where α ∈ M# X and γ ∈ M (X × R+ ) satisfies R+ (1 − e for all bounded B ∈ BX . [Hint: Kallenberg (1983a, Chapter 7) gives an alternative proof. Compare also with Proposition 10.2.IX.]

10.1.6 Verify the assertion that a completely independent MPP has a simple ground process Ng if and only if (10.1.12) holds for every dissecting system T for bounded A ∈ BX . Without complete independence (10.1.12) need not hold [see (9.3.18) and Proposition 9.3.XII].

10.2.

Infinitely Divisible Point Processes

87

10.1.7 Cox processes directed by stationary G-processes. Let ξ be a stationary Grandom measure on Rd as in Example 10.1(c) so that in the notation of the example, κ(A) = κ(A) for some finite positive constant κ. (a) Show that if ξ itself is used as the directing measure of a Cox process, then the realizations are stationary but a.s. not simple. (b) In the case d = 1, suppose the directing process is not ξ but the smoothed y version X(y) = −∞ φ(y − x) ξ(dx) for some continuous nonnegative integrable kernel function φ(·); X(·) has a density and can be regarded as a type of general shot-noise process [cf. Examples 6.1(d) and 6.2(a)]. Show that the Cox process is well-defined, stationary, a.s. simple, with finite mean rate m = θα−1 κ R φ(u) du and reduced factorial covariance + density c[2] (u) = κ θα−2 (1 − α)

φ(x) φ(x + u) dx. R+

10.2. Infinitely Divisible Point Processes Our aim in this section is to characterize the class of infinitely divisible point processes and random measures. In the point process case, this question is intimately bound up with characterizations of Poisson cluster processes. The role of infinitely divisible point processes and random measures in limit theorems for superpositions is taken up in the next chapter (Section 11.2). Definition 10.2.I. A point process or random measure is infinitely divisible if, for every k, it can be represented as the superposition of k independent, identically distributed, point process (or random measure) components. In symbols, a point process N is infinitely divisible if, for every k, we can write (k) (k) (10.2.1) N = N1 + · · · + Nk , (k)

where the Ni (i = 1, . . . , k) are i.i.d. components. Using p.g.fl.s, the condition takes the form (in an obvious notation) G[h] = (G1/k [h])k

h ∈ V(X ) .

(10.2.2)

The p.g.fl. is nonnegative for such h, so we can restate (10.2.2) as follows. A point process is infinitely divisible if and only if, for every k, the uniquely defined nonnegative kth root of its p.g.fl. is again a p.g.fl. Similarly for random measures the defining property can be restated as follows. A random measure is infinitely divisible if and only if, for every integer k > 0, the uniquely defined kth root of its Laplace functional is again a Laplace functional. From these remarks we may immediately verify that, for example, a Poisson process is infinitely divisible (replace the original parameter measure µ by the measure µ/k for each component), as, more generally, are the Poisson cluster

88


processes studied in Section 6.3 (replace the parameter measure µc for the cluster centre process by (µc )/k and leave the cluster structure unaltered). In the point process case, any fidi distribution has a joint p.g.f. expressible as G[hA ], where hA (·) is of the form hA (x) = 1 −

n

(1 − zi )IAi (x)

(10.2.3)

i=1

for appropriate subsets Ai of the set A. Then from (10.2.2) it follows that the fidi distributions of an infinitely divisible point process are themselves infinitely divisible. Conversely, when a point process has its fidi distributions infinitely divisible, (10.2.2) holds for functions h of the form (10.2.3). Because such functions are dense in V(X ), it follows by continuity as in Theorem 9.4.V that p.g.f.s like G[hA ] and G1/k [hA ] define p.g.fl.s. Similar arguments apply to the case of a random measure, thereby proving the following lemma. Lemma 10.2.II. A point process or random measure is infinitely divisible if and only if its fidi distributions are infinitely divisible. We now embark on a systematic exploitation of this remark and the results set out in earlier sections concerning the representation of infinitely divisible discrete distributions (see, in particular, Exercises 2.2.2–3). We first consider the case of a finite point process. Proposition 10.2.III. Suppose that the point process N with p.g.fl. G[·] is a.s. finite and infinitely divisible. Then there exists a uniquely defined, a.s. , such that Pr{N = ∅} = 0, and a finite positive number finite point process N α such that − 1) G[h] = exp α(G[h] h ∈ V(X ) , (10.2.4) is the p.g.fl. of N and ∅ denotes the null measure. where G Conversely, any functional of the form (10.2.4) represents the p.g.fl. of an a.s. finite point process that is infinitely divisible. Proof. It is clear that any functional of the form (10.2.4) is a p.g.fl. and that the point process to which it corresponds is infinitely divisible (replace is a.s. finite, because α by α/k and take kth powers). It is also a.s. finite if N if G[ρIX ] → 1 as ρ increases to 1, then also G[ρIX ] → 1, implying N is a.s. finite (see Exercise 9.4.5). Suppose conversely that N is infinitely divisible and a.s. finite, and consider n its p.g.fl. When h has the special form i=1 zi IAi (·), where A1 , . . . , An is a measurable partition of X , we know from Exercise 2.2.3 that G[h], which then reduces to the multivariate p.g.f. P (z1 , . . . , zn ) of the random variables N (A1 ), . . . , N (An ), can be represented in the form P (z1 , . . . , zn ) = exp α[Q(z1 , . . . , zn ) − 1] ,

10.2.


89

where Q is itself a p.g.f. with Q(0, . . . , 0) = 0 and α is positive, independent of the choice of the partition and equal to − log(Pr{N (X ) = 0}). Now consider the function = 1 + α−1 log G[h]. G[h] reduces to the multivariate p.g.f. Q. When h has the above special form, G inherits continuity from G. Hence, it is a p.g.fl. by Theorem 9.4.V. To Also, G show that the resulting process is a.s. finite consider the behaviour of G[ρIX ] as ρ increases to 1. Because N itself is a.s. finite, G[ρIX ] → 1 by Exercise X ] → 1, showing that G is the 9.4.5. But then log G[ρIX ] → 0 and so G[ρI p.g.fl. of a point process N that is a.s. finite. The representation (10.2.4) has a dual interpretation. It shows that any a.s. finite and infinitely divisible process N can be regarded as the ‘Poisson randomization’ [borrowing a phrase from Milne (1971)] of a certain other . In this interpretation, the process N is constructed by point process N first choosing a random integer K according to the Poisson distribution with probabilities pn = e−α αn /n! , and then, given K, taking the superposition of K i.i.d. components each . having the same distribution as N On the other hand, the process N can also be related to the cluster pro in terms of the cesses of Section 6.3. To see this, first represent the p.g.fl. G Janossy measures for N , so that (10.2.4) becomes log G(h) = α

∞ k=1

1 k!

···

X (k)

h(x1 ) · · · h(xk ) Jk (dx1 × · · · × dxk ) − 1 .

This infinite sum can be rewritten as log G(h) =

∞ k=1

···

X (k)

[h(x1 ) · · · h(xk ) − 1] Qk (dx1 × · · · × dxk ),

(10.2.5)

where Qk (·) = (α/k!)Jk (·). Observe finally that this last form is the log p.g.fl. of a Poisson cluster process, as in Proposition 6.3.V. Both interpretations above coexist for an a.s. finite process: they represent alternative constructions for the same process. To investigate the behaviour when the a.s. finite condition is relaxed, we first observe that any infinitely divisible process remains infinitely divisible but becomes a.s. finite when we consider its restriction to any bounded Borel set. Its local representation therefore continues to have the form (10.2.4). Rewrite (10.2.4) for the special case that the bounded set is a (large) sphere, n say, of the process N Sn say, and introduce explicitly the distribution, P

90


restricted to B(Sn ). Writing Gn [ · ] for the corresponding p.g.fl. of N , we have from (10.2.4) that Gn [h] = exp αn

∈N # (Sn ) N

exp

(dx) − 1 P n (dN ) , log h(x) N Sn

(10.2.6) where we recall the convention that the inner exponential term is to be counted has no points in the region where h differs from unity, and as as unity if N zero if N has any points in the region where h vanishes. Bearing this in mind, we have in particular, from (10.2.6), : N (Sn ) > 0} ≡ αn P : N (Sn ) > 0} = − log P{N (Sn ) = 0}, n {N n {N Q (10.2.7) n and we continue the convention that n is an abbreviation for αn P where Q (Sn ) = 0} = 0, so that e−αn is just the probability that the original n {N P process has no points in Sn . ∗ say, on the n may be used to induce a similar measure, Q Each measure Q n # class of cylinder sets in the full space NX determined by conditions on the behaviour of the counting process on Sn . Specifically, for C a set in NS#n of the form (Ai ) = ri , Ai ⊆ Sn , i = 1, . . . , k}, ∈ N# : N C = {N Sn where the ri are nonnegative integers not all zero, we associate the set C ∗ in ∈ N #: N (Ai ) = ri , Ai ⊆ Sn , i = 2, . . . , k}, and put NX# given by C ∗ = {N X n (C). ∗n (C ∗ ) = Q Q (Sn ) = 0: for this reason This construction fails for the set in NX# for which N ∗n not on the full sub-σ-algebra of cylinder sets with base we have to define Q determined by conditions in Sn , but on the sub-σ-algebra generated by those (Sn ) > 0. Let us denote this subcylinder sets incorporating the condition N σ-algebra by Bn . Then it is clear that the Bn are monotonic increasing and that ∞ Bn = B N0# (X ) , σ n=1

where N0# (X ) denotes the space NX# with the null measure ∅(·) omitted. On ∞ ∗ , the projective the union n=1 Bn , we can consistently define a set function Q ∗ limit of {Qn }, by setting ∗ (A) = Q ∗ (A) Q n ∗ reduces to Q∗ whenever whenever A ∈ Bn . This is possible because Q m n m > n and we restrict attention to sets in Bn . The set function Q∗ is countably additive on each of the Bn but not obviously so on their union. The situation, however, is similar to that of the Kolmogorov extension theorem

10.2.


91

for stochastic processes, or to the extension theorem considered in Section 9.2, where countable additivity is ultimately a consequence of the metric assumptions imposed on the space X . The same argument applies here also; we ∗ has a unique extension leave the details to Exercise 10.2.1. It implies that Q # to a measure Q on the σ-algebra B(N0 (X )). In addition to the fact that it is defined on the sets of N0# (X ) rather than # NX , Q enjoys one further special property. Equation (10.2.7) implies that for any bounded set A, : N (A) > 0} < ∞. Q{N (10.2.8) defined on the Borel Definition 10.2.IV. A boundedly finite measure Q sets of N0# (X ) = NX# \ {N (X ) = 0}, and satisfying the additional property (10.2.8), is called a KLM measure. The measure is so denoted for basic contributions to the present theory in Kerstan and Matthes (1964) and Lee (1964, 1967). Theorem 10.2.V. A point process on the c.s.m.s. X is infinitely divisible if and only if its p.g.fl. can be represented in the form G[h] = exp

N0# (X )

exp

X

log h(x) N (dx) − 1 Q(dN )

(10.2.9)

When such a representation exists, it is unique. for some KLM measure Q. Proof. Suppose that the point process N is infinitely divisible, and let be the KLM measure constructed as above. When it is equal to unity Q outside the sphere Sn , the representation (10.2.9) reduces to (10.2.6) from and so the functional G in (10.2.9) must coincide with the construction of Q, the p.g.fl. of the original process. is given, and consider (10.2.9). If Conversely, suppose a KLM measure Q we set N : N (Sn ) > 0} αn = Q{ (finite by assumption in Definition 10.2.IV), (10.2.9) can again be reduced to the form (10.2.6) for functions h that equal unity outside Sn , and therefore, by Proposition 10.2.III, it is the p.g.fl. of a local process defined on Sn . In particular, therefore, (10.2.9) reduces to a joint p.g.f. when h has the form k 1 − i=1 (1 − zi )IAi . The continuity condition follows from the remark already made that (10.2.9) defines a local p.g.fl. when we restrict attention to the behaviour of the process on Sn . Thus, (10.2.9) is itself a p.g.fl. Infinite divisibility follows from Lemma 10.2.II and the remarks already made concerning the local behaviour of G[h]. Finally, uniqueness follows from the construction and the uniqueness part of Proposition 10.2.III. Example 10.2(a) Poisson process [see Example 9.4(c)]. If (10.2.9) is to reduce must be simple and have a to the p.g.fl. (9.4.17) of a Poisson process, each N

92


N (X ) = 1} = 0, because single point as its support; that is, we must have Q{ must for given N the integrand for the integral at (10.2.9) with respect to Q reduce to h(x) − 1, where {x} is the singleton support of N . In fact, the KLM must be related to the parameter measure µ by measure Q N (A) = 1} = Q{ N : N (X ) = 1 = N (A)} = µ(A) Q{

(bounded A ∈ BX ).

Further insight into the structure of such processes can be obtained from a classification of the properties of their KLM measures. In particular, we make the following definitions. Definition 10.2.VI. An infinitely divisible point process is regular if its KLM measure is carried by the set : N (X ) < ∞} Vr ≡ {N

(10.2.10a)

and singular if its KLM measure is carried by the complementary set : N (X ) = ∞}. Vs ≡ {N

(10.2.10b)

We now have the following decomposition result. Proposition 10.2.VII. Every infinitely divisible point process can be represented as the superposition of a regular infinitely divisible process and a singular infinitely divisible process, the two components being independent. Proof. This follows from the representation (10.2.9) on writing =Q r + Q s , Q where for each A ∈ B(N0# (X )), r (A) = Q(A ∩ Vr ), Q

s (A) = Q(A ∩ Vs ). Q

r (·) and Q s (·) is again a KLM measure, and because the original Each of Q p.g.fl. appears as the product of the p.g.fl.s of the two components, the corresponding components themselves must be independent and their superposition must give back the original process. Further characterizations of various classes of infinitely divisible point processes can be given in terms of their KLM measures, as set out below. Some refinements for the stationary case are given in Section 12.4. Proposition 10.2.VIII. (i) An infinitely divisible point process is a.s. finite if and only if it is regular and its KLM measure is totally finite.

10.2.


93

(ii) An infinitely divisible point process can be represented as a Poisson cluster process, with a.s. finite clusters, if and only if it is regular. (iii) An infinitely divisible point process can be represented as a Poisson randomization if and only if its KLM measure is totally finite. Proof. Part (i) is a restatement of Proposition 10.2.III, regularity coming from the assertion that the process N is a.s. finite, and the total boundedness of the KLM measure from the fact that it can be represented in the form αP, is a probability measure. where 0 < α < ∞ and P Part (ii) follows from the representation of Poisson cluster processes in Proposition 6.3.V, taking X = Y there, so that the measures Qk (·) in that proposition can be combined to give a measure on the space of all finite counting measures, as in Proposition 5.3.II. That this is a KLM measure follows from the absence of any Q0 term, and the condition (6.3.33), which in the terminology of the present section can be rewritten [see (6.3.35)] as N : N (A) > 0} < ∞ µ(A) = Q{


Conversely, we can split the KLM measure of any regular infinitely divisible : N (X ) = k}, in each of which process into its components on the sets Vk = {N it induces a measure Qk with the properties described in Proposition 6.3.V. Finally, part (iii) follows from the observation that here also the KLM = αP, where α is the parameter of measure can be written in the form Q the Poisson randomizing distribution and P is the distribution of the point process being randomized, which we assume adjusted if necessary so that N : N (X ) = 0}) = 0. P({ Extensions to infinitely divisible multivariate and marked point processes are considered in Exercises 10.2.2 and 10.2.5. An analogous representation holds also for infinitely divisible random measures, and is set out below. Proposition 10.2.IX. A random measure on the c.s.m.s. X is infinitely divisible if and only if its Laplace functional can be represented in the form for all f ∈ BM+ (X ) − log L[f ] = 1 − exp − f (x) α(dx) + f (x) η(dx) Λ(dη), M# 0 (X )

X

X

(10.2.11) where α ∈ = − {∅}, and Λ is a σ-finite measure on M# (X ) satisfying, for every bounded Borel set B ∈ BX and distribution 0 F1 (B; x) ≡ Λ{η: η(B) ≤ x}, (1 − e−x ) F1 (B; dx) < ∞. (10.2.12) M# X,

M# 0 (X )

M# X

R+

A proof involving the inductive limit of the corresponding representations for the Laplace transforms of the fidi distributions can be given along the same

94


lines as that of Theorem 10.2.V, with (10.2.11) reducing to (10.2.9) when ξ is a point process. See Exercise 10.2.5. Results about the convergence of infinitely divisible distributions and their role in limit theorems are reviewed in Chapter 11, notably Section 11.2. Results for the stationary case are outlined in Section 12.4.

Exercises and Complements to Section 10.2 10.2.1 Kolmogorov extension theorem analogue for Q∗ . Show that the measure Q∗ defined below (10.2.7) admits consistent fidi distributions in the sense that Q∗ {N : N (Ai ) = ki , i = 1, . . . , n} satisfy the two consistency Conditions 9.2.V, namely, marginal consistency ˜ ∗ is finitely additive and symmetry under permutations. Show also that Q and continuous in the sense that for disjoint Ai , Q

∗

N: N

n

Ai

=

i=1

and

Q∗ {N : N (An ) > 0} → 0

n

N (Ai )

= 0,

i=1

when An ↓ ∅.

[For the latter, write Vn = {N : N (An ) > 0}. Because N (An ) ↓ 0 for all N ∈ NX , {Vn } is a monotonic decreasing sequence of sets, say Vn ↓ V . Supposing N0 ∈ V , then N0 (An ) > 0 for all n giving a contradiction. Q∗ (Vn ) → 0 follows from the countable additivity of Q∗ (·) on Sk , because we may assume the existence of some k for which An ⊆ Sk (n = 1, 2, . . .).] The same arguments as used in the proof of Lemma 9.2.IX now show that there exists a countably additive set function Q defined on B(NX# ) such that Q(C) = Q∗ (C) Q∗ -a.s.; that is, Q∗ admits a countably additive extension Q as required. 10.2.2 For an infinitely divisible multivariate point process [see Definition 6.4.I(a)], ˜ is defined on X × {1, . . . , m} and satisfies for show that the KLM measure Q bounded A ∈ BX Q{N = (N1 , . . . , Nm ): N1 (A) + · · · + Nm (A) > 0} < ∞. 10.2.3 Show that the KLM measure of the Gauss–Poisson process of Example 6.3(d) is the sum of two components, namely, measures Qj concentrated on realizations with N (X ) = j for j = 1, 2. 10.2.4 Let N be an infinitely divisible marked point process on the space X with marks in K, so Ng (A) < ∞ for each bounded A ∈ BX . (a) Write out the representation of N as an infinitely divisible point process on Y ≡ X × K. (b) Observe that the ground process Ng is infinitely divisible, and write down its representation as an infinitely divisible point process on X . (c) Investigate the relation between the KLM measures Q on B(Y0 ) for N and Qg on B(X0 ) for Ng . In particular, investigate whether the KLM measure Q has Q{N : N (A × K) > 0} < ∞ for such A. (d) Investigate the cluster process representation of a regular infinitely divisible marked point process.

10.3.

Point Processes Defined by Markov Chains

95

10.2.5 Proof of Proposition 10.2.IX. To establish (10.2.11), show the following. (a) ξ is infinitely divisible if and only if its fidi distributions are infinitely divisible. (b) Each fidi distribution has a standard representation similar to that of (10.2.11) subject to the constraint (10.2.12). (c) An inductive limit argument as in the proof of Proposition 10.2.III holds. [Hint: See Kallenberg (1975, Theorem 6.1).]

10.3. Point Processes Defined by Markov Chains In many applications, point processes arise as an important—and observable —component of the process of primary interest, but not necessarily as the primary process itself. Very commonly, the principal object of study is a Markov process. Our aim in this section is to examine some of the ways Markov processes can give rise to point processes in time, and some of the issues arising in the discussion of such models. We show first that a Markov or semi-Markov process on finite or countable state space can be regarded equivalently as an MPP {(tn , κn )}, where each tn is the time of a state transition, and the associated mark κn denotes the state entered when the transition occurs. In the other direction, every point process with a conditional intensity can be represented as a Markov process, via its history, which can be thought of as a Markov process of jump–diffusion type on a general state space, the points of the state space representing past histories as viewed from the present. The process moves continuously between states during the intervals between events, and jumps to a new state whenever an event occurs. It is Markovian because both the timing of the jump (determined by the conditional intensity) and the nature of the jump (determined by the conditional mark distribution) are functions of the current state. In such great generality, this observation may not have great value, although we shall return to it in Section 12.5 in developing a framework for the discussion of convergence to equilibrium for point processes. However, many important models arise as special cases, when the relevant history can be condensed into a compact and manageable form. A renewal process provides the simplest example, where the relevant history is just the backward recurrence time, which increases at unit rate between events, resets to zero whenever a jump occurs, and constitutes a simple diffusion–jump-type Markov process. For a Wold process, the backward recurrence time and the length of the last complete interval together constitute a Markov process driving the point process. Some other simple examples are described following the discussion of Markov and semi-Markov processes. We then note an important distinction between models such as the renewal process, in which the underlying Markov process can be directly constructed from observations on the point process, and those for which the point process

96


forms only part of a more complex Markovian system, which therefore remains at best partially observable if the only available information comes from the point process observations. This latter class includes hidden Markov models (HMMs) for point processes, such as the so-called Markov-modulated Poisson process (MMPP), and the Markovian arrival process (MAP) and batch Markovian arrival processes (BMAPs) developed by Neuts and co-workers, and now widely used in modelling internet protocol (IP) traffic and elsewhere. Difficult issues of parameter estimation arise for such processes: we outline the use of the expectation–minimization (E–M) algorithm for this purpose. Example 10.3(a) Point process structure of Markov renewal and semi-Markov processes [e.g. Asmussen (2003, Section VII.4); C ¸ inlar (1975, Chapter 10); Kulkarni (1995, Chapter 9)]. The probability structure of a Markov renewal or semi-Markov process on finite or countable state space X = {i, j, . . .} is defined 0 by the following ingredients: an initial distribution {pi : i ∈ X}; a matrix of transition probabilities (pij ) with j∈X pij = 1 (all i ∈ X); and two families of distribution functions Fij0 (u) and Fij (u) [(i, j) ∈ X × X and u ∈ (0, ∞)], defining the initial and subsequent lengths of time the process remains in state i, given that the next transition takes it to j. These ingredients can be used to construct a bivariate sequence {(κn , τn )} satisfying, for u ∈ R+ and sequences xr ∈ R+ and kr ∈ X for r = 0, 1, . . . , Pr{κ0 = k0 } = p0k0 ,

(10.3.1a)

Pr{τ1 ≤ u, κ1 = k1 | κ0 = k0 } = Fk00 k1 (u) pk0 k1 , and for n = 2, 3, . . . ,

Pr{τn ≤ u, κn = kn | κ0 = k0 , (κr , τr ) = (kr , xr ) (r = 1, . . . , n − 1)} (10.3.1b) = Pr{τn ≤ u, κn = kn | κn−1 = kn−1 } = Fkn−1 kn (u) pkn−1 kn . Letting u → ∞ here we see that {κn : n = 0, 1, . . .} is a discrete-time X-valued Markov chain with one-step transition matrix (pij ), and that for each n ≥ 1, τn conditional on (κn , κn+1 ) is independent of {(κr , τr ): r = 0, . . . , n − 1}. We assume that the process has neither instantaneous states nor explosions (see Exercise 10.3.1). Sufficient conditions, ensuring in particular that tn ≡ t0 + τ1 + · · · + τn → ∞ a.s., are that the matrix (pij )should be irreducible and have nontrivial invariant measure {πi } satisfying i πi µi < ∞, where µi =

j∈X

pij

∞

u Fij (du) = 0

j∈X

∞

u Gij (du) =

0

u Gi (du) 0

is the mean sojourn time in state i and Gi (·) =

j∈X

Gij (·) ≡

j∈X

∞

pij Fij (·).

10.3.


97

When card(X) ≥ 2 and pii = 0 (i ∈ X), there is a one-to-one measurable mapping between sequences {(tn , κn )} and X-valued, right-continuous, piecewise-constant functions X(t) with at most finitely many change-points on any bounded interval, where t is a change-point of X if and only if X(t −) = X(t ) = X(t +). Clearly, any such X(t) ∈ X for all t for some countable set of states X = {i, j, . . .}, and all change-points on [0, ∞) can be ordered as 0 = t0 < t1 < · · · < tn → ∞ (n → ∞). Thus any such X(·) determines a marked point sequence {(tn , κn ): n = 0, 1, . . .} for κn ∈ X for all n. Conversely, such a marked point sequence determines X(·) via the relation X(t) =

∞

κn I[tn ,tn+1 ) (t),

(10.3.2)

n=0

that is, X(t) = X(tn + 0) = κn (tn ≤ t < tn+1 ). Write τn = tn − tn−1 for all n for which tn < ∞ (see Figure 9.2). We call the sequence of marked points {(tn , κn )} a Markov renewal process, and {X(t)} a semi-Markov process. The discussion around (10.3.2) implies that the two are equivalent under the stated conditions. Observe the following. (1) When X is a one-point set we have a renewal process, delayed if F 0 = F . (2) When all τn = 1, {X(n): n = 0, 1, . . .} is a discrete-time Markov chain on X with one-step transition probability matrix (pij ). (3) When Fij0 (u) = Fij (u) = 1 − e−qi u (0 < qi < ∞) for all j, and qij = qi pij , X(·) is a conservative continuous-time Markov chain with Q-matrix (qij ). A Markov renewal process {(tn , κn )} as defined can just as well be interpreted as an MPP with mark space K = X and ground process defined by Ng (A) = #{n: tn ∈ A} for bounded A ∈ BR+ . By the equivalence just noted, a semi-Markov process can also be treated as an MPP. Consider first its conditional intensity. For any finite t > 0, since Ng is finite on bounded subsets, either Ng [0, t) = 0 or there exists a largest tn ∈ [0, t), defining tprev say with associated mark1 κprev . Suppose that tprev is defined, and the transition distribution functions Fij (u) have densities fij (u), so that there also exist transition-time density functions gij (t) dt = Pr{τn ∈ (t, t + dt), κn = j | κn−1 = i} = pij fij (t) dt.

(10.3.3)

Then the conditional intensity function λg (t) for the ground process depends only on the current state X(t−) (= κprev ) and when it was entered, and we have λg (t) = gκprev (t − tprev )/[1 − Gκprev (t − tprev )], The conditional intensity function λ∗ itself is expressible as λ∗ (t, κ | Ht ) = 1

gκprev ,κ (t − tprev ) = λg (t)f (κ | t, Ht ), 1 − Gκprev (t − tprev )

(10.3.4a)

In terms of the pair (tprev , κprev ), a Poisson process depends on neither element, a renewal process depends on tprev only, a Markov process depends on κprev only, and a semi-Markov process depends on the complete pair. See Exercise 10.3.2.

98


where the conditional mark distribution is given by f (κ | t, Ht ) =

gκprev ,κ (t − tprev ) . j∈X gκprev ,j (t − tprev )

For k ∈ X, each Nk (A) = #{n: tn ∈ A and κn = k} is a point process on R for which Nk (t) ≡ Nk (0, t] counts the number of entries into the state k during (0, t]. In the Markov case, the Markov property implies that the conditional intensity λ∗ (t, k) dt = E[dNk (t) | Ht ] depends on the past only through the state last entered, so that the ground intensity is given by λ∗g (t) = qX(t−) = qκprev , whereas for the conditional mark distribution f (k | t, Ht ) = pκprev ,k = pX(t−),k

(10.3.4b)

[cf. equation (7.3.3)]. The trajectories of the Markov process are easy to reconstruct from this MPP, because it carries details of which states were entered and for how long. The likelihood for a realization {(tn , kn ): n = 1, . . . , N (T )}, starting in state k0 at t = 0 and extending over an observation period (0, T ), can be written in the form LT = e−qk0 t1 qk0 k1 e−qk1 (t2 −t1 ) qk1 k2 . . . e

−qk

N (T )

(T −tN (T ) )

.

(10.3.5)

In the semi-Markov case, the MPP again contains complete information about the evolution of the process, and allows the likelihood to be written down explicitly in terms of the gij (·); details are set out in Exercise 10.3.5. As in the simple renewal process [see (4.1.5–10)], an important role in describing the properties of the semi-Markov process is played by the Markov renewal operator H(·) with elements Hij (t) = E[Nj [0, t] | (t0 , κ0 ) = (0, i)]. H is given by the sum of the series of convolution powers, in which G(t) = Gij (t) , H = I + G + G ∗ G + G ∗ G ∗ G + ···, or equivalently from the Markov renewal equation H = I + G ∗ H = I + H ∗ G, where convolution of matrices of nondecreasing functions is defined elementwise by t dAik (u) Bkj (t − u) (A ∗ B)(t) ij = k∈X

0

[if instead of nondecreasing functions we have matrices of nonnegative den t sities then define (a ∗ b)(t) ij = k 0 aik (u) bkj (t − u) du]. In particular,

10.3.


99

assuming densities as above and that Fij0 = Fij , the factorial moment densities for the ground process are given by the matrix products mg[r] (u1 , . . . , ur ) = p 0 H (u1 )H (u2 − u1 ) . . . H (ur − ur−1 )1

(10.3.6)

p0 = (p0i ) is the vector of initial probabilities, H (t) = for r =1, 2, . . . , where hij (t) = Hij (t) , and 1 is a vector of ones. Ball and Milne (2005) generalize this result to a wide range of point processes defined by transitions between states or groups of states; see also Exercises 10.3.3–4. Let {πi } be left-invariant for (pij ). Then Ug (x) = i∈X πi j∈X Hij (x) is the analogue of the renewal function at (4.1.5). See Exercise 13.4.4. The next three examples illustrate further cases where a point process can be expressed in terms of a relatively simple Markov process. Example 10.3(b) Hawkes process with exponential decay [see also Exercise 7.2.5 and Example 7.3(c)]. This is the simplest of several such examples considered t in Chapter 7. The governing Markov process is of shot-noise type Y (t) = 0 e−α(t−u) N (du), and is a linear function of past observations; the conditional intensity function takes the form λ∗ (t) = λ + ν

t

α e−α(t−u) N (du) = λ + ναY (t)

0

when we add in a background rate λ. Exercise 7.2.5 details the forward Kolmogorov equation, the stationary distribution, and the likelihood function, assuming the process starts from the stationary distribution. Examples 7.3(b)–(c) consider various extensions, in particular to situations where the conditional intensity can be a more general function of Y (t). Example 10.3(c) Birth-and-death process. Consider a simple birth-anddeath process, with birth rate λ per individual and death rate µ per individual. Let N + (t) and N − (t) denote the numbers of births and deaths recorded up to time t. Then the population size at time t is just the difference N (t) = N (0) + N + (t) − N − (t). We can treat this as an MPP by supposing that the instants of births and deaths are recorded separately, together forming the ground process Ng (t) = N + (t) + N − (t), with the marks ‘+’ denoting a birth and ‘−’ a death. The process N (t) forms an ergodic Markov process if λ < µ and there are also new arrivals, or ‘births from external source’ (i.e., immigrants) that occur according to a Poisson process of constant rate ν > 0. This is a classic example of a continuoustime Markov process, and it is well known that, for example, a stationary

100


distribution exists when µ > λ and is negative binomial [see, e.g., Bartlett (1955, p. 78)] with generating function G(z) =

∞

j

z Pr{N (t) = j} =

j=0

µ − λz µ−λ

−ν/µ .

In point process terms, the conditional mark distribution takes the form ! f ∗ (+ | t) = [ν + λN (t−)] λ∗g (t), ! f ∗ (− | t) = [µN (t−)] λ∗g (t),

where

λ∗g (t) = ν + (λ + µ)N (t−)

[recall (7.3.3)]. The likelihood can be written down from the standard point process formulae (Proposition 7.3.III) provided either N (0) is known (e.g., the population starts from size zero), or its initial distribution is known [e.g., in the stationary regime, it should be the stationary distribution for N (t)]. Exercise 10.3.5 gives some details. Example 10.3(d) Pure death process for software reliability [Jelinski and Moranda (1972); Singpurwalla and Wilson (1999)]. Consider a new or partially tested piece of software containing a number of errors (‘bugs’). Every time the software fails, one of the bugs is discovered and repaired. Suppose that every undiscovered bug contributes a constant component µ to the total risk; denote the number of bugs still undiscovered at time t by X(t). Then the conditional intensity at time t is given by µX(t). The process X(t) is clearly Markovian; in fact it constitutes a pure death process with constant death rate µ. In principle, it is directly observable only if N = X(0) is observable, a somewhat unlikely circumstance in the given context. More commonly therefore, N is treated as an unknown parameter (see Exercise 10.3.5 and, e.g., Singpurwalla and Wilson). But unless µ is known, very little information about N is retrievable from the data on observed failure times, and standard likelihood estimates of N are either unstable or unobtainable. Hence a Bayesian approach is often preferred, with a prior distribution for the initial state which may be either given subjectively or constructed from experience of previous studies. The posterior distribution of N can then be used to obtain an estimate of the number of remaining bugs. The last example shows very clearly, as is also true in earlier examples, that the underlying Markov process cannot be considered as fully observable unless the initial state is known. If it is not known, the initial state plays a role similar to that of an unknown parameter, which can be either estimated along with the other parameters from the likelihood function (this rarely leads to a satisfactory estimate because the information in the data with any bearing on the initial value is usually limited), or specified in terms of a prior distribution. Such examples form simple cases of the more general situation where the observed point process carries only partial information about the underlying

10.3.


101

Markov process. More commonly, not only the initial state but also the subsequent evolution of the Markov process remains unobserved. The birthand-death process illustrates also how simple it is for a point process driven by an observable Markov process to transform into an example of this kind. In this example, the situation would be radically altered if the ‘+/−’ marks were no longer observed: for a history with n observed points, there would then be 2n different ways in which the marks might be assigned, and obtaining the overall likelihood would entail averaging over 2n likelihoods conditional on a given ordering. At this point we enter the territory of hidden Markov models. The observed history, usually the internal history for the point process, is insufficient to allow the reconstruction of the Markov process driving the point process. Consequently the simple likelihoods characterizing the previous examples, which were dependent on knowing the driving process, are no longer available. Put in other terms, the conditional intensity based on the internal history is complex and difficult to handle directly, in contrast to the conditional intensity given the history of the driving process, which commonly has a simple form. Starting from the 1960s, these ideas led to a filtering theory for point processes, motivated by the analogy with the Kalman filter, and making use of the martingale constructions described in Chapter 14. More recently, a new focus on the problems of parameter and state estimation for such processes has developed through the use of the E–M (estimation–maximization) algorithm, an approach we outline shortly. First, however, we explore directly the simplest example of an HMM with point process observations. It played a key role in the early discussion of point process filtering [see, e.g., Yashin (1970); Galchuk and Rosovskii (1971); Snyder (1972); Jowett and Vere-Jones (1972); Rudemo (1972, 1973); Vere-Jones (1975); Brémaud (1981, Chapter 4)], and is also the starting point for the more elaborate HMMs which have come to be used extensively in communication theory and elsewhere. Example 10.3(e) Cox process directed by a simple Markov chain; ‘telegraph signal’ process; Markov modulated Poisson process. We sketched this model briefly in Exercise 7.2.8. The more extended discussion here generally follows Rudemo (1972). Suppose given a Markov process {X(t): t ≥ 0} on the finite state space K {1, . . . , K} with Q-matrix Q = (qij ) so that j=1 qij = 0 (i = 1, . . . , K), and qi ≡ −qii , assumed positive for all i, governs the exponential holding times in state i, and pij = qij /qi represents the probability that when a jump occurs from state i it is into a state j = i. Then the matrix of transition probabilities P (t) ≡ (pij (t)) satisfies P (0) = I and the forward and backward equations dP = QP (t) = P (t)Q, dt with solution P (t) = exp(tQ). Suppose further that while this Markov process is in state j, points of a Poisson process are generated at rate λj , and that

102


the observational data consist only of these points. The simplest nontrivial case of this set-up occurs with a process X(·) on two states with λ2 ≈ 0 and λ1 somewhat larger: we then have a model of a ‘telegraph signal’ process. Several estimation problems arise in connection with this process. Let us consider in particular the problem of ‘tracking’ the unobserved Markov process X(·), given the observations on the point process. This requires maintaining and updating the family of probabilities πi (t) ≡ P({X(t) = i} | Ht ).

(10.3.7)

Suppose the points observed on (0, T ) are t1 < · · · < tN (T ) , with tN (T ) < T . To obtain the πi (t) above, we consider the evolution of the ‘joint statistics’ defined, on any subinterval of the form (0, t) for 0 < t < T and with tN (t) < t ≤ tN (t)+1 , by pi (t; t1 , . . . , tN (t) ) dt1 . . . dtN (t) = Pr{X(t) = i and points occur in (tj , tj + dtj ), j = 1, . . . , N (t)}. (10.3.8) Call ‘either X(·) changes state or there is a point at t’ an event at t. Then, conditional on X(t+) = i, the time τ elapsing until the next event is exponentially distributed with Pr{τ > u} = e−(qi +λi )u , and, independent of τ , the event is either a point with probability λi /(qi +λi ) or a transition of X(·) from i to j with probability qij /(qi + λi ). Between observed points, therefore, the joint statistics evolve in a similar manner to the basic transition probabilities but with the matrix Q − Λ in place of Q, where Λ ≡ diag(λ1 , . . . , λK ). When a jump occurs, the joint statistics are weighted by factors λi , corresponding to multiplying the vector of joint statistics by the matrix Λ. It then follows that the vector p(·) ≡ p1 (·), . . . , pK (·) of joint statistics can be expressed as the matrix product (10.3.9a) p(t) ≡ p(t; t1 , . . . , tN (t) ) = p(0) R (0, t]; t1 , . . . , tN (t) , where R (0, t] = J(t) if N (t) = 0 and otherwise R (0, t] = J(t1 ) Λ J(t2 −t1 ) Λ . . . J(tN (t) −tN (t)−1 ) Λ J(t−tN (t) ), (10.3.9b) is the vector of initial probabilities for the process p(0) = p1 (0), . . . , pK (0) X(·) and J(x) = exp (Q − Λ)x . The probability πi (t) that at any time t ∈ (0, T ] the process is in state i, given the observations t1 , . . . , tN (t) up to time t and the initial distribution, is the ratio πi (t) =

pi (t) . p(t) 1

(10.3.7 )

Although (10.3.9) gives an explicit representation of the joint statistics, it may be just as convenient, particularly if an updating procedure is envisaged,

10.3.


103

to represent their evolution in terms of the differential equations they satisfy between events, and the discrete jumps that occur at the events tn on the trajectory. In terms of the pi (t) these equations are linear in form as below, with Dt ≡ ∂/∂t: Dt pi (t) = −(λi + qi )pi (t) +

pj (t)qji

(t = any tn ),

(10.3.10a)

j =i

∆pi (tn ) = pi (tn +) − pi (tn −) = (λi − 1)pi (tn −).

(10.3.10b)

Similar equations can be written down for the conditional probabilities πi (t) but in view of the ratios involved these are nonlinear, having the form Dt πi (t) = −πi (t)[λi + qi − λH (t)] + ∆πi (tn ) =

where

λ i − 1 πi (tn −), λH (t)

λH (t) =

πj (t)qji

(t = any tn ),

(10.3.11a)

j =i

(10.3.11b)

K i=1

λi pi (t)

(10.3.12)

is the conditional intensity at time t, given only the internal history H. If we consider the same example from the point of view of parameter estimation, (10.3.9) immediately yields the likelihood for observations over the observation period (0, T ) in the form L(t1 , . . . , tN (T ) ) = p(T ) 1 = p(0) R (0, T ] 1,

(10.3.13)

where 1 is the column vector of 1s. Because the likelihood is represented here as an explicit function of the Q-matrix of the hidden Markov process and the rates λi , it could in principle be used directly in maximization routines to find likelihood estimates of these quantities. However, this is a cumbersome and unstable process at best, and various techniques have been suggested to help stabilize the estimation procedures. One option, treated in detail in Vere-Jones (1975), is to extend the difference/differential equations from the joint probabilities (and hence the likelihood) to the means and variances of the parameter estimates; the results are still cumbersome and awkward to implement in practice. A more effective approach, indicated already in Exercise 7.2.8, is to bring the E–M algorithm to bear on the problem. This approach also reveals structural features that are important in discussing more general classes of models. The virtues of the E–M algorithm in this context are that it allows us to return to the simpler likelihood structures that occur when the underlying Markov process is known, thus making maximization (the M-step) easy, and replaces the rather unstable direct maximization of the likelihood by a more stable iterative procedure. Nevertheless, numerical issues

104


remain a major concern, and for the point process models many hundreds or even thousands of observations may be needed to gain stable parameter estimates, and in unfavourable cases (e.g., when some state transitions appear only rarely), much more may be required. To introduce these ideas, we first outline the basic steps in applying the E–M algorithm to an HMM in discrete time and space. The immediate point process application, treated in Example 10.3(f), assumes Poisson observations such as would result from binning the observed points in Example 10.3(d). Standard references for the E–M algorithm are Dempster et al. (1977), Elliott et al. (1995), MacDonald and Zucchini (1997), and particularly relevant examples are discussed, for example, in Qian and Titterington (1990), Deng and Mark (1993), Turner et al. (1998). The pioneering work goes back to papers by Baum and Petrie (1966) and Baum and Eagon (1967). To describe an HMM in discrete time and space we suppose given the matrix P = (pij ), i, j = 1, . . . , K, of one-step transition probabilities of the underlying Markov chain which for simplicity we assume is aperiodic and irreducible. We also suppose given a family of probability distributions: when the chain is in state j it generates an observation with density fj (z) (j = 1, . . . , K and z real). The procedures start by introducing the forward and backward probabilities defined, respectively, by αn (j) = Pr{Xn = j; Z1 = z1 , . . . , Zn = zn } βn (j) = Pr{Zn+1 = zn+1 , . . . , ZN = zN | Xn = j}

(10.3.14a) (10.3.14b)

[strictly, αn (j) = α(j; z1 , . . . , zn ) and βn (j) = β(j; zn+1 , . . . , zN )]. For n = 1, . . . , N − 1, these probabilities satisfy the recurrence relations αn+1 (j) =

K

αn (i)pij fj (zn+1 ),

(10.3.15a)

pjk fk (zn+1 )βn+1 (k),

(10.3.15b)

i=1

βn (j) =

K k=1

where the pij are the one-step transition probabilities of the discrete-time chain (so (pij ) = P∆t in our case), the fj (·) are probability densities (Poisson probabilities in our case) for the observations when the chain is in state j, and α1 (·) is the vector of initial probabilities and βN (·) = 1. Matrix versions of these equations are discussed in Exercise 10.3.7. It is assumed that the fj are unchanging over the observation period, and that successive observations are conditionally independent, given the corresponding states. For every 1 ≤ n ≤ N , the likelihood LN of the N observations, obtained by averaging over the possible state sequences, can be written in the form LN = K α (i)β n n (i); in particular, i=1 LN =

K i=1

αN (i).

(10.3.16)

10.3.


105

In practice it is generally desirable to renormalize the forward and backward probabilities at each step to avoid computations with extremely small numbers (see Exercise 10.3.7). Although the recurrence equations allow the forward and backward probabilities to be computed quickly, their underlying importance is embodied in the following lemma. Lemma 10.3.I (State Estimation Lemma). For an HMM with P and {fj (·)} as above, the conditional probabilities dn (i) = Pr{Xn = i | Z1 , . . . , ZN } and en (i, j) = Pr{Xn = i, Xn+1 = j | Z1 , . . . , ZN } are given, respectively, by αn (i)βn (i) , n = 1, . . . , N, k αn (k)βn (k) αn (i) pij fj (Zn+1 ) βn+1 (j) en (i, j) = , n = 1, . . . , N − 1. k, αn (k) pk f (Zn+1 ) βn+1 () dn (i) =

(10.3.17a) (10.3.17b)

Proof. The proof of (10.3.17a), as of the likelihood representation (10.3.16), depends on the fact that we can use the Markovian property to break open the joint probability Pr{Xn = i; Z1 = z1 , . . . , ZN = zN } and write it as the product of the two terms αn (i), βn (i) for any n in 0 ≤ n ≤ N . Start with Pr{Xn = i; Z1 = z1 , . . . , ZN = zN } = Pr{Z1 = z1 , . . . , ZN = zN | Xn = i} Pr{Xn = i}. Now, given the state i at time n, the distribution of the observations beyond n is independent of those before n and is nothing other than the backward probability βn (i). That is because the Markovian character of the transition probabilities means that no relevant information about the future states can be transmitted past the present state i, once that is given. Likewise the distribution of the observations up to time n is independent of those which follow, given the state at time n. The right-hand side of the above equation therefore reduces to [αn (i)/Pr{Xn = i}] × βn (i) × Pr{Xn = i} = αn (i)βn (i). Rewriting the joint probability in terms of the conditional distribution of the state, given the observations, and normalizing, yields (10.3.17a). Similar reasoning gives (10.3.17b). See also Exercise 10.3.7 for a matrix proof. The E–M algorithm for this setting involves finding those parameter values for the full model [i.e., both pij and fj (·)] that maximize the expected value of the likelihood, given the observations and an initial set of parameter values. Finding the expected value of the full likelihood, given the observations, is the E-step; finding the parameter values which maximize this expected likelihood is the M-step. An argument based on Jensen’s inequality shows that each iteration of the algorithm can only increase the likelihood. To illustrate the algorithm we turn to a discrete-time version of Example 10.3(e) [see also Fischer and Meier-Hellstern (1993), Davison and Ramesh (1993), and Rydén (1994, 1996)].

106


Example 10.3(f) Discrete-time HMMs with Poisson observations: E–M algorithm analysis. Because the standard procedures apply to discrete-time processes, a preliminary step in adapting them to the context of Example 10.3(e) is to bin the observations into small time intervals of length ∆t (preferably smaller than the mean interval length). The model then reduces to the discrete-time Markov chain with transition matrix P∆t = eQ∆t , whereas the observations consist of the sequence of counts Z1 , . . . , ZN in successive bins n = 1, . . . , N . The counts can be modelled conveniently as Poisson random variables having parameter µi = λi ∆t when the Markov chain is in state i. It is assumed that changes of state occur only on the boundaries of the ∆t intervals. We take the vector of parameters for this problem to be θ = {π1 , . . . , πK ; p11 , . . . , pij , . . . , pKK ; µ1 , . . . , µK }, where {µi } are the parameters of the distributions fi (·). The complete likelihood Lc , given successive states {i1 , . . . , iN } and observations {Z1 , . . . , ZN }, is given by N −1 N log pin ,in+1 + log fin (Zn ). log Lc (θ) = log πi1 + n=1

n=1

In our case, inasmuch as the distributions are Poisson, log fin (Zn ) = Zn log µin − µin − log(Zn !) . It is convenient here to rewrite the likelihood by collecting the quantities multiplying a particular term such as log pij ; then log Lc (θ) equals i

δi0 i log πi +

i,j

Eij log pij +

Gi log µi − Di µi + C,

(10.3.18)

i

where Di counts the total visits to state i, Eij the number of transfers from state i to state j, and Gi the number of points emitted while the chain is in state i, and C is a function of the observations Zn only. Equation (10.3.18) shows that {δi0 i , Di , Eij , Gi } is a set of sufficient statistics for the parameters θ. Next, take expectations conditional on the observations Z = {Z1 , . . . , ZN } and some initial parameter set θ∗ , treating the running parameter values in (10.3.18) as fixed numbers. This requires taking an average over state histories of the log-likelihood (10.3.16) (specifically, the various sufficient statistics), conditional on the observations and initial parameters. But these conditional expectations can all be written down in terms of the conditional probabilities of visits to or transfers between successive states, given the observations, and

10.3.


107

hence in terms of the quantities appearing in Lemma 10.3.I. Specifically, we find N −1 E[Eij | Z] = n=1 e∗n (i, j) ≡ e∗ij , E[δi0 ,i | Z] = πi∗ , N N E[Gi | Z] = n=1 Zn d∗n (i) ≡ gi∗ , E[Di | Z] = n=1 d∗n (i) ≡ d∗i , where ∗ indicates dependence on the initial parameters. Substitution leads to an expression for the conditioned log-likelihood which exactly replicates the form of (10.3.18), namely, E[log Lc (θ) | Z] equals gi∗ log µi − d∗i µi + C. (10.3.19) πi∗ log πi + e∗ij log pij + i

i,j

i

Equation (10.3.19) constitutes the E-step. To implement the maximization step we have to find the values of the current parameters that maximize (10.3.19) for given values of the observations and of the starred expressions. Recalling the constraints i πi = 1 = j pij , we find for the new estimates e∗ij g∗ π î = d∗1 (i), pîj = ∗ , µ î = i∗ . (10.3.20) di di These equations, which are commonly referred to as the Baum–Welch reestimation equations, constitute the M-step. Together with the results of Lemma 10.3.I, they summarize the application of the E–M algorithm in the given discrete context. The algorithm proceeds by successive application of the E- and M-steps, using the forward and backward probabilities to evaluate the quantities appearing on the right-hand sides in (10.3.20), and the equations themselves to evaluate the revised parameter estimates. In practice, a special problem again revolves around estimating the initial distribution, as there is not usually enough information in the data to provide stable estimates for the probabilities πi . If the πi are taken to be the stationary probabilities for the chain being estimated, and hence functions of the pij , the simplicity of the updating equations is lost, and they become essentially intractable. A reasonable compromise [cf. the discussion in Turner et al. (1998)] is to evaluate the stationary distribution for the chain with the initial parameters θ1 , and regard this distribution as fixed in the M-step. Then only the transition probabilities and the distribution parameters need to be re-estimated, resulting in modified forms in (10.3.20) with the given initial distribution replacing the estimated initial distribution. The E–M algorithm may converge rather slowly as it approaches the maximum, in which case it may be more efficient to switch over for the final steps to direct maximization of the likelihood (10.3.16), using the E–M estimates as starting values. The parameters for the continuous-time model which motivated Example 10.3(f) can be related approximately to the parameters of the discrete-time version by the relations λi ≈ µi /∆t;

qii ≈ −(1 − pii )/∆t;

qij ≈ −pij /∆t (i = j).

108


It is also possible to apply the E–M algorithm directly to the continuoustime process [see Rydén (1996)], and we now outline this alternative approach. Example 10.3(g) Cox process directed by a finite Markov chain: Direct E– M analysis [continued from Example 10.3(e)]. We start from the definition (10.3.9a) for the joint statistic pi (t), which we can identify as the continuoustime analogue of the forward probability αn (i). To emphasize the analogy, we write for the remainder of this discussion, using N (t) as before, αt (i) = Pr{X(t) = i and points occur in (0, t] at times t1 , . . . , tN (t) } = p(0) R (0, t] δi = pi (t), where δi is a K-vector of 0s except for 1 in the ith component. Similarly, we can write down an analogue to the backward probability βn (i), namely, βt (i) = Pr{points occur in (t, T ] at times tN (t)+1 , . . . , tN (T ) | X(t) = i} = δ i R (t, T ] 1. The differential versions (10.3.10–11) may be regarded as continuous-time analogues of the forward recurrence equations (10.3.15a) in the Baum–Welch formulation of HMMs. The quantities αt (i), βt (i) can be computed from the differential equations, or by direct evaluation of (10.3.9a) and its analogue for the backward probabilities. Once again the likelihood can be evaluated as αt (i)βt (i) = αT (i) (every t in 0 < t < T ). Lc = i

i

The Markov property again allows us to write down state estimation probabilities, analogous to those in Lemma 10.3.I. These take the forms dt (i) = Pr{X(t) = i | points at t1 , . . . , tN (T ) } = αt (i)βt (i)/Lc , (10.3.21a) et (i, j) dt = Pr{transition i to j occurs in (t, t + dt) | points at t1 , . . . , tN (T ) } = αt (i)qij βt (j)/Lc . (10.3.21b) The complete likelihood can be written down directly in terms of the epochs {s : = 1, . . . , M } of state transitions, s0 = 0, the state transitions themselves {i−1 → i }, and the event times {tn }. Recalling the notation qi = −qii , πij = qij /qi , and taking logarithms, we can write the log-likelihood as log Lc = log πi0 +

M −1 =0

log qi ,i+1 + [N (s+1 ) − N (s )] log λi

− (qi + λi )(s+1 − s ) − (qiM + λiM )(T − tM ) δi0 ,i log πi − Di (qi + λi ) + j =i Eij log qij − Gi log λi , = i

10.3.


109

where Di accumulates the time spent in state i during (0, T ], Eij counts the number of transitions from i to j, and Gi counts the number of events which occur while the chain is state i. Taking expectations after conditioning on the observations yields, for example, E[Di | t1 , . . . , tN (T ) ] ≡ Di∗ =

T

dt (i) dt, 0

∗ and G∗i but with dt replaced by with analogous integral expressions for Eij dN (t) (i.e., to give sums rather than integrals). Hence we obtain for the E-step, after substituting for the starred quantities,

E(log Lc | H(0,T ) ) =

T

d0 (i) log πi − (qi + λi )

i

+

j =i

dt (i) dt 0

T

et (i, j) dt − log λi

log qij 0

T

dt (i) dN (t) ,

0

where N (·) denotes the observed counting process. Then maximizing over the parameters πi , qi , qij , λi , we obtain for the M-step estimates T

π î = d0 (i), T î = λ

0

dt (i) dN (t)

T 0

dt (i) dt

,

et (i, j) dt , qîj = 0 T dt (i) dt 0 T j =i 0 et (i, j) dt qî = . T d (i) dt t 0

(10.3.22)

The integrals in these equations can be evaluated with the aid of (10.3.9) by summation of integrals over intervals between the points 0, t1 , . . . , tN (T ) , T . From the numerical point of view, a serious problem is the calculation of the matrix exponentials J(t) in (10.3.9b). If a simple discretization is used for this purpose, one is effectively reverting to the discrete-time model previously discussed, and there are greater advantages in keeping to the discrete set-up all through. One alternative is to use matrix diagonalization methods; a further alternative, suggested by Rydén (1996), is the ‘uniformization’ algorithm outlined in Exercise 10.3.9. A recent discussion of numerical aspects can be found in Roberts, Ephraim, and Dieguez (2006). This model has been the subject of considerable extension and elaboration. One generalization, which combines Example 10.3(g) with elements of the discrete HMM considered above, is to allow the observed process to be an MPP rather than a simple point process; a brief outline is given in Exercise 10.3.9. Example 10.3(f) can also be considered as a simple special case of the more general situation where the counting process N (t) is not itself Markovian, but forms one component in a more complex process which is

110


Markovian. Rudemo (1973) appears to be the first to have studied point processes of this kind. He considered a bivariate system X(t), N (t) , where N (t) is the observed counting process, X(t) is a K-valued unobserved Markov process with Q-matrix Q, and counts may be produced either between state transitions, as in the previous example, or at the transitions themselves. The system remains Markovian, and similar issues arise: tracking the current state of the unobserved process X(t) from observations on the counting process, and estimating the parameters of the bivariate process. This framework also covers the extensive series of studies by Neuts and co-workers on Markovian and Batch Markovian Arrival Processes (so-called MAPs and BMAPs); see inter alia Neuts (1978, 1979, 1989), Ramaswami (1980), Asmussen et al. (1996), and Klemm et al. (2003). An important feature of these models is that, if they are used as input streams for singleserver and other queueing systems, many characteristics such as distributions of queue lengths and waiting times, can be represented as matrix-analytic analogues of the forms which they take in the simpler systems with Poisson or renewal input. Here, we concentrate on the point process properties, referring the reader to accounts of the wider range of applications in the cited references. The Q-matrix for a two-component process X(t), N (t) has a block structure of the form ⎫ ⎧ Q00 Q01 Q02 . . . ⎪ ⎪ ⎪ ⎪ ⎪ Q11 Q12 . . . ⎪ Q ⎪ ⎪ 10 ⎪ ⎪ , ⎪ ⎪ ⎪ ⎪ Q Q . . . Q ⎪ ⎪ 20 21 22 ⎭ ⎩ ... ... ... where each Qrs is a K × K matrix that describes the transition rates of the unobserved process X(t) which may occur while the counting variable moves from r to s. In most cases backward transitions are not possible, so that Qrs = 0 for r > s; also, the process X(t) has stationary transition probabilities and N (t) has stationary increments. The Q-matrix then becomes a block-type version of the Q-matrix of a pure birth process, ⎧ ⎫ Q0 Q1 Q2 . . . ⎪ ⎪ ⎪ ⎪ 0 Q0 Q1 . . . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (10.3.23) ⎪ ⎪ ⎪ 0 ⎪. . . . 0 Q ⎪ ⎪ 0 ⎩ ⎭ ... ... ... Example 10.3(g) is recovered if we take Q0 = Q − Λ, Q1 = Λ, Qj = 0 for j > 1, where Q is the Q-matrix of X(t). That is to say, N (t) increases by 1 each time a point occurs, but the value of X(t) is unaltered, and the transitions of X(t) do not directly affect the value of N (t). More generally, processes of the type (10.3.23) are characterized by the matrix Q0 , which describes the transitions of the process X(t) in the absence of jumps, and the matrices Q1 , . . . , QL describing the transitions of X(t) which accompany jumps of size 1, . . . , L, respectively. One of the simplest point processes of this type, often used as a building block in constructing more complex models, is described below.

10.3.


111

Example 10.3(h) Renewal process of phase type; PH-distributions [Neuts (1978)]. This model is motivated by the situation in which points are recorded only when an underlying Markov process enters a particular state, say 0. It then becomes an example of an alternating renewal process, with one type of interval corresponding to sojourns in state 0, and the other to the periods while the process is traversing a path through the remaining states. As such it can be treated as a modified version of Example 10.3(a) as in Exercise 10.3.4. However, we wish to consider the limiting situation in which sojourns in state 0 are instantaneous, so that the observed process is actually a renewal process. This limiting process is then described by two components: a defecK tive K × K Q-matrix with row sums j=1 qkj = −δk < 0, and a vector of re-entry probabilities, say πk , describing the probability of entering state k at the same time as a renewal occurs. This corresponds to a model governed by a matrix of type (10.3.23) in which Q0 = Q, Q1 = δ π, Q = 0 for > 1. The interval between successive renewals can be represented as the sum of a random number of exponentially distributed waiting times, corresponding to the holding times in each of the states passed through before the next renewal, and mixed over the starting state. Its distribution takes the form 1 − F (t) = π eQt 1,

(10.3.24)

which in Neuts’ terminology represents a PH-distribution with representation (Q, π). Special cases, such as mixtures or sums of exponentials, correspond to giving special forms to Q and π; see Exercise 10.3.10 for examples. Expressions for the Laplace transform of the above density, as well as for the associated renewal function, are outlined in Exercise 10.3.10. Consider next the BMAP, originally called a versatile Markovian point process by Neuts (1979). This can be considered both as a Markov process with Q-matrix of the form (10.3.23), and an MPP. Both points of view are helpful in developing properties of the process, as we indicate below. The process accommodates not only batch arrivals, but also some dependence between the lengths of the intervals between arrivals, because the lengths of two consecutive intervals both depend on the state of X(t) at the arrival time by which they are separated. Variations on this model are extensively used in information-processing networks, where the batch size approximates packet length, and the arrival rate the packet frequency [see Klemm et al. (2003)]. Example 10.3(i) BMAP representations: E–M analysis [Neuts (1979); Rydén (1996); Klemm et al. (2003)]. We suppose again that the process is driven by an unobserved Markov process X(t) with K states and generator Q. In this case, however, the observed points correspond to batches of size 0 < ≤ L, and may or may not be associated with a change of state. To accommodate the latter possibilities, thematrix Q is split among the different batch sizes, L and written as a sum Q = =0 Q , where the Q appear in the representation (10.3.23), so the Q govern the transitions associated with arrivals of batch

112


() () size (or no arrivals for Q0 ). Write Q = qij , so positive elements qkk with > 0 correspond to arrivals for which no change of state occurs. Although the BMAP is the archetypal process with a representation as at (10.3.23), for estimation purposes it is more fruitful to consider it as an MPP with bivariate marks, (k, ) say, the two components recording the state k entered and the size of the associated arrival batch, including the possibility = 0. Marks of the form (k, 0) do not occur when X(t) is in state k, because they would correspond to transitions from state k to state k with no accompanying arrivals, and hence to no effective transition of any kind. Inasmuch as the structure of (10.3.23) already implies that the transition rates depend only on the state k and not on the accumulated number of arrivals, the conditional intensity for the associated MPP {tn , (kn , n )} depends on the history only through the current state X(t), which we denote by ξ to ease the notation. In terms of the representation (10.3.23), we have then for the conditional intensity, for k = 1, . . . , K and = 0, . . . , L, $ 0 if = 0 and ξ = k, λ∗k, (t) = () qξk otherwise. The ground intensity for the overall process (including transitions not associated with arrivals) is given by λ∗g (t) = E[dNg (t) | ξ]/dt = −(Q0 )ξξ = qξ ,

(10.3.25a)

and the conditional mark distribution takes the form f ∗ (k, | t) = πk (ξ) ≡ qξk /qξ , ()

= 0, 1, . . . , L.

(10.3.25b)

The complete process is stationary if and only if the Markov process X(t) is stationary, becasue X(t) determines the occurrence probabilities for all types of transitions. Because the marginal process X(t) is Markovian, and governed by the matrix Q, X(t) is stationary if and only if it starts with initial distribution π, satisfying π Q = 0 and which we assume to be welldetermined and unique. Taking expectations under this distribution, we find for the mean rate of occurrence of all transitions, K

¯ = E[λ∗ (t)] = λ g

(0)

πk qkk =

k=1

K

πk q k

(all t).

k=1

The mean rate of arrival of batches of size > 0 is λ = and the overall rate of arrival of batches is λ=

K k=1

πk

K L =1 j=1

()

¯ − λ0 , qkj = λ

K k=1

πk

K

() j=1 qkj ,

10.3.


113

(0) where λ0 = k πk j:j =k qkj is the expected rate of transitions not accompanied by arrivals. Other characteristics of the process, including interval distributions and correlations, can be represented in matrix exponential terms along the lines of Example 10.3(h) (see Exercise 10.3.11). The BMAP models share with the E–M algorithm the feature that elaborations of the forward–backward equations can be used for parameter estimation. Indeed, the steps follow a very similar pattern to those of Example 10.3(f), and we indicate them only in summary form; Klemm et al. (2003) can be consulted for further details and numerical aspects. Suppose that over the observation interval (0, T ), the process has initial probability distribution {πk0 }, starts from state k0 , and jumps at times {tn } with marks {kn , n }, corresponding to transitions into states kn associated with arrival batches of size n (including the possibility n = 0). Then the complete likelihood takes the form Lc = πk00 qk01,k1 e−qk0 t1 qk12,k2 e−qk1 (t2 −t1 ) . . . . ( )

( )

(10.3.26)

Grouping together the terms associated with particular transitions, or sojourns in particular states, this can be rewritten as % Lc =

πk00

() N qjk j,k

()

& e−

qk Dk

,

(10.3.27)

j,k, ()

where, with reference to the observation period (0, T ), Nj,k counts the total number of transitions from state j into state k associated with an arrival batch of size , and Dk is the total length of time that X(t) is in state k. We now turn to extensions of the backwards and forwards probabilities. In the present model, given the initial distribution π0 and the set of observed arrival times and batch sizes {(τn , n ): n = 1, . . . , N (T )} (and note that the {τn } form in general only a subset of the set {tn } used in describing the complete likelihood), the forward probabilities take the form αt (k) = (π0 ) eQ0 (τ1 ) Q1 eQ0 (τ2 −τ1 ) Q2 . . . eQ0 (t−τN (t) ) ek .

(10.3.28)

Likewise the backward probabilities βt (i) can be written Q0 (τN (t)+1 −t) βt (k) = e QN (t)+1 eQ0 (τN (t)+2 −τN (t)+1 ) QN (t)+2 . . . eQ0 (T −τN (T ) ) 1. ke

It is evident that here also, for every t in (0, T ), the incomplete likelihood can be expressed as αt (k)βt (k) = αT (k) = αT 1. (10.3.29) L(T ) = k

k

114


This expression can be directly maximized to find suitable parameter estimates. Alternatively, as in Example 10.3(h), we can seek more stable procedures based on the E–M algorithm. Adapting the latter approach, the forward and backward probabilities reappear in the appropriate extension of the state estimation Lemma 10.3.I. Equation (10.3.21a) retains the form dt (i) ≡ Pr{X(t) = i | S} = (0) Nij (t) denotes the number of

αt (i)βt (i)/L(T ),

whereas if transitions from i to j with no associated arrivals in time t, (10.3.21b) becomes (0) (0) et (i, j) dt ≡ E[dNij (t) | S] = αt (i)qij βt (j)/L(T ) dt. For the conditional rate at time t of i → j transitions associated with arrivals of batch size > 0 we have () αt (j)qjk βt (k) () . et (j, k) = LT Turning finally to the E- and M-steps, we obtain from (10.3.27), log Lc =

log πi00

−

Dk log qk +

k

L

()

()

Nij log qjk .

(10.3.30)

j,k =0

Taking expectations conditional on the observed sequence S ≡ {(tn , n )} constitutes the E–step, and leads to an expression similar to (10.3.30) but with () E[Sk | S] and E[Njk | S] replacing the corresponding expressions without the expectations. Maximizing with respect to the parameters is again straightforward, and leads to the updated estimates ()

π î0

= Pr{X(0) = i | L},

() qˆjk

=

E(Njk | S) E(Sj | S)

,

qˆj =

L k

()

qˆjk .

(10.3.31)

=0

Thus the crucial difficulties are again in evaluating the conditional expectations which appear in these equations, and again these can be represented in terms of the forward and backward probabilities. We find Pr{X(0) = i | S} = d0 (i), T T (10.3.32a) (0) E(Sk | S) = dt (i) dt, E(Njk | S) = et (j, k) dt. 0

0

The conditional expectations for the cases with > 0 have a slightly different form because they correspond to known times and sizes of arrivals. However, similar reasoning leads to the results T () () et (j, k) dN (t), (10.3.32b) E(Njk | S) = 0

where N (t) counts the number of batches of arrivals of size . Evaluation of the matrix exponentials which arise in these formulae can again be tackled by diagonalization or by the uniformization approximation described in Exercise 10.3.8.

10.3.


115

Exercises and Complements to Section 10.3 10.3.1 Existence. Any state i of a continuous time Markov chain on countable state space with time-homogeneous transition probabilities (an MC, say) as in Part II of Chung (1967) is either stable or instantaneous according as the corresponding diagonal matrix element qi = −qii is finite or infinite, respectively. An MC with only stable states and all of them nonabsorbing can nevertheless have infinitely many jumps in a finite interval, but for an MC to be regarded as a point process as defined in Chapter 9 we wish to exclude such a possibility. By using the role of the diagonal matrix elements qi = −qii as intensities, argue that a stochastic condition for an MC to have a.s. finitely many jumps {tn } in bounded subsets of R is that 1/qX(tn −) = ∞

a.s.

tn ∈R+

Verify that the pure birth process with quadratic birth rates fails this condition. [Hint: See, e.g., Feller (1968, Section XVII.4).] 10.3.2 (a) Use (tprev , κprev ) of Example 10.3(a) with the transition probabilities and distribution functions specifying a semi-Markov process to describe the conditional intensity at time t of (i) a Poisson process; (ii) a renewal process; (iii) a Markov process (on countable state space); and (iv) a semiMarkov process. Verify the assertion of the footnote in that Example. (b) Given a Markov renewal process {(tn , κn )} around (10.3.1) with card(X) ≥ 2 but allowing κn = κn+1 (so pii > 0 for at least one i ∈ X), X(·) at (10.3.2) is still well-defined. Define a subset {(tn , κn )}, still a Markov renewal process, such that κn+1 = κn (all n); the relation (10.3.2) still holds (and the sample functions remain the same), and when the original transition matrix (pij ) is irreducible, the transition matrix (pij ), where pij = pij /(1−pii ), now leads to a one-to-one measurable mapping between the latter Markov renewal process and the semi-Markov process X(·). 10.3.3 Markov renewal or semi-Markov process observed on a subset [cf. C ¸ inlar (1975, 10(1.13))]. Let Y = {(tn , κn )} be a realization of an irreducible Markov renewal process on state space X, and let X be a nonempty proper subset of X. Construct the subset Y of Y via Y = {(tn , κn ): κn ∈ X } and relabel it sequentially as {(tn , κn )}. (a) Show that Y is an irreducible Markov renewal process on X . (b) Let Y have matrix renewal function H(t) = (Hij (t)). Show that the components of the matrix renewal function of Y are {Hij (t): i, j ∈ X }. (c) When card(X ) ≡ R say is finite, find an expression for the R × R matrix GX (t) of transition distributions for the process observed only in X in terms of the original matrix G(t) = (Gij (t)). In the special case where the original process is Markovian, show that the transition distributions are of the phase-type discussed in Example 10.3(h). [Hint: {κn } is a discrete-time Markov chain on X, ensuring that {κn } is a similar process on X , inheriting irreducibility from Y. Its one-step transition probabilities (pij ) say, for i, j ∈ X , can be found from the pij via taboo probability versions of the Chapman–Kolmogorov equations, or else directly

116

10. Special Classes of Processes from a renewal-type equation. The same is true of the components of the matrix of transition distributions GX .]

10.3.4 (Continuation). When Y = {(tn , κn )} is a Markov renewal process, the process Y2 = {(tn , (κn−1 , κn ))} is also a Markov renewal process, now with state space X(2) . The subset Y2 of observations of transitions of Y restricted to a subset X2 ⊆ X(2) , is again a Markov renewal process by Exercise 10.3.3(a); for simplicity confine attention to the case t0 = 0. Then the first moment H(i,j),(k,) (t) say, for the number of jumps tn in (0, t] for which k → given a jump i → j at 0, is independent of i and satisfies the Markov renewal equation t

Hj,(k,) (t) = δjk Gk (t) + h∈X

0

dGjh (u) Hh,(k,) (t − u).

Then with H(k,) = (H1,(k,) · · · Hj,(k,) · · · ) and δk = (δ1k · · · δjk · · · ) , H(k,) (t) = (H ∗ Gk )(t)δk , provided (G(n∗) ∗ H(k,) )(t) → 0 (n → ∞). The analogue mg[r] for Y2 of the factorial moment densities at (10.3.6) when G has density g = (gij ), in terms now of gX2 (·) = (δ(i,j),X2 gij (·)) and with u0 = 0 < u1 < · · · < ur , is the product of matrices and vectors mg[r] (u1 , . . . , ur ) = p 0

r

(H ∗ gX2 (us − us−1 ))1.

s=1

[Hint: Deduce the equation for Hj,(k,) from the Markovian nature of {κn } and a backwards Chapman–Kolmogorov decomposition. See Ball and Milne (2005); Darroch and Morris (1967) considered the Markovian case earlier.] 10.3.5 Likelihoods for semi-Markov processes. In the notation of Example 10.3(a), with t0 = 0 and X(0) = κ0 , show that the likelihood L of a semi-Markov process X(t) observed on (0, T ] as having successive jumps at t1 , . . . , tN (T ) into states κ1 , . . . , κN (T ) is expressible as

L=

N (T )

gκn−1 κn (tn − tn−1 ) [1 − GκN (T ) (T − tN (T ) )],

n=1

u

where Gk (u) = j 0 gkj (u) du = j pkj Fkj (u). Write down the conditional intensity for the corresponding MPP, and verify that the above expression coincides with the usual form of the likelihood for an MPP. 10.3.6 Alternative treatments of the Jelinski–Moranda process [Example 10.3(d)]. (a) Formulate the likelihood when the initial state is treated as an unknown parameter. (b) Outline a Bayesian approach to the estimation of parameters in the process by finding the form of the posterior distribution for both the initial state and the parameters of the death process. 10.3.7 Forward and backward equations for discrete time HMMs. (a) Use induction to verify formally the forward and backward equations (10.3.17) for estimation in HMM [cf. MacDonald and Zucchini (1997)].

10.3.


117

(b) Alternatively, show that the forward and backward equations reduce to matrix iterations such as α n+1 = α n P Dn+1 , where P is the transition probability matrix, and Dn = diag{f1 (zn ), f2 (zn ), . . . , }. Hence, for example, we have the explicit form α n = π 0 P D1 P D2 . . . P Dn , with a similar expression for the backward probabilities. Use these to give straightforward proofs of the likelihood equation (10.3.16) and (10.3.17) from the state estimation Lemma 10.3.I. (c) To introduce normalized forms of the forward and backward probabilities, † (i) = αn (i)/( j αn (j)), and similarly βn† (i) = βn (i)/( j βn (j)). set αn Reformulate equations (10.3.13) for these normalized forms. [Remark: These quantities are numerically more stable; constants ρn say are needed to recover the original αn (·), and similarly for βn (·).]

10.3.8 Uniformization approximation for calculating matrix exponentials [Gross and Miller (1984)]. Show that if m > max{−qii } is chosen just larger than the maximum diagonal element, exp (Qt) can be represented in the form eQt = e−mt emtA , where A denotes the nonnegative matrix Q/m + I, and hence that ∞ ∞ (mt)n n pn (t)An , A = exp (Qt) = e−mt n! n=0 n=0 where the {pn (t)} are the probabilities in a Poi(mt) distribution. Sufficient iterates of the fixed matrix A, and a sufficient range of values of the Poisson probabilities, can then be computed to give an effective algorithm for determining values of the matrix exponential to high precision. 10.3.9 MPP extension of Cox process driven by Markov chain [see Examples 10.3(e) and 10.3(g)]. Let Q be the matrix of transition rates for a K-state Markov process X(t). Suppose that for k = 1, . . . , K, when X(t) = k, points are generated at rate λk with conditional mark distribution fk (x). Write down the conditional intensity λ(t, x) of the corresponding MPP in terms of the current state ξ, and use it to find the complete likelihood for observations over an interval (0, T ). Verify that the forward probabilities αt (k) for the observations up to time t, given the sequence {(tn , xn ): n = 1, . . . , N (t)} can be represented as a matrix product α t = π 0 R(0, t] where R(0, t] = J(t1 )ΛD(x1 )J(t2 − t1 )ΛD(x2 ) . . . J(t − tN (t) ) and J(t) has the same interpretation as in (10.3.6b), Λ = diag(λ1 , . . . , λK ), and D(x) = diag(f1 (x), . . . , fK (x)). Use these representations to find a set of sufficient statistics for the complete likelihood, and the appropriate extension of the E- and M-steps of Example 10.3(g). 10.3.10 PH-distributions and their Laplace transforms. (a) Write out the distribution (10.3.24) explicitly in the special case that Q is a 2 × 2 matrix. (b) Show that both sums and mixtures of exponentials can be represented as PH-distributions, and find the elements in their representation. [Hint: Restrict Q to the diagonal and superdiagonal; for a mixture of exponentials Q has pure diagonal form.]

118

10. Special Classes of Processes (c) Show that the distribution (10.3.24) has Laplace transform (s) = π [sI − Q]−1 1. Use this representation to find the renewal function. Generalize to a matrix renewal function as in Example 10.3(a).

10.3.11 Interval distributions and correlations for the stationary BMAP process. Find a matrix exponential (phase-type) representation for the time interval between one batch arrival and the next, first assuming the batches (of sizes 1 and 2 say) are associated with transitions into states k1 and k2 respectively, and then without making this assumption. Does this representation imply that the observed process can be regarded as a semi-Markov process with states = 1, . . . , L? Find also the correlations between successive intervals.

10.4. Markov Point Processes Markov processes in time heavily influenced the growth of applied probability modelling in the second half of the twentieth century. Dobrushin (1968) successfully described a spatial Markov property, and the 1970s saw Hammersley and Clifford (unpublished, 1971), Moran (1973), and Besag (1974), for example, describing processes on two-dimensional lattices with a Markovian property, exploiting adjacency of points on a lattice to limit the range of stochastic dependence [Isham (1981) presents a broad review]. Ripley and Kelly (1977) gave a definitive description of Markov point processes, with an important sequel by Baddeley and Møller (1989) and subsequent expansion by Baddeley and co-workers; there is a consolidated account in van Lieshout (2000), and a broad exposition with examples in Møller and Waagepetersen (2004, Chapter 6 and Appendices F and G). Many mathematical properties of Gibbs distributions were anticipated earlier in statistical physics [Ruelle (1969, Chapter 3), Preston (1976)]. Georgii (1988) gives a probabilistic approach to Gibbs measures on multidimensional lattices. The practical appeal of Markov models lies in the form of the joint probability distribution in many variables: it is expressible as the product of many conditional probabilities each in a small number of variables defined only on ‘adjacent’ time points. This then raises the possibility of specifying the model purely in terms of local conditional probabilities. The Papangelou conditional intensity function in Definition 10.4.I plays this role in point process modelling when coupled with an algebraic relationship property of ‘neighbourliness’ denoted below by ∼ , although the paradigm example of a renewal process fails this relationship in general [see around (10.4.16)]. Consider then simple finite point processes on a c.s.m.s. X , often a bounded subset of R2 or R3 . Proposition 5.3.II gives a canonical space for finite point processes as the union X ∪ of all product spaces {X (n) : n = 0, 1, . . .} with generic element x = {x1 , . . . , xn } for which n = card{x} = n(x). In Section 5.3 we used Janossy measures {Jn (·): n = 0, 1, . . .} to describe finite point processes; here we use their density functions {jn (x): n = 1, 2, . . .} with respect to n-fold products of Lebesgue measure on X with (X ) < ∞, symmetric as

10.4.

Markov Point Processes

119

around (5.3.1–2). It is sometimes more convenient to describe the distributions through the density function f = {fn } with respect to the distribution π = {πn } of a totally finite unit-rate Poisson process on the measurable space (X , BX ); if more generally this Poisson process has probability measure Pµ , then f is the likelihood of the process relative to Pµ as in (7.1.7). When X = Rd and µ = these two descriptions are virtually equivalent inasmuch as jn (x)/fn (x) = e−(X ) -a.e. (see Exercise 10.4.1). For Janossy densities, always symmetric, we have the interpretation as at (5.4.13), rewritten here in notation closer to the present setting, namely $ ' exactly n points in a realization: . (10.4.1) jn (x) dx1 . . . dxn = Pr one in each subset (xi , xi + dxi ) (i = 1, . . . , n), and none elsewhere Even more generally, f (= a collection {fn } of symmetric functions) may be a density with respect to any totally finite measure on the space X ∪∗ . For example, if X is a finite set and we use counting measure on X as the reference measure, then for each n = 0, 1, . . . , #(X ) we should want fn (x) equal to n! times the probability mass associated with the point set x for which n = n(x). Realizations of finite point processes with densities as at (10.4.1) are a.s. simple, and for them we have x ∈ X ∪∗ a.s., where X ∪∗ denotes the subset of X ∪ containing in each component X (n) only those x = {x1 , . . . , xn } for which xi = xj (i = j). This description proves more convenient here than the integer-valued measures N used in Chapter 9, but they are equivalent because n(x) N (·) = i=1 δxi (·), and N ∈ NX∗ when x ∈ X ∪∗ a.s. (cf. Proposition 9.1.X). We use x to denote both an element of X ∪∗ and a subset of X . For y ∈ X we usually write x ∪ y rather than x ∪ {y}. For the difference we write variously x \ y = x \ {y} = xy . An inclusion written y ⊂ x is strict. With this understanding, a nonnegative measurable function f : X ∪∗ → R+ is a density function of a simple finite point process when f (x) = fn(x) (x) for x ∈ X ∪∗ and f is integrable as at (10.4.2b). For such a process it is convenient to write (with mixed notation), when the reference measure is π(·), for a measurable function g, ∞ g(x) f (x) π(dx) = g(x) fn (x) πn (dx), (10.4.2a) E[g(N )] = X ∪∗

and f satisfies X ∪∗

f (x) π(dx) ≡

n=0 ∞ n=0

X (n)

X (n)

fn (x) πn (x) = 1.

(10.4.2b)

Often, the process of interest is on a subset of some Euclidean space Rd and it is regular in the sense of Definition 7.1.I, whereas the reference probability measure is a unit rate Poisson process, in which case this Poisson process is on a compact subset of Rd .

120


Definition 10.4.I. Given a simple finite point process on X with a density f : X ∪∗ → R+ ≡ {fn (x): x ∈ X (n)∗ , n = 1, 2, . . .}, the function ⎧ (y ∈ X , x = ∅), ⎨ f1 (y)/f0 (∅) ρ(y | x) = fn+1 (x ∪ y) f (x ∪ y) ⎩ ≡ (y ∈ X \ x, x ∈ X (n)∗ , n = 1, 2, . . .) fn (x) f (x) (10.4.3) defines its Papangelou conditional intensity [set ρ(y | x) = 0 if fn (x) = 0]. We remark that in view of our earlier comments, ρ(y | x) can just as easily be given in terms of Janossy densities: see Exercise 10.4.2. Ripley and Kelly’s definition of a Markov point process involves a concept of ‘adjacency’ of pairs of points and an analogue of a ‘neighbourhood’ based on such a notion of adjacency. This concept for points y, z ∈ X is embodied in some reflexive symmetric relation ∼ (meaning that y ∼ y and for z = y, y ∼ z if and only if z ∼ y), as, for example, y ∼ z if and only if |y − z| ≤ R for some finite positive R (but, see Exercise 10.4.3). Any such relation ∼ defines a (∼)-neighbourhood (or just neighbourhood for short) b∼ (y) of any y ∈ X by (10.4.4) b∼ (y) = {z ∈ X : z ∼ y}. This definition is easily extended to y ∈ X ∪∗ by setting b∼ (y) = {z ∈ X : z ∼ y for some y ∈ y}. Definition 10.4.II. A simple finite point process with density function f : X ∪∗ → R+ is a Markov point process if for every x with f (x) > 0 its Papangelou conditional intensity ρ(y | x) = f (x ∪ y)/f (x) satisfies (y ∈ X \ x), (10.4.5) ρ(y | x) = g y, x ∩ b∼ (y) where g: X × X ∪∗ → R+ . Call such f a Markov density function. In other words, for f to be the density function of a Markov point process, we require that, for all x ∈ X ∪∗ , y ∈ X \ x and writing n = n(x), the (n + 1)-dimensional joint density function fn+1 (x ∪ y) must be expressible as a product of the n-dimensional joint density function fn (x) and some function g(·, ·) that depends only on y and those elements of x that lie in the (∼)neighbourhood b∼ (y) of the ‘extra’ point y, i[i.e., g(y, ·) is independent of all elements of x and X that are not (∼)-neighbours of y]. In particular, for y, z ∈ x such that z ∼ y (hence, z ∈ / b∼ (y)), we have ∼ ∼ xy ∩ b (y) = (xz )y ∩ b (y). Thus, for such y and z, the relations f (xz ) f (x) = g y, xy ∩ b∼ (y) = g y, (xz )y ∩ b∼ (y) = f (xy ) f (xz )y

if z ∼ y (10.4.6)

hold for the density function f of a Markov point process, so that f (x) =

f (xy ) f (xz ) f (xz )y

(y, z ∈ x, z ∼ y).

(10.4.6 )

10.4.


121

In (10.4.6), f (x) is a compact notation for fn(x) (x), so f (xy ) = fn(xy ) (xy ) = fn(x)−1 (xy ). Theorem 10.4.V shows that there are far-reaching consequences of this condition which states that the conditional intensity of adding an ‘extra’ point y to a set xy is independent of any point z that is not in the neighbourhood b∼ (y). Example # 10.4(a). An inhomogeneous Poisson process with density µ(·) has jn (x) = xi ∈x µ(xi ), so ρ(y | x) = f (x ∪ y)/f (x) = µ(y), which, being independent of x, is clearly of the required form (10.4.5) for the process to be a Markov point process. A major property of Markov point processes is the Hammersley–Clifford Representation Theorem 10.4.V below. It is important practically because it expresses the joint density function of a point set x as a product of (conditional) probability density functions of many smaller subsets y ⊂ x. When it was first proved, the result also identified classes of Markov random fields on one hand and Gibbs states with nearest neighbour potentials, the latter being already well known in statistical physics [cf. Clifford (1990)]. Specifically, (10.4.7) expresses the density function f of a Markov point process as products of terms involving another function φ: X ∪∗ → R+ which is simpler than f in that φ(x) = 1 as soon as the set x includes a pair of distinct elements, y, z say, for which y ∼ z. In other words, φ(x) can differ from 1 only when x is a clique defined in 10.4.III(a) below. Definition 10.4.III (Cliques). Let x, y be finite nonempty subsets of X (equivalently, x, y ∈ X ∪∗ ), and ∼ a reflexive symmetric relation on elements of X . (a) y is a clique (write (∼)-clique if distinction is needed) if it is the empty set or a singleton set or else y ∼ z for every two-point subset {y, z} ⊆ y. (b) A clique y ⊂ x is a maximal clique of the set x when y ∪ {z} is not a clique for every z ∈ x \ y. (c) Clq(x) (or (∼)-Clq(x) if needed) is the family of all cliques y ⊆ x. To understand cliques better, we list below some of their properties, leaving their proof to Exercise 10.4.5. In this list, ∼ is a reflexive symmetric binary relation on elements of the c.s.m.s. X containing points y, z, . . . and finite nonempty subsets x, y, . . . . Note [compare (v) and (vi)] that cliques do not in general yield equivalence relations (this was wrongly claimed on p. 219 of the first printing of Volume I). (i) The empty set and all one-point sets {y} are cliques. (ii) The two-point set {y, z} with y = z is a clique if and only if y ∼ z. (iii) If y ⊂ x and x is a clique, then so also is y. (iv) ∼ is transitive within a clique. (v) Distinct maximal cliques can overlap when ∼ is not transitive.

122


(vi) If ∼ is transitive then distinct maximal cliques cannot overlap, and the maximal cliques that are subsets of x provide a decomposition of x into equivalence classes. Definition 10.4.IV. Let h, φ be nonnegative real-valued functions on X ∪∗ . (a) A family H of subsets of X ∪∗ is hereditary if x ∈ H implies y ∈ H whenever y ⊂ x. (b) h is an hereditary function if h(x) > 0 implies h(y) > 0 for every y ⊂ x. (c) φ is a (∼)-interaction function if φ(x) = 1 whenever x is not a (∼)-clique. Møller and Waagepetersen (2004, Example F.2) describe a process defined on a space with each element of H a strict subset of X ∪∗ [i.e., H ⊂ X ∪∗ = H]. The following theorem is a key result for Markov point processes. Its first version is unpublished [see Besag (1974), discussion by its originators there, and Clifford (1990)]; this version is due largely to Ripley and Kelly (1977). Theorem 10.4.V (Hammersley–Clifford Representation). A probability density function f : X ∪∗ → R+ is the density function of a Markov point process if and only if there is a (∼)-interaction function φ such that for nonempty sets x,

φ(y) = φ(z) (x ∈ X ∪∗ ). (10.4.7) f (x) = y⊆x

z∈Clq(x)

Remark. Because ∅ ∈ Clq(x), φ(∅) appears as a factor of f (x) for every x, and hence acts as a multiplicative normalizing constant in this representation. Proof. For a (∼)-interaction function φ, φ(z) = 1 when z is not a clique so in (10.4.7) the second equality is trivial and only the first needs proof. Given φ, define f˜(·) by either product in (10.4.7), and suppose that f˜ is integrable and hence can be and is normalized to be a density function on X ∪∗ . # Then for x with f˜(x) > 0 and z ∈ X \ x, f˜(x ∪ z)/f˜(x) = y⊆x φ(y ∪ z). Because φ is an interaction function, it is possible#for φ(y ∪ z) = 1 only when y ∪ z is a clique, so that y ∪ z ⊆ b∼ (z), hence y⊂x, y∪z∈Clq(x) φ(y ∪ z) = g z, x ∩ b∼ (z) for some function g. We thus have the form at (10.4.5), and f˜ as defined is the density function of a Markov point process. Conversely, let f be the density function of a Markov point process, and define ψ iteratively by ⎧ ⎪ ⎨ f (∅) ψ(x) = 1 ⎪ ⎩ f (x)! #

if x = ∅, if nonempty x is not a clique, y⊂x ψ(y) otherwise,

(10.4.8)

taking 0/0 = 1 if need be. Then ψ is an interaction function, and it remains to show that the representation (10.4.7) holds with φ = ψ. Suppose x is given, with n(x) = r ≥ 2, and that (10.4.7) has been proved for all x with n(x ) ≤ r − 1; note that it is true for r − 1 = 1.

10.4.


123

# # First, if f (x) = 0 and y⊂x ψ(y) = 0, then y⊆x ψ(y) = 0 also and (10.4.7) holds. # Next,#if y⊂x ψ(y) > 0, then for any z ⊂ x (hence, n(z) ≤ r − 1), we have f (z) = y⊆z ψ(y) > 0 because y ⊆ z ⊂ x, and the right-hand side of (10.4.6) is positive. If also f (x) = 0, then the left-hand side of (10.4.6) is zero so we have reached a contradiction if x not a clique, whereas if x is a clique then by the last case of (10.4.8), (10.4.7) holds. # Finally, when both f (x) > 0 and y⊂x ψ(y) > 0, either x is a clique and by the last case of (10.4.8), (10.4.7) holds, or else x is not a clique and therefore there exist y, z ∈ x such that z ∼ y. Because f is the density function of a Markov point process, (10.4.6 ) holds with arguments in the right-hand side there having at most r − 1 points, so that # # f (xy )f (xz ) w⊆xy ψ(w) w⊆xz ψ(w) = # . (10.4.9) f (x) = f (xz )y w⊆(xz )y ψ(w) Now ψ(x) = 1 because x is not a clique, and for any other w ⊂ x, either (i) w contains both y and z (and ψ(w) = 1 because it is not a clique), or (ii) w contains neither, in which case w ⊆ (xy )z , or (iii) w contains exactly one of y possibilities and z so it is of the form w ∪ y or w ∪ z for w ⊆ (xy )z . These # and facts imply that the right-hand side of (10.4.9) equals w⊆x ψ(w); that is, (10.4.6) holds when n(x) = r. In typical applications, the Papangelou conditional intensity or the clique density function φ(·) may be known, in particular, when n(x) is ‘small’ for most, if not all, cliques x. Example 10.4(b) Strauss process [continued from Example 7.1(c) and Exercise 7.1.8]. In the notation of this chapter, the Janossy density of the Strauss model of Example 7.1(c), for which x ∼ y if and only if x − y ≤ R, is given for 0 < β < ∞, 0 < γ ≤ 1, and α a normalizing constant, by jn (x) = αβ n(x) γ m(x,R) , where m(x, R) is the number of distinct elements y, z ∈ x for which y ∼ z. Then ρ(y | x) = β n(x∪y)−n(x) γ m(x∪y,R)−m(x,R) = βγ n(x∩SR (y)) , the exponent of γ being equal to the number of elements of x within distance R of y. Then ρ(y | x) is of the required form (10.4.5) for a Markov process, and therefore the representation (7.1.5) for the Janossy density follows from the Hammersley–Clifford theorem. Kelly and Ripley (1976) showed that the Strauss process is uniquely characterized by two properties: its density function is hereditary, and its Papangelou conditional intensity is of the form, for x ∈ X ∪∗ and y ∈ X \ x, ρ(y | x) = g n(x ∩ SR (y)) , where g: Z+ → R+ and SR (y) is the closed ball with centre y and radius R.

124


Example 10.4(c) Area-interaction point process [Baddeley and van Lieshout (1995), van Lieshout (2000, Section 4.3)]. Suppose a simple finite point process on a compact subset X ⊂ Rd has density with respect to a Poisson process at unit rate on X given by f (x) = αβ n(x) γ −(X ∩UR (x))

(x ∈ X ∪∗ ; α, β, γ > 0), (10.4.10) n(x) where denotes Lebesgue measure on Rd , UR (x) = i=1 SR (xi ) is the union of n(x) spheres with centres xi ∈ x and common radii R > 0, and α is a normalizing constant. In this form it is also called the penetrable spheres model used by Widom and Rowlinson (1970) and others to study liquid– vapour equilibrium questions (note also the next example); the model in R is tractable as a Kingman regenerative phenomenon [Hammersley et al. (1975)]. Because f (·) at (10.4.10) is a density function with respect to a unit-rate Poisson distribution, Pr{N (X ) = n} lies between e−(X )

[(X )]n · αβ n γ −(X ) n!

and

e−(X )

[(X )]n · αβ n , n!

hence α lies between (eβ−1 /γ)−(X ) and e−(β−1)(X ) , so N (X ) is finite-valued a.s. Thus f is indeed the density function of a simple finite point process, being Poisson when γ = 1. Its Papangelou conditional intensity is given by ρ(y | x) = βγ −(X ∩[SR (y)\UR (x)]) , and because the set difference here is a function of R, y and those xi ∈ x for which |y − xi | < 2R, a function g(y, x ∩ S2R (y)) can be constructed to satisfy (10.4.5); that is, a process with density (10.4.10) is a Markov point process, with x ∼ y if and only if |x − y| ≤ 2R. It is attractive or repulsive as γ ≥ or ≤ 1, respectively, and Poisson for γ = 1, where a simple point process with Papangelou conditionial intensity ρ is called attractive (respectively, re/ y, ρ(z | x) ≤ ρ(z | y) pulsive) whenever, for all x, y with x ⊂ y and z ∈ ρ(z | x) ≥ ρ(z | y) . Baddeley and van Lieshout allow a general version of this model by replacing Lebesgue measure in the exponent of (10.4.10) by a totally finite Borel ⊂ X regular measure ν say and the spheres SR (xi ) by compact sets S(x) where |S|: X → R+ is continuous and bounded. They also show how a birthand-death process can have the model as a stationary distribution. Observe that for γ < 1, the process is not as ‘aggressively’ repulsive as the Strauss process in which the exponent of γ increases quadratically in n(x) whereas in (10.4.10) it changes at most linearly in n(x), with such changes becoming closer to zero with greater overlap between different spheres SR (xi ) as n(x) increases. Example 10.4(d) Penetrable spheres mixture model [Widom and Rowlinson (1970), van Lieshout (2000, Examples 2.8, 2.11)]. Consider a bivariate point process (x1 , x2 ) constructed on a bounded set X by superposing two independent Poisson processes Nj at rates βj (j = 1, 2) subject to every x ∈ x1 being

10.4.


125

at a minimum distance R from every y ∈ x2 ; that is, d(x1 , x2 ) > R, where for nonempty finite point sets x, y, d(x, y) = minx∈x, y∈y d(x, y). Then the density relative to a unit rate Poisson process on X equals n(x1 ) n(x2 ) β2 I{d(x1 ,x2 )>R}

f (x1 , x2 ) = αβ1

(10.4.11)

for some normalizing constant α. This density is positive when, given x1 , all n(x2 ) of the points of the second component avoid the union UR (x1 ) [cf. (10.4.10)] of circles of radius R around all the points of the first component. It follows, using Poisson process properties, that the marginal density of the first component equals ∞ i e−(X ) n(x ) n(x ) X \ UR (x1 ) β2i = αβ1 1 e(β2 −1)(X ) e−β2 (X ∩UR (x1 )) . αβ1 1 i! i=0 Thus, the marginal distributions in this mixture model are just the cases 1 < γ = eβ2 or eβ1 of the area-interaction model of Example 10.4(c). Given a space X , a multitude of possible symmetric reflexive relations can i be defined. When two such relations ∼ (i = 1, 2) are given, the intersection ∩ ∩ 1 2 relation ∼ defined by y ∼ z if and only if both y ∼ z and y ∼ z, and the ∪ ∪ 1 union relation ∼ defined by y ∼ z if and only if at least one of y ∼ z and 2 y ∼ z holds, are both well-defined symmetric reflexive relations. When the 1 2 1 2 relations ∼ and ∼ are ordered in the sense that (say) y ∼ z implies y ∼ z for ∩ ∪ 1 2 all y, z, it follows that we can identify ∼ and ∼ with ∼ and ∼, respectively. Now suppose that {fn } and {fn } are Markov density functions for point 1 2 sets on X with respect to ∼ and ∼, respectively, and that 1 = f (x)f (x) π(dx) < ∞ (10.4.12) c ∪∗ X for some finite constant c > 0, so that cf = cf f is a density function, where (x)fn(x) (x). f (·) = {fn (·)} = {fn (·)fn (·)}, and f (x) = fn(x) Proposition 10.4.VI. Let f = f f be the product of two Markov density functions. When cf is a density function for some finite positive c, it is a Markov density function. Proof. The Hammersley–Clifford representation applied to the densities f and f for n = n(x) implies that

fn (x) = φ (y) φ (z), 1

y∈(∼)-Clq(x)

2

z∈(∼)-Clq(x)

where φ and φ are the interaction functions determined by the densities f and f . Observe that if we define the function φ on X ∪∗ by ⎧ ∩ ⎪ ⎪ φ (x)φ (x) x is a (∼)-clique, ⎨ 1 2 x is a (∼)-clique but not a (∼)-clique, φ (x) φ(x) = 2 1 ⎪ x is a (∼)-clique but not a (∼)-clique, ⎪ ⎩ φ (x) ∪ 1 otherwise (i.e., x is not a (∼)-clique),

126

10. Special Classes of Processes ∪

then φ is a (∼)-interaction function and the function f is expressible as in equation (10.4.8). The Hammersley–Clifford theorem now implies that there exists some finite positive c such that cf is the density function of a Markov point process on X . Proposition 10.4.VI enables us to extend the Strauss model in such a way as to allow degrees of interaction that may depend on the distance between points, as sketched in Exercise 10.4.6. Such a model is still a Gibbs model. Example 10.4(e) Spatial birth-and-death process [Preston (1977); see also van Lieshout (2000, pp. 83–87)]. This is a continuous-time space–time Markov process with state space X ∪∗ satisfying the following. (a) The only transitions are ‘births’ (x → x ∪ y) and ‘deaths’ (x ∪ y → x), where x ∈ X ∪∗ and y ∈ X \ x. (b) The probability of more than one transition in (t, t + h) is o(h). (c) Given the state x at t, the probability of a death x → x \ y (y ∈ x) during (t, t + h) equals D(x \ y, y)h + o(h), where D(·, ·): X ∪∗ × X → R+ is a BX ∪∗ × BX -measurable function. (d) Given the state x at t, the probability of a birth x → x ∪ y in (t, t + h), where y ∈ F ∈ BX , equals B(x, F )h dy + o(h), where B(x, ·) is a finite measure on (X , BX ). Assume that B(x, ·) has a density b(x, ·) with respect to the finite measure λ(·) on (X , BX ), so that intuitively, b(x, y) is the transition rate for a birth x → x ∪ y. Let f be a Markov function that is the density of a finite point process on X . Ripley (1977) observed that if there exists a spatial birth-and-death process such that whenever f (x ∪ y) > 0 it is true that the detailed balance relation (10.4.13) b(x, y)f (x) = D(x, y)f (x ∪ y) > 0 (x ∈ X ∪∗ ) holds, then the birth-and-death process is indecomposable and time-reversible, and its unique equilibrium distribution is the point process with density f . Existence and convergence are guaranteed by the following amalgamation of Preston’s (1977) Proposition 5.1 and Theorem 7.1 [see, e.g., Preston or van Lieshout (2000) for proof]. Proposition 10.4.VII. Let B(·, ·): X ∪∗ × BX ∪∗ → R+ and D(·, ·): X ∪∗ × X → R+ be such that B(x, ·) is a finite measure on (X , BX ) for each x ∈ X ∪∗ , B(·, F ) is BX ∪∗ -measurable for each F ∈ BX , and D(·, ·) is BX ∪∗ × BX measurable. Define βn =

sup B(x, X ),

x∈X (n)∗

δn =

inf

x∈X (n)∗

D(x \ y, y).

(10.4.14)

y∈x

Suppose that either (a) βn = 0 for all sufficiently large n ≥ 0 and δn > 0 for all n ≥ 1; or

10.4.


127

(b) βn > 0 for all n > 0, δn > 0 for all n ≥ 1, and ∞ β0 . . . βn−1 < ∞, δ1 . . . δn n=1

∞ δ1 . . . δn = ∞. β . . . βn n=1 1

(10.4.15)

Then there exists a unique spatial birth-and-death process for which B and D are the transition rates (the backwards equations involving B and D have a unique solution). The process converges in distribution as t → ∞ to its unique equilibrium measure, independent of the initial state. When either the birth- or death-rate is constant, the equilibrium distribution is an area-interaction process (see Exercise 10.4.8). Powerful as it is, Ripley and Kelly’s definition of a Markov point process, Definition 10.4.II, poses problems for a stationary renewal process which, seemingly, should be the simplest nontrivial case of a Markov point process on R. To see this, suppose that on the interval (0, t) there are n points x = {xi : i = 1, . . . , n} with 0 < x1 < · · · < xn < t say, coming from a stationary renewal process whose lifetimed.f. F has support a(0, a), density function ∞ f , and finite mean lifetime λ−1 = 0 xf (x) dx [so, 0 f (x) dx = F (a) = 1 > F (a − h) for any h > 0]. Then it is a standard result (see, e.g., Exercise 7.2.3) that the Janossy density function jn (x) is given by n−1

f (xi+1 − xi ) λ[1 − F (t − xn )]. jn (x) = λ[1 − F (x1 )]

(10.4.16)

i=1

Consequently, for any x such that jn (x) > 0 [and then, necessarily, max{x1 , max2≤i≤n (xi − xi−1 ), t − xn } ≤ a], we have ⎧ if y < x1 , [1 − F (y)]f (x1 − y)/[1 − F (x1 )] ⎪ ⎪ jn+1 (x ∪ y) ⎨ f (y − xi−1 )f (xi − y)/f (xi − xi−1 ) if xi−1 < y < xi , = i = 2, . . . , n, ⎪ jn (x) ⎪ ⎩ f (y − xn )[1 − F (t − y)]/[1 − F (t − xn )] if y > xn . (10.4.17) It is evident from (10.4.17) that if such a process is to be Markovian in terms of some ‘adjacency’ relation ∼ and therefore have a Hammersley–Clifford representation as in Theorem 10.4.V, that cliques on which interaction functions are defined can have at most two elements of a set x, and that these elements must be nearest neighbours either to the right or left of a given element x ∈ x. Furthermore, even supposing that {xi , xi+1 } is a clique in x, adjoining a point y∈ / x for which xi < y < xi+1 , would then change the status of {xi , xi+1 } so that it would no longer be a clique. Within the setting of the earlier part of this section, this ‘argument’ suggests that a stationary renewal process does not generally fit the Ripley–Kelly definition of a Markov point set. However, a word of caution is apposite: a Poisson process in R is a renewal process,

128


and it also satisfies the Ripley–Kelly definition (because of the complete independence property). We turn our attention therefore to the relation y ∼ z (or its negation) used earlier between elements y, z ∈ x ⊂ X . In all the examples noted here and in the literature generally, the relation is defined in fact for any pair y, z ∈ X independent of any set x to which either (or both or neither) may belong. The setting of a renewal process suggests we restrict attention to describing the x elements of the pair {y, z} as satisfying a reflexive symmetric relation y ∼ z only when the pair is a subset of x and that the relation may depend crucially x /x on x in the sense that we may have, for y, z ∈ x, y ∼ z but, although for w ∈ x∪w we necessarily have x ⊂ (x ∪ {w}), it need not be the case that y ∼ z. For x example, when y, z ∈ x ⊂ R and y ∼ z means that z is the nearest right- or left-neighbour of y from the set x, incrementing the set x by a point w lying between y and z destroys this nearest right- or left-hand neighbour property. x Suppose then that such a reflexive symmetric relation ∼ is defined for all ∪∗ two-point subsets of x ∈ X , subject to x being in an hereditary family x H (see Definition 10.4.IV). When for y, z ∈ x the property y ∼ z holds, x say that y and z are neighbours within x, or (∼)-adjacent. For y ⊂ x the x (∼)-neighbourhood is x

bx (y) = {z ∈ x: z ∼ y for some y ∈ y}. x

x

(10.4.18) x

The set y is a (∼)-clique if y ∼ z for all y, z ∈ y, and the (∼)-clique indicator function is x 1 if y ⊂ x is a (∼)-clique, (10.4.19) I x (y) = 0 otherwise x

x

[for strict analogy with (10.4.4) the notation b∼ (y) and I ∼ (y) would be used here]. Using this notation, we have for example I x ({y, z}) = 1

x

if and only if y ∼ z.

Observe the status of the sets y and x in (10.4.18–19): y provides the points that are ‘targeted’ for adjacency, and x the ‘environment’ within which ‘adx jacency’ is defined via the reflexive symmetric relation ∼ . x x Notice that for ∼ but not ∼, expanding x can destroy the (∼)-adjacency x x∪u of points y, z ∈ x (i.e., there can exist u ∈ / x such that y ∼ z but y ∼ z). x

Definition 10.4.VIII. A function f : H → R+ is a (∼)-Markov function if for all x in the hereditary class H, the function f is hereditary, and for y ∈ X and x ∪ y ∈ H with f (x) > 0, the ratio ρ(y | x) depends only on y, bx∪y (y), x∪y x and the relations ∼ and ∼ restricted to the neighbourhood set bx∪y (y). Recall that the Hammersley–Clifford Theorem 10.4.V gives a representation of joint densities in terms of simpler ‘interaction’ functions. For this x set-dependent adjacency relation ∼ the analogous function, and the extended theorem, are as follows; its proof is similar to that of Theorem 10.4.V and can be found in Baddeley and Møller (1989).

10.4.


129

Definition 10.4.IX. Let the function φ: H → R+ be hereditary and be such that for y ∈ / x ∈ H, if φ(x) > 0 and φ(bx∪y (y)) > 0, then φ(x ∪ y) > 0. x A (∼)-interaction function is a function Φ defined in terms of such φ by a relation of the form x (10.4.20) Φ(y | x) = φ(y)I (y) , where 00 = 0. x

Theorem 10.4.X (Hammersley–Clifford, extended). Let the relation ∼ satisfy the consistency conditions, for finite x ∈ H, w ⊂ z ∈ H, y, z ∈ X but y, z ∈ / z, and x = z ∪ {y, z} ∈ H, (C.1) I z (w) = I z∪y (w) implies w ⊂ bz∪y (y); and x (C.2) when y ∼ z, I z∪y (w) + I z∪z (w) = I z (w) + I x (w). x Then f is a (∼)-Markov function if and only if

f (x) = Φ(y | x) (10.4.21) y⊆x x

for all x ∈ H, where Φ is a (∼)-interaction function. Baddeley, van Lieshout and Møller (1996) considered Poisson cluster processes and showed them to be nearest-neighbour Markov processes as above when the clusters are uniformly bounded, or if the cluster centre process is Markov or nearest-neighbour Markov and the clusters are both uniformly bounded and a.s. non-empty. Thus, the nearest-neighbour Markov property is preserved under random translation but not under random thinning. Other extensions of ∼ have been suggested: Ord’s process in Exercise 10.4.10 gives one, Chin and Baddeley (1999, 2000) looked first at a relation based on components exhibiting pairwise-connectivity and then at interactions between components of point configurations, and van Lieshout (2006a) has considered a sequential definition in association with space–time processes.

Exercises and Complements to Section 10.4 10.4.1 Suppose that a simple finite point process on (a subset of) Rd has Janossy density {jn (x)}, and that its density with respect to an inhomogeneous Poisson process with intensity λ(x) (x ∈ X ) is {fn (x)}. Show that jn (x) = e−Λ(X ) fn (x) xi ∈x λ(xi ) where Λ(X ) = X λ(u) (du).

10.4.2 Let the finite point process on X have Janossy densities {jn (·)} as in Sections 5.4 and 7.1. Then the Papangelou conditional intensity of Definition 10.4.I is expressible for some finite c > 0 c j1 (y) (y ∈ X , x = ∅), ρ(y | x) = jn+1 (x ∪ y)/jn (x) (y ∈ X \ x, x ∈ X (n)∗ , n = 1, 2, . . .).

10.4.3 Suppose the reflexive symmetric relation ∼ of a Markov point process is given by y ∼ z if and only if |y − z| ≤ R for R = 0. Because the only cliques are then singletons, deduce that the point process must be Poisson. 10.4.4 Verify the properties (i)–(vi) of cliques listed after Definition 10.4.III, providing in particular a counterexample to property (v).

130


10.4.5 Check conditions for Examples 10.4(b) and (c) to be repulsive or attractive. 10.4.6 Extended Strauss models; multiscale processes. Suppose given for a finite positive integer k, 0 = R0 < R1 < · · · < Rk < Rk+1 = ∞, βj ∈ (0, ∞) and γj ∈ (0, 1] (j = 1, . . . , k), and X a bounded Borel subset of Rd . For x ∈ X ∪∗ and R ∈ R+ let m(x, R) = #{xi , xj ∈ x : xi − xj < R} as in Example 10.4(b), and ∆m(x, R , R ) = m(x, R ) − m(x, R ) for 0 ≤ R < R ≤ ∞. n(x) m(x,Rj ) (a) Let αj be such that f (j) (x) = αj βj γj is a Markov density function for each j = 1, . . . , k. Use Proposition 10.4.VI to deduce that k

(γj . . . γk )∆m(x,Rj−1 ,Rj )

f (x) = αβ n(x)

j=1

is a Markov density function for suitable α and β. ∆m(x,R

,R )

j−1 j is a Markov density (b) More generally, f (x) = αβ n(x) kj=1 γj function for some finite positive α and β; display its Papangelou intensity function. [Hint: Penttinen (1984) or Møller and Waagepetersen (2004, Example 6.2).] (c) Define stochastic monotonicity of Markov point sets for such models.

10.4.7 (a) In the setting of Example 10.4(f) verify that a nonnegative function f defined for all x ∈ X ∪∗ by (10.4.13) and integrable as below (10.4.2) is a Markov density function when conditions such as (10.4.15) are satisfied. (b) When f is a Markov function, with (∼)-interaction √function √ φ say, a repre√ sentation such √ as (10.4.7) but with φ holds for f ≡ { fn }. Conclude that when f is integrable as at (10.4.2b) it is a Markov function.

10.4.8 Suppose a birth-and-death process as in Example 10.4(e) has constant birthrate B(x, A) = (X ∩ A) for A ∈ BX and death-rate D(x) = y∈x D(x \ y, y) for a BX ∪∗ × BX -measurable function D(·, ·) that satisfies condition (a) or (b) of Proposition 10.4.VII. Show that the equilibrium measure is an areainteraction process as in Example 10.4(c). [Hint: Baddeley and van Lieshout (1995, Section 4) also give a constant death-rate (but variable birth-rate) analogue of this property.] 10.4.9 Let X = {x1 , y1 , x2 , y2 , z, y3 } and suppose that x1 ∼ y1 ∼ x2 ∼ y2 ∼ z ∼ y3 ∼ x1 but u ∼ v for all other pairs {u, v} ⊂ X . For any {u, v} ⊂ X and w w ⊂ X define u ∼ v if either u ∼ v or else u ∼ w and w ∼ v for some w ∈ w. x

y3 . Then (a) Consider the two sets x = X \ z and y = {y1 , y2 , y3 }, so y2 ∼ x x∪z x x∪z (y) = 1, so I (y) = I (y). But y1 ∈ / bx∪z (z), so I (y) = 0 and I w condition (C.1) does not hold for such X and ∼ as defined. (b) Use y as above, but now put z = y ∪ z and let u, v = x1 , x2 . Then I z (y) = I z∪u (y) = I z∪v (y) = 0 but I z∪{u,v} (y) = 1, so (C.2) also fails.

x 10.4.10 Ord’s process. Consider the function f (x) = αβ n n i=1 g(area of C (xi )), where x ∈ bounded subregion of R2 , n = n(x), C x (xi ) denotes the Voronoi cell associated with xi ∈ x as in Example 10.4(c), and g a function described shortly. If g is not constant then f is not a Markov function, because f (x ∪ y)/f (x) depends on neighbours of neighbours of y, but for positive x x bounded g, f is a Markov function w.r.t. ∼2 defined by xi ∼2 xj if either x x x xi ∼ xj or else there exists xk ∈ x such that both xi ∼ xk and xk ∼ xj .

CHAPTER 11

Convergence Concepts and Limit Theorems

11.1 11.2 11.3 11.4

Modes of Convergence for Random Measures and Point Processes 132 Limit Theorems for Superpositions 146 Thinned Point Processes 155 Random Translations 166

When random measures and point processes are regarded as probability mea# sures on the appropriate c.s.m.s. M# X or NX , they may be associated with concepts of both weak and strong convergence of measures on a metric space. In this chapter we examine these concepts more closely, finding necessary and sufficient conditions for weak convergence, relating this concept to other possible definitions of convergence, and applying it to some near-classical questions concerning the convergence of superpositions, thinnings, and translations of point processes. A common theme in the limit theorems described in this chapter is the emergence of the Poisson process as the limit of repeated applications of some stochastic operation on an initial point process. In a loose sense, each of the operations of superposition, thinning, and random translation is entropy increasing; it is not surprising then that among point processes with fixed mean rate, the Poisson process has maximum entropy (see Section 7.6 and the further discussion in Chapter 14). These limit theorems help not only to explain the ubiquitous role of the Poisson process in applications but also to reveal its central place in the structural theory of point processes. Of course these applications far from exhaust the role of convergence concepts in the general theory of random measures and point processes. Other important applications arise in the discussion of ergodic theorems and convergence to equilibrium in Chapter 12, and in various questions related to Palm theory in Chapter 13 and conditional intensities in Chapter 14. In this chapter we mostly restrict attention to X = Rd , even though extensions to more general locally compact groups are usually possible. Many of these extensions are covered in MKM (1978) and especially MKM (1982), giving systematic extensions of earlier work to the context of a general locally compact group. 131

132

11. Convergence Concepts and Limit Theorems

11.1. Modes of Convergence for Random Measures and Point Processes In this section we examine different possible modes of convergence for a family of point processes or random measures. We generally suppose that the # processes involved are to be thought of as distributions on M# X or NX , where X as usual is a general c.s.m.s. Then the question is this: given a sequence of probability measures {Pn } on M# X , in what sense should the statement Pn → P be understood? Three types of convergence suggest themselves for immediate consideration: strong convergence of probability distributions on M# X (i.e., Pn − P → 0, where P is the variation norm as defined at the end of Section A1.3); weak convergence of probability distributions on M# X [i.e., Pn → P weakly, meaning Pn → P in the sense of weak convergence of measures on the metric space M# X ; see Definition A2.3.I(i)]; and convergence of the finite distributions [i.e., for all suitable finite families of bounded Borel sets A1 , . . . , Ak , the joint distributions of the random variables ξ(A1 ), . . . , ξ(Ak ) under Pn converge weakly to their limit distribution under P]. The consequential matter of convergence of moments is noted before considering in the final part of this section some further questions that arise when we try to relate convergence of measures in M# X and convergence of the associated cumulative processes (distribution functions) in the function space D(0, ∞) or its relatives [see discussion following Example 11.1(c)]. We sometimes adopt a common abuse of terminology by stating that the random measures ξn converge weakly (or strongly) to a limit random variable ξ when all that is meant is the weak (or strong) convergence of their distributions in M# X ; in fact there are no requirements for convergence of the random measures themselves (as elements of or mappings into M# X ). The same abuse applies to “point processes Nn converge to N .” We note first that strong convergence implies weak convergence. Indeed, for any set U ∈ B(M# X ), we have in the notation of Section A1.3 Pn − P ≥ VPn −P (U ) ≥ |Pn (U ) − P(U )|. It follows that strong convergence implies Pn (U ) → P(U ) for all U ∈ B(M# X ), which then implies weak convergence by Theorem A2.3.II. The converse is not true; Example 11.1(a) below serves as a counterexample. Indeed, strong convergence implies that any fixed atom for the limit probability must also be a fixed atom for its approximants. One of the most important applications of convergence in variation norm concerns convergence to equilibrium, or stability properties, of stochastic processes, in which context it is frequently established through the concept of coupling. Two jointly defined stochastic processes X(t) and Y (t) are said to couple if there exists an a.s. finite random variable T (the coupling time) such that X(t) and Y (t) are a.s. equal for all t ≥ T . The basic lemma is as below [see, e.g., Lindvall (1992) or Thorisson (2000) for more extended discussion].

11.1.

Modes of Convergence

133

Lemma 11.1.I. Let X(t, ω), Y (t, ω), be two stochastic processes, defined on a common probability space (Ω, E, P), and taking their values in a common c.s.m.s. V . Denote by Pt , Qt the distributions of X(t), Y (t), respectively, on (V, BV ). Suppose that X and Y couple, with coupling time T . Then Pt − Qt ≤ 2 P{T > t}. Proof. To establish convergence in variation norm, we first note that " " " " " Pt − Qt = sup " f (v) Pt (dv) − f (v) Qt (dv)"" , f :f ≤1

the supremum being taken over all bounded measurable functions f on (V, BV ) with f = supv∈V |f (v)| ≤ 1. (Indeed, the supremum is achieved when f = IU + − IU − in the Jordan–Hahn decomposition of Pt − Qt : see Theorem A1.3.IV.) We then have " " " " " " "f X(t, ω) − f Y (t, ω) " P(dω) " f (v) Pt (dv) − f (v) Qt (dv)" ≤ " " Ω " " "f X(t, ω) − f Y (t, ω) " P(dω) = T ≤t " " "f X(t, ω) − f Y (t, ω) " P(dω) + T >t

≤ 2 f P{T > t} ≤ 2 P{T > t}. This lemma can be applied to point processes by associating X(t) ∈ NX# with the shifted version St N of a point process initially defined on R+ : see the further discussion on convergence to equilibrium in Section 12.5 where the weaker concept of shift-coupling is also discussed (see around Lemma 12.5.IV). Although convergence in variation norm is generally the more difficult to establish, once available it is very convenient to use. This is because in addition to its properties as a norm, it also respects convolution in the sense that µ ∗ ν ≤ µ ν (see Exercise 11.1.1). In practice, it is often convenient to work not with the norm on the full space M# X , but rather with the family of norms on each of the spaces of totally finite measures MA for bounded A ∈ BX . We have already met examples of this approach in the discussion of convergence to equilibrium of renewal and Wold processes (see in particular Corollary 4.4.VI). Yet another possibility is to look at norm convergence rather than weak convergence for the fidi distributions, an issue that arises in applying the Stein–Chen approach to establishing convergence to Poisson distributions. Although the main emphasis in our discussions is on weak convergence, an illustration of the sort of analysis required is given in the second half of

134


Section 11.3, where some of the preliminary inequalities are derived and used to strengthen the convergence statements in the discussion of thinning. Other examples are given from time to time in the exercises and elsewhere. We turn now to the main topic of this section, namely the weak convergence of random measures and point processes, and its relation to the weak convergence of the finite-dimensional distributions. In connection with the latter concept, we call the Borel set A a stochastic continuity set for the measure P if P{ξ(∂A) > 0} = 0, equivalently P{ξ(∂A) = 0} = 1. Without a restriction to sets that are continuity sets for the limit measure, convergence of the fidi distributions would be too strong a concept to be generally useful as the following example shows. Example 11.1(a) Convergence and continuity sets. Let ξn consist of exactly one point in each interval (k, k + 1), k = 0, ±1, ±2, . . . , with each such point uniformly distributed over (k, k + 1/n). Then as n → ∞, we would like to say that the sequence converges to the deterministic point process with one point at each integer. However, if A = (0, 1), we have Pn {ξ(0, 1) > 0} = 1 but P{ξ(0, 1) > 0} = 0. Thus, we can expect difficulties to arise in the definition if the limit random measure has fixed atoms, and these atoms lie on the boundary of the set A considered in the finite-dimensional distribution. Similar but more general examples can readily be constructed. Granted the need for the restriction it is important to know that there are ‘sufficiently many’ stochastic continuity sets. Given P, let SP denote the class of stochastic continuity sets for P. From the elementary properties of set boundaries (see Proposition A1.2.I), it is clear that SP is an algebra (see Exercise 11.1.2). The following lemma is then sufficient for most purposes. Lemma 11.1.II. Let X be a c.s.m.s., P a probability measure on B(M# X ), and SP the class of stochastic continuity sets for P. Then for all x and, given x, for all but a countable set of values of r > 0, Sr (x) ∈ SP . Proof. It is enough to show that for each finite positive ε, δ, and R, the set of r in [0, R] satisfying

P ξ ∂Sr (x) > δ > ε is finite. Suppose the contrary, and let ε, δ, and R be such that for some countably infinite set {r1 , r2 , . . .} of distinct values of r in 0 ≤ r ≤ R < ∞, P(Bi ) > ε for i = 1, 2, . . . , where Bi = ξ: ξ ∂Sri (x) > δ . Then

ε ≤ lim sup P(Bi ) ≤ P(lim sup Bi ) ≤ P ξ SR (x) = ∞ i→∞

because

i→∞

ξ SR (x)

∞ ≥ ξ ∂Sri (x) = ∞ i=1

whenever ξ ∂Sri (x) ≥ δ > 0 for an infinite number of values of i. This contradicts the bounded finiteness of ξ.

11.1.


135

Corollary 11.1.III. The stochastic continuity sets of P form an algebra that contains both a dissecting ring and a covering ring. Proof. Inspection shows that the constructions of a dissecting system at Proposition A2.1.IV and of a covering ring before Corollary A2.3.III are not affected by replacing any sphere Sα (di ) that is not a stochastic continuity set by a marginally smaller sphere Sα (di ) that is. Because the remaining stages of the constructions involve only finite unions, intersections, and differences, they do not lead out of the algebra of such continuity sets. We can now state the following more formal definition. Definition 11.1.IV. The sequence {ξn } converges in the sense of convergence of fidi distributions if for every finite family {A1 , . . . , Ak } of bounded continuity sets Ai ∈ BX the joint distributions of {ξn (A1 ), . . . , ξn (Ak )} converge weakly in B(Rk ) to the joint distribution of ξ(A1 ), . . . , ξ(Ak ). The mapping that takes a general element ξ of M# X into ξ(A), where A is a bounded Borel set, is measurable, essentially by definition of the σ-algebra B(M# X ), but need not be continuous. To see this last point it is enough to consider variants on Example 11.1(a), where the sequence of measures {ξn } converges in the w# -topology in M# X to a limit measure ξ that has an atom on the boundary ∂A. However, only those measures ξ giving nonzero mass to ∂A can act in this way as discontinuity points of the mapping ξ → ξ(A), for if ξ(∂A) = 0 and ξn →w# ξ, then by Theorem A2.3.II (see also Proposition A2.6.II), ξn (A) → ξ(A) . Now let {Pn } be a sequence of probability distributions, and suppose that Pn → P weakly, and that A is a stochastic continuity set for P. This last is just another way of saying that the set D of discontinuity points for the mapping fA : ξ → ξ(A) satisfies the condition P(D) = 0. It then follows from the extended form of the continuous mapping theorem (Proposition A2.3.V) that Pn (fA−1 ) → P(fA−1 ), or in other words that the distribution of ξ(A) under Pn converges to its distribution under P. A similar argument applies to any finite family of bounded Borel sets {A1 , . . . , Ak } satisfying P{ξ(∂A) = 0} = 1 and hence leads to the following lemma. Lemma 11.1.V. Weak convergence implies weak convergence of the finitedimensional distributions. What is more surprising is that the converse of this statement is also true, so that for random measures and point processes, the concepts of weak convergence and convergence of fidi distributions are equivalent. This result, which constitutes the main theorem of this section, is proved at Theorem 11.1.VII. In preparation for this result, we set out in explicit form the conditions for a family of probability measures on B(M# X ) to be uniformly tight (cf. Appendix A2.4, in particular Theorem A2.4.I). In the proposition below T refers to an arbitrary index set, not necessarily countable.

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Proposition 11.1.VI. For a family of probability measures {Pt , t ∈ T } on B(M# X ) to be uniformly tight, it is necessary and sufficient that, given any closed sphere S ⊂ X and any ε, δ > 0, there exists a real number M < ∞ and a compact set C ⊆ S such that, uniformly for t ∈ T , Pt {ξ(S) > M } < ε,

(11.1.1)

Pt {ξ(S \ C) > δ} < ε.

(11.1.2)

If X is locally compact, and in particular if X = Rd , the second condition is redundant. Proof. Uniform tightness means that, for each ε > 0, there exists a compact set K ∈ B(M# X ) such that Pt (K) > 1 − ε for all t ∈ T . From Proposition A2.6.IV and Theorem A2.4.I, K is compact if there exists a sequence of closed spheres S n ↑ X such that for each δ > 0 and n < ∞ there exist constants Mn and compact sets Cn,δ ⊆ S n such that for all ξ ∈ K, (a) ξ(S n ) ≤ Mn , and (b) ξ(S n \ Cn,δ ) ≤ δ. Effectively, (11.1.1) and (11.1.2) are just reformulations of (a) and (b). Indeed, supposing first that (11.1.1) and (11.1.2) are satisfied, choose any sequence of closed spheres S n ↑ X such that each S n is a stochastic continuity set for P. From (11.1.1) we choose Mn such that Pt {ξ(S n ) > Mn } < ε/2n+1 , ⊆ S n so that and from (11.1.2) we choose the compact set Cmn Pt {ξ(S n \ Cmn )} < ε/2m+n+2 .

Define the sets, for n, m = 1, 2, . . . , Qmn = {ξ: ξ(S n \ Cmn ) ≤ m−1 }, ∞ ∞ ( ( K= (Qn ∩ Qmn ).

Qn = {ξ: ξ(S n ) ≤ Mn },

n=1 m=1

By construction, (a) and (b) are satisfied so K is compact, and Pt (K c ) ≤

∞

Pt (Qcn ) +

n=1

∞ ∞ ε ε = ε. Pt (Qcmn ) ≤ + 2n+1 m=1 2m+n+2 m=1 n=1 ∞

Thus, K satisfies all the required conditions. Suppose conversely that the measures Pt , are uniformly tight. Given ε, choose compact K ⊂ M# X and hence deduce the existence of spheres S n ↑ X such that there exist constants Mn for which (a) holds, and, given δ, there exist Cn,δ such that (b) holds. Given any S, choose n so that S ⊆ S n , set

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137

M = Mn so that (11.1.1) is true, and, given δ, set C = Cn,δ ∩ S so that ξ(S \ C) ≤ ξ(S \ Cn,δ ) and hence (11.1.2) holds. Example 11.1(b) Convergence of one-point processes. Let ξn be the degenerate point process in R in which all the mass is concentrated on the counting measure with a single atom at the point n. Then (11.1.1) holds trivially for all S and ε with M = 2. In fact εn → ε∞ weakly, where ε∞ has all its mass concentrated on the zero random measure 0. Thus, it is important to bear in mind that weak convergence here does not preclude the possibility that the limit point process be everywhere zero. may n Next let ηn = k=1 ξk . Then for all n we have ηn (0, m] ≤ m so that (11.1.1) still holds with M equal to the radius of the sphere S. In this case ηn → η∞ weakly, where η∞ is the deterministic point process at unit rate with an atom ateach positive integer. n Finally, let ζn = k=1 kζn−k . Here condition (11.1.1) fails and no weak limit exists. The next theorem is the main result of this section. It is a striking consequence of the integer-valued character and locally bounded nature of a point process. Theorem 11.1.VII. Let X be a c.s.m.s. and P, {Pn : n = 1, 2, . . .} distri# butions on (M# X , B(MX )). Then Pn → P weakly if and only if the fidi distributions of Pn converge weakly to those of P. Proof. The first part of the theorem has already been proved in Lemma 11.1.V. Because the set of all fidi distributions determines a probability measure uniquely, in order to prove the converse it suffices to show that the family {Pn } is uniformly tight, for then every sequence contains a weakly convergent subsequence, and from the convergence of the fidi distributions this must be the limit measure P; thus, all convergent subsequences have the same limit, and so the whole sequence converges. To establish tightness, we use the assumption that the fidi distributions converge for stochastic continuity sets of P to show that (11.1.1) and (11.1.2) hold for any given S, ε, and δ. We start by choosing S ⊇ S to be a stochastic continuity set not only for P but also for each of the Pn , n = 1, 2, . . . . (Because by Lemma 11.1.II only countable sets of exceptional radii are involved, this choice can always be made.) Furthermore, we can choose values M that are continuity points for the distribution of ξ(S ) under P and for which P{ξ(S ) > M } < 12 ε and Pn {ξ(S ) > M } → P{ξ(S ) > M } as n → ∞. Thus, for n > n0 say, we have Pn {ξ(S) > M } < ε, and by increasing M if necessary we can ensure that this inequality holds for all n. This establishes (11.1.1).

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Again working only with spheres that are stochastic continuity sets for P, choose spheres Srj (xi ) ≡ Sij centred on the points xi (i = 1, 2, . . .) of a separability set and with radii rj ≤ 2−j . Define Cij = S ij ∩ S . Because ξ

K

Cij

↑ ξ(S )

(K → ∞),

i=1

we can choose Kj so that, with Cj =

Kj i=1

Cij ,

P{ξ(S − Cj ) ≥ δj } ≤ ε/2j+1 , where δj ≤ δ/2j is chosen to be a continuity point of the distribution of ξ(S −Cj ) under P. Again using the weak convergence of the fidi distributions, and increasing the value of Kj if necessary, we can ensure as before that the similar inequality (11.1.3) Pn {ξ(S − Cj ) ≥ δj } ≤ ε/2j ∞ holds for all n. Now define C = j=1 Cj . Then C is closed, and by construction it can be covered by a finite number of ε-spheres for every ε > 0, so by Proposition A2.2.II, C is compact. We have moreover from (11.1.3) that, for every n, ∞ (S − Cj ) > δ Pn {ξ(S ) − ξ(C) > δ} = Pn ξ j=1

≤

∞

Pn {ξ(S − Cj ) > δ/2j }

j=1

≤

∞ j=1

Pn {ξ(S − Cj ) > δj } ≤

∞ ε = ε, j 2 j=1

thereby establishing (11.1.2). Thus, both conditions of Proposition 11.1.VI are satisfied, and we conclude that the family {Pn } is tight. Several equivalent conditions for weak convergence can be derived as corollaries or minor extensions to the above theorem. The last condition represents a minor weakening of the full strength of convergence of fidi distributions. Proposition 11.1.VIII. Each of the following conditions is equivalent to the weak convergence Pn → P weakly, where in (i) and (ii), f ranges over the space of continuous functions vanishing outside a bounded set. (i) The distribution of X f dξ under Pn converges weakly to its distribution under P. (ii) The Laplace functionals Ln [f ] ≡ EPn exp − X f (x) ξ(dx) converge pointwise to the limit functional L[f ]. (iii) For point processes, the p.g.fl.s Gn [h] converge to G[h] for each continuous h ∈ V0 .

11.1.


139

(iv) For every finite family {A1 , . . . , Ak } from a covering semiring of bounded continuity sets for the limit random measure ξ, the joint distributions of {ξn (A1 ), . . . , ξn (Ak )} converge weakly in B(Rk ) to the joint distribution of {ξ(A1 ), . . . , ξ(Ak )}. Proof. For any f as described, the mapping defined by Φf (ξ) =

f (x) ξ(dx); X

is continuous at ξ provided ξ Z(f ) = 0, where Z(f ) is the set of discontinuities of f . Hence, in particular, Φf is continuous for all ξ whenever f itself is continuous. Thus, the distributions of Φf (ξ) under Pn converge weakly to its distribution under P. Now suppose that f is a function of the form i ci IAi (x), where i |ci |< ∞ contiand {Ai } is a bounded family of bounded Borel sets that are stochastic nuity sets for P. Convergence of the distributions of the integrals X f dξ for all such functions f is equivalent to the joint convergence in distribution of ξ(A1 ), . . . , ξ(Ak ) for every finite integer k, that is, to fidi convergence. Because such functions can be approximated by continuous functions, as, for example, in the proof of Theorem A2.3.II, it follows that (i) implies convergence of the fidi distributions and hence weak convergence. Condition (ii) is equivalent to (i) by well-known results on Laplace transforms. Because f (x) = − log h(x) is a function as in (i) if and only if h is continuous and h ∈ V0 , (iii) is equivalent to (ii) when the distributions Pn correspond to point processes. Establishing the sufficiency of the last condition is a matter of verifying that the constructions in the proof of Theorem 11.1.VII can be carried through with sets Ai drawn from the covering semiring (or, more generally, from the ring it generates, in as much as it is clear that the convergence carries over to sets taken from this generated ring). Because by definition of a covering ring each open sphere can be approximated by sets in the ring, both constructions in the first part of the proof can be so modified, implying that the sequence {Pn } is uniformly tight. If {Pnk } is any weakly convergent subsequence from {Pn }, with limit P say, then P and P must have the same fidi distributions for sets drawn from the covering semiring. Then from Proposition 9.2.III it follows that P = P , and hence as before that Pn → P weakly. In the case of point processes, even sharper versions of condition (iv) are possible when the processes are simple, through the use of the avoidance function (see Theorem 9.2.XII). In this case the limit measure must correspond to a simple point process if it is to be uniquely characterized by the avoidance function, and condition (ii) below is such an additional requirement about asymptotic orderliness. Several variants are now possible; the following is perhaps the simplest.

140


Proposition 11.1.IX. Let {Pn : n = 1, 2, . . .}, P be distributions on NX# with P corresponding to a simple point process, and suppose that R ⊆ SP is a covering dissecting ring. In order that Pn → P weakly, it is sufficient that (i) Pn {N (A) = 0} → P{N (A) = 0} as n → ∞ for all bounded A ∈ R; and (ii) for all bounded A ∈ R and partitions Tr = {Ari : i = 1, . . . , kr } of A by sets of R, kr Pn {N (Ari ) ≥ 2} = 0. (11.1.4) lim sup sup n→∞

Tr i=1

Proof. In view of Theorem 9.2.XII it is enough to show that under the stated conditions the family {Pn } is uniformly tight and that the limit of any weakly convergent subsequence must be a simple point process. Let S be a closed sphere in R, and in (ii) take A = S. Observing that {N (S) > kr } implies {N (Ari ) ≥ 2 for at least one i}, kr

Pn {N (Ari ) ≥ 2} ≥ Pn {N (S) > kr }.

i=1

Given ε > 0, (11.1.4) implies that the sum on the left-hand side here is bounded by ε for n ≥ n0 , hence (by adjusting kr if necessary) for all n, and thus that the first condition for uniform tightness holds. Condition (11.1.2) here can be stated in the following form. Given ε > 0, there exists a compact set C such that Pn {N (S − C) = 0} > 1 − ε for n = 1, 2, . . . . Choose C so that for the limit distribution we have P{N (S − C) = 0} > 1 − 12 ε. From assumption (i) we have Pn {N (S − C) = 0} → P{N (S − C) = 0} as n → ∞, from which the required inequality (11.1.2) holds for all sufficiently large n, and hence (by increasing C if necessary) for all n. Thus, both conditions for uniform tightness of {Pn } are satisfied. Now let {Pnk } be any weakly convergent subsequence from the family {Pn }, with limit P say, so that from Theorem 11.1.VII all fidi distributions converge to those of P ; hence, from (i) of the proposition, for A ∈ R, P{N (A) = 0} = P {N (A) = 0}. From this result it follows not necessarily that P = P but merely that P{N (A) = 0} = P {N ∗ (A) = 0}

or

P = (P )∗ ,

where N ∗ is the support point process of N and (P )∗ is its distribution (see Corollary 9.2.XIII). However, we have kr

P {N (Ari ) ≥ 2} =

i=1

ki i=1

lim Pnk {N (Ari ) ≥ 2}

k→∞

so that from (11.1.4) lim sup Tr

kr i=1

P {N (Ari ) ≥ 2} ≤ lim sup

kr

k→∞ Tr i=1

Pnk {N (Ari ) ≥ 2} = 0,

11.1.


141

from which it follows (Proposition 9.3.XII) that P is simple, and hence that P = (P )∗ = P. So all weakly convergent subsequences have limit P , and thus Pn → P weakly. Thus far we have considered essentially convergence of probability measures of point processes, but what of their moments? Uniform integrability is an analogous ‘bounding’ condition that ensures that a sequence of moment measures may converge, as in the assertion below (the proof is standard and left to the reader; see also Exercise 11.1.3). Proposition 11.1.X. Let {Nn } be a weakly convergent sequence of point processes on the c.s.m.s. X with limit N and for which the first moment measures {Mn (A)} = {E[Nn (A)]} are finite for bounded A ∈ BX . These moment measures converge in the w# -topology to the first moment measure M of N if and only if for some sequence of spheres Sk ↑ X , E[Nn (Sk )I{Nn (Sk ) ≥ a}] → 0 (a → ∞)

uniformly in n.

As an illuminating example of the use of weak convergence arguments, we outline below the proof used by Brémaud and Massoulié (2001) to establish the existence of a type of Hawkes process [Examples 6.3(c), 7.2(b)] with no immigration component: the population is both maintained and balanced purely via its own offspring and their locations. Example 11.1(c) Existence of Hawkes process without ancestors [Brémaud and Massoulié (2001)]. We consider a sequence of ordinary Hawkes processes with infectivity functions (intensity measure of offspring process) of the form µ (x) = (1− )µ(x) , µ(x) dx = 1, with associated immigration rates ν = ν. All these processes have the same mean rate 1−

ν ν = = ν, 1 − (1 − ) µ (x)dx

so it is plausible that as → 0, the processes may converge to some limit process with mean rate ν and conditional intensity of the form

t

λ(t) = −∞

µ(t − u) N (du).

(11.1.5)

Our aim is to establish that such convergence does indeed occur. In view of Theorem 11.1.VI, it is sufficient to show that the finite dimensional distributions converge to a consistent limit. Fix an interval [a, b] and consider the total number N (a, b) of points of the approximating process falling within this interval. From stationarity of the approximating process we have from Markov’s inequality, uniformly for all > 0, Pr{N (a, b) > M } ≤

ν(b − a) E[N (a, b)] = , M M

142


so that the left-hand side converges uniformly to 0 as M → ∞. It follows that the distributions of N (a, b) are uniformly tight. It is easily seen from this that all fidi distributions for the processes N restricted to (a, b) are uniformly tight. We can therefore extract a subsequence such that the fidi distributions on (a, b) converge weakly to some limit process on (a, b). By covering the real line with a family of such intervals, we can even find a subsequence along which all the fidi distributions converge weakly. Then it follows from Theorem 11.1.VI that the point processes themselves converge weakly to some limit point process. Exercise 11.1.4 gives a more general version of this argument. In this model, the weak convergence just established also implies convergence of the expressions for the conditional intensities. To see this, consider expressions of the form which define the conditional intensity, namely E[N (a, b)IA ] = E

b

IA λ(u) du ,

(11.1.6)

a

where A ∈ Ha , the internal (minimal) history for the process (see the discussion in Chapter 7, or later in this volume in Chapter 14). In particular, it is enough to consider A of the form A = {N : N (C1 ) = n1 , N (C2 ) = n2 , . . . , N (Ck ) = nk } for integers k, n1 , . . . , nk and sets Ci ∈ (−∞, a), because sets of this kind generate Ha . In this model, the existence of densities for the Poisson processes of new immigrants and offspring implies that all bounded Borel sets C ∈ R are continuity sets. It follows that the function N → N (a, b)IA is continuous in the weak# topology in BN (see the discussion following Lemma 11.1.III), and hence, from the continuity theorem (Proposition A2.3.V), that E[N (a, b)IA ] → E[N (a, b)IA ]. Similar arguments apply also to the more complex expressions on the right-hand side of (11.1.6), which for the approximating process N we can write as E a

b

IA λ (x) dx = E a

b

IA ν +

u

−∞

µ (x − u) N (du) dx .

Again the expectations converge to the corresponding form for the limit process, and serve to identify the conditional intensity for the limit process with the form (11.1.5). Nothing in the argument so far precludes the possibility that the limit point process is a.s. equal to the zero counting measure. Indeed, Brémaud and Massoulié show that if the function µ(x) is ‘light-tailed’ (decays at an exponential rate or faster) then the only possible limit processes are degenerate (zero or infinite; see Exercise 11.1.5). Because E[N (a, b)] = ν(b − a), a sufficient condition to ensure that the limit process is nontrivial is that the limit of the first moment measures should be the first moment measure of

11.1.


143

the limit process. For this, a uniform integrability condition is needed for the quantities N (a, b)Pr{N (a, b) > M }, as indicated in Exercise 11.1.3. Because for any random variable X ≥ 0, E[XIX>M ] ≤ E[X 2 ]/M , a sufficient further condition is boundedness of the variances var[N (a, b)]. This requires a careful examination of the spectral properties of N ; for details see Exercise 11.1.6, where it is shown that the variances remain bounded provided the infectivity function satisfies the ‘heavy-tail’ conditions that for some 0 < α < 12 , t1+α µ(t) is bounded on t ≥ 0, and t1+α µ(t) approaches a finite limit as t → ∞. We conclude this section with a few remarks concerning the relation between weak convergence of random measures with state space X = R+ and weak convergence of the associated cumulative processes as elements in D(0, ∞). Here the cumulative function associated with a measure µ on R+ is defined by Fµ (x) = µ{(0, x]} 0 < x < ∞. Such functions are monotonic increasing, right-continuous with left limits, and therefore define a subspace of D(0, ∞). Because the metrics in M# (R+ ) and D(0, ∞) are obtained from compounding the analogous metrics over finite intervals, it is sufficient to compare the behaviour of the two metrics over a common finite interval, which for convenience we take as the interval (0, 1]. In both cases we are effectively concerned with the distance between two cumulative functions F , G over (0, 1). Weak convergence of a family of measures on (0, 1] is equivalent to convergence of the cumulative functions with respect to the Lévy metric ρL , where ρL (F, G) is defined as the infimum of values ε such that for all x ∈ (0, 1], G(x − ε) − ε ≤ F (x) ≤ G(x + ε) + ε [we take G(−y) = 0, G(1 + y) = G(1) (y > 0) for the purposes of this definition]. On the other hand convergence of the distribution functions in D(0, 1) is equivalent to convergence with respect to the Skorohod metric ρS , where ρS (F, G) is defined as the infimum of values ε such that there exists a continuous mapping λ of [0, 1] onto [0, 1], with λ(0) = 0, λ(1) = 1, for which sup |x − λ(x)| < ε, 0≤x≤1

sup |F λ(x) − G(x)| < ε. 0≤x≤1

The statements ρL (F, G) < ε and ρS (F, G) < ε both require F and G to be close in the sense that uniformly for x ∈ (0, 1], the value of F (x) differs from a possibly slightly shifted value of G(x) by less than ε, the shift also not being allowed to exceed ε. In the case of the Skorohod metric, the degree of shift is controlled by the function λ(x), whereas in the Lévy case it is constrained not to exceed ε but is otherwise not related from one x to any other. In both cases the statement ρ(Fn , F ) → 0 is equivalent to the requirement that Fn (x) → F (x) at all continuity points of x (see Exercises 11.1.7–8).

144


Provided therefore that discussion is restricted to the subspace of cumulative processes, the two types of convergence are equivalent. Equivalently, the mapping from M(0, 1) into D(0, 1), which takes the measure ξ into its cumulative function Fs , is both ways continuous. The continuous mapping Theorem A2.3.V therefore yields the following result. Lemma 11.1.XI. A sequence of random measures {ξn } on M# (R+ ) converges weakly to a random measure ξ if and only if the corresponding sequence of cumulative processes Fξn converges weakly in D(0, ∞) to the cumulative process Fξ . Extensions to Rd can be obtained in terms of the concepts described by Straf (1972). From the point of view of weak convergence, it is therefore immaterial whether we deal with the random measures directly or the stochastic processes defined by the associated cumulative functions. When rescaling is involved, as, for example, in the central limit theorem, the limits need no longer correspond to random measures, and functional limit theorems can be obtained. An example is given in Proposition 12.3.X.

Exercises and Complements to Section 11.1 11.1.1 For totally finite signed measures µ, ν on a c.s.m.s. X show that the variation norm · (see Section A1.3) has the following properties: (a) αµ = |α| µ for every scalar α. (b) µ + ν ≤ µ + ν, with equality if the supports of µ and ν are disjoint. (c) µ ∗ ν ≤ µ ν, with equality if both measures are of constant sign (and thus µ = |µ(X )|). 11.1.2 (a) Let N be a point process on R and R the ring generated by half-open intervals such as (a, b], so that for A ∈ R, ∂A consists of the finite set of endpoints of the constituent intervals of A. Deduce that A is a stochastic continuity set unless any of the endpoints of its constituent intervals happen to be fixed atoms of the process. (b) Deduce that if N on R is stationary, it has no fixed atoms. (c) In general, when P is a probability measure on M# X , the stochastic continuity sets for P form an algebra. 11.1.3 Convergence of moment measures. Let N be a point process that is the limit of the weakly convergent sequence {Nn } of point processes. Show that M (A) = E[N (A)] ≤ lim inf n→∞ E[Nn (A)] = lim inf n→∞ Mn (A). In order to have equality here, some condition such as the uniform integrability condition of Proposition 11.1.X is needed: weak convergence and existence and boundedness of the moment measures are not enough to ensure their vague convergence. Suppose that the point processes Nn are on R and defined as follows. Choose with probability 1 − n−1 the randomized unit lattice point process with points at {r + U : r = 0, ±1, . . .} and U a random variable uniformly distributed on (0, 1), and with probability n−1 the point process with points

11.1.


145

on the lattice {r/n: r = 0, ±1, ±2, . . .} in which the centre of the lattice is located uniformly at random over the unit interval. This sequence {Nn } has the following properties. (i) It converges weakly to the randomized unit lattice point process. (ii) E[Nn (A)] = (2 − n−1 )(A) for n = 1, 2, . . . . (iii) E[N∞ (A)] = (A) for all bounded Borel sets A. 11.1.4 When X = Rd , every sequence of random measures with locally uniformly bounded first moment measures is relatively weakly compact (i.e., contains a weakly convergent subsequence). In particular, any sequence of random measures with uniformly bounded mean densities is relatively weakly compact. [Hint: Use Markov’s inequality to show that for each bounded Borel set A there exists a constant CA < ∞ such that Pr{ξ(A) > M } ≤ CA /M for all M > 0.] 11.1.5 For a Hawkes process as in Example 11.1(c), show that if xµ(x) dx ≤ ∞ then a process with conditional intensity (11.1.5) cannot have a finite mean. [Hint: Pr{N (R+ ) = 0} = E[Pr{N (R+ ) = 0 | H0 }] ≥ exp(−λ t µ(t) dt); this yields a contradiction if the process is ergodic and nonzero.] 11.1.6 Hawkes process without ancestors [see Example 11.1(c)]. ∞ (i) Suppose the density µ satisfies the conditions 0 µ(t) dt = 1 and for finite 1+α µ(t) ≤ R and limt→∞ t1+α µ(t) = r. Show positive R and r, supt>0 t that ∞ iu e −1 µ(ω) − 1]ω −α = r du. lim [ˆ ω→0 u1+α 0 (ii) Let V (T ) denote the variance of the approximating process N over a fixed interval (0, T ). Show that as → 0, V (T ) remains bounded, and V (T ) →

λ 2π

∞

−∞

|eiωT − 1|2 dω. ω 2 |1 − µ ˆ(ω)|2

(iii) Show that the limit variance obtained above is, in fact, the variance of the limiting process described in the example. [Hint: For (ii), use (i) and equation (8.2.10) for the spectral density of a Hawkes process. For (iii), see Lemma 1 of Brémaud and Massoulié (2001).] 11.1.7 Let ρL , ρS refer to the Prohorov and Skorohod metrics, respectively, on the space of finite measures on (X , BX ) (the Prohorov metric reduces to the Lévy metric on R). Let {Nn }, N be finite counting measures on X . Prove that ρL (Nn , N ) → 0 if and only if for all sufficiently large n, Nn has the same number of atoms as N , and the locations of the atoms in N , converge to their locations under N . Deduce that ρL (Nn , N ) → 0 implies ρS (Nn , N ) → 0. [Hint: See, e.g., Straf (1972).] 11.1.8 Write d0 (F, G) = supx∈R |F (x) − G(x)| for the sup metric on the space of d.f.s F , G on R, and write µF , µG for the measures generated by such d.f.s. Prove that 2d0 (F, G) ≤ µF − µG .

146


11.2. Limit Theorems for Superpositions Limit theorems for superpositions of point processes go back at least as far as Palm (1943) and continued in Khinchin (1955) in developing a simple version of Proposition 11.2.VI below for the superposition of a large number of independent identically distributed stationary point processes on R. Under suitable conditions, rescaled versions of the resulting processes have a Poisson process limit. Extensions through work of Ososkov (1956), Franken (1963) and Grigelionis (1963) led ultimately to Theorems 11.2.III and 11.2.V in which rescaling is subsumed by convergence of sums in a uniformly asymptotically negligible array. The formal setting for studying the sum or superposition of a large number of point processes or random measures is a triangular array {ξni : i = 1, . . . , mn ; n = 1, 2, . . .} and its associated row sums ξn =

mn

ξni ,

n = 1, 2, . . . .

i=1

If for each n the processes {ξni : i = 1, . . . , mn } are mutually independent, we speak of an independent array, and when they satisfy the condition that for all ε > 0 and all bounded A ∈ BX lim sup P{ξni (A) > ε} = 0,

(11.2.1)

n→∞

the array is uniformly asymptotically negligible, or u.a.n. for short. In the case of a triangular array of point processes, the u.a.n. condition (11.2.1) reduces to the simpler requirement that lim sup P{Nni (A) > 0} = 0.

n→∞

(11.2.2)

i

Note that an independent u.a.n. array is called infinitesimal in MKM (1978, Section 3.4), a null-array in Feller (1966) and Kallenberg (1975, Chapter 6), and holospoudic in Chung (1974, Section 7.1). The terminology u.a.n. comes from Loève (1963) (Loève in fact wrote uan). Although this formal setting can be extended (see Exercise 11.2.3) and the notation simplified by taking mn = ∞, from which the finite case is obtained by assuming all but finitely many elements to be zero, we retain the setting with mn < ∞ for the sake of familiarity. In the discussion that follows the reader will doubtless observe the very close analogy between the results developed for point processes and the classical theory for sums of i.i.d. random variables in R. This is hardly surprising, for a point process is just a particular type of random measure and a random measure is just a random variable taking its values on the metric Abelian group of boundedly finite signed measures on the state space X . As such it comes under the extension of the classical theory developed, for example, in Parthasarathy (1967, Chapter 4). We develop results for point processes

11.2.

Limit Theorems for Superpositions

147

directly; however, the reader may find it useful to bear the classical theory in mind as a guide, as for example at the end of this section in reviewing the corresponding results for random measures. We start with a preliminary result on the convergence of infinitely divisible n : n = point processes, continuing with notation from Section 10.2. Let {Q a 1, 2, . . .} denote a sequence of KLM measures (Definition 10.2.IV) and Q limit measure. We cannot immediately speak of the weak convergence of firstly because the KLM measures are only σ-finite in general, and n to Q, Q secondly because they are only defined on the Borel subsets of the space N0# (X ) ≡ NX# \{N (X ) = 0}, which is not complete. To define an appropriate n }, recall that for each modification of weak convergence for the sequence {Q A bounded Borel set A the KLM measure Q induces a totally finite measure Q # A on the space N0 (A) of nonzero counting measures on A. Extend Q to the (A) = Q A on N # (A) and Q (A) {N : N (A) = 0} = 0. whole of NA# by setting Q 0 n : n = 1, 2, . . .} conDefinition 11.2.I. The sequence of KLM measures {Q verges Q-weakly to the KLM measure Q (i.e., Qn → Q Q-weakly), if for every the extended bounded Borel set A that is a stochastic continuity set for Q, (A) (A) measures Qn converge weakly to Q . This requirement can be spelt out more explicitly in terms of fidi distributions or the convergence of functionals. Thus, it is equivalent to requiring that for every finite family of bounded Borel sets Ai ∈ SQ (i = 1, . . . , k; k ≥ 2), n N : N (Ai ) = ji (i = 1, . . . , k), k ji > 0 Q i=1

: N (Ai ) = ji (i = 1, . . . , k)}. → Q{N

(11.2.3)

A more convenient form for our purposes is the following. For every continuous function h ∈ V(X ) equal to one outside some bounded Borel set A, as n → ∞, n (dN ) exp log h(x) N (dx) Q {N (A)>0}

X

→

exp {N (A)>0}

log h(x) N (dx) Q(dN ). (11.2.4)

X

These restatements are immediate consequences of the conditions for weak (A) convergence developed in Section 11.1 applied to the measures Q n , which, although not probability measures, are totally finite so that the framework for weak convergence remains intact. Proposition 11.2.II. (a) The set of infinitely divisible distributions is closed in the topology of weak convergence in NX# . (b) If {Pn : n = 1, 2, . . .} and P are infinitely divisible distributions on NX# , the corresponding KLM measures, then Pn → P weakly if and n }, Q and {Q Q-weakly. only if Qn → Q

148


Proof. Suppose that Pn → P weakly and that the Pn are infinitely divisible. Take any integer k, and observe that if Pn has p.g.fl. Gn [·], then (Gn [·])1/k n (·), where Q n is is also a p.g.fl. and corresponds to the KLM measure k −1 Q the KLM measure of Pn . When Gn [h] → G[h] for all continuous h ∈ V(X ), it follows that (Gn [h])1/k → (G[h])1/k , so that (G[h])1/k is a p.g.fl. for every integer k, and hence P is infinitely divisible. Using the p.g.fl. representation (10.2.9), we have for all continuous h ∈ V(X ), n (dN ) exp log h(x) N (dx) − 1 Q N0# (X )

X

→

N0# (X )

exp X

log h(x) N (dx) − 1 Q(dN ).

we can approximate the step Taking A to be a stochastic continuity set for Q, function h(x) = 1 − IA (x) arbitrarily closely by h ∈ SQ and thus conclude that : N (A) > 0}, n {N : N (A) > 0} → Q{N Q as the exponential term above vanishes if N (A) > 0. By subtraction we then have that the integrals n (dN ) exp log h(x) N (dx) Q {N (A)>0}

X

converge as required at (11.2.4). n → Q Q-weakly, the argument can be reversed. Conversely, when Q Given a point process with distribution P and p.g.fl. G[ · ], define its Poisson approximant [corresponding to the accompanying law in the classical theory— see, e.g., Parthasarathy (1967, VI.6)] to be the Poisson randomization with distribution P ∗ and p.g.fl. G∗ [ · ] given by all h ∈ V(X ) . (11.2.5) G∗ [h] = exp(G[h] − 1) More generally, given a triangular array {Nni }, the corresponding Poisson ∗ ∗ , with distributions Pni and p.g.fl.s G∗ni [ · ]; approximants are given by Nni ∗ when the Nni are independent, take the Nni to be independent also. Because P ∗ {N (A) > 0} = 1 − exp[−P{N (A) > 0}] ≤ P{N (A) > 0}, ∗ } is u.a.n. whenever {Nni } is a u.a.n. array (see also the triangular array {Nni Exercises 11.2.1–2). The following theorem is basic for point processes.

Theorem 11.2.III. Let {Nni : i = 1, . . . , mn ; n = 1, 2, . . .} be an indepen∗ } an independent array of corresponding Poisson apdent u.a.n. array, {Nni proximants, and N an infinitely divisible point process with KLM measure Then the following assertions are equivalent. Q.

11.2.

(i) (ii) (iii)


mn i=1

mn

149

Nni → N weakly.

∗ Nni → N weakly. (0) i=1 Pni → Q Q-weakly. i=1

mn

(0)

[In (iii), Pni is the restriction of Pni , the distribution of Nni , to N0# (X ).] Proof. Recall the simple inequalities, valid for 0 ≤ 2αi ≤ 1 (i = 1, . . . , mn ) 0 ≤ − log

mn

(1 − αi ) −

i=1

mn

αi ≤

i=1

mn

αi2

≤

mn

i=1

αi

max αi . (11.2.6) i

i=1

We apply these inequalities with αi = Gni [h], where h(x) = 1 for x outside some stochastic continuity set A for P, the distribution of N . From the u.a.n. condition, 1 − Gni [h] ≤ Pr{Nni (A) > 0} ≤ 12 (i = 1, . . . , mn ) for n sufficiently large, and thus mn i=1

mn mn 1 − Gni [h] ≤ 1 − Pr{Nni (A) > 0} Pr{Nni (A) > 0} ≤ − log i=1

mn Nni (A) = 0 = − log Pr

i=1

i=1

→ − log Pr{N (A) = 0} < ∞, so that the left-hand sum here is uniformly bounded (over subsets of A) for n sufficiently large. It follows from (11.2.6) that if one of mn

i=1

Gn [h]

and

exp

mn

1 − Gni [h]

i=1

converges to a finite nonzero limit, then so does the other, and the limits are equal. This implies the equivalence of (i) and (ii) of the theorem. (0) ∗ are infinitely divisible, with KLM measures Pni , so The processes Nni (0) ∗ Pni . By the row sum Nni is infinitely divisible, with KLM measure appealing to Proposition 11.2.II, the equivalence of (ii) and (iii) follows. The arguments used in the proof lead to an alternative formulation of the result in terms of p.g.fl.s (see Exercise 11.2.3). The following result is an easy corollary. Proposition 11.2.IV. A point process is infinitely divisible if and only if it can be represented as the limit of the row sums of a u.a.n. array. The most important application of Theorem 11.2.III is to finding conditions for convergence to a Poisson process.

150


Theorem 11.2.V. The triangular u.a.n. array {Nni : i = 1, . . . , mn ; n = 1, 2, . . .} converges weakly to a Poisson process with parameter measure µ if and only if for all bounded Borel sets A with µ(∂A) = 0, mn

Pr{Nni (A) ≥ 2} → 0

(n → ∞)

(11.2.7)

i=1

and

mn

Pr{Nni (A) ≥ 1} → µ(A)

(n → ∞).

(11.2.8)

i=1

Proof. Recall from Example 10.2(a) that for a Poisson process the KLM is related to the parameter measure µ(·) by measure Q(·) : N (A) > 0}, µ(A) = Q{N itself is concentrated on one-point realizations, so that and that Q : N (X ) > 1} = 0. Q{N It follows from Theorem 11.2.III that if the array converges to a Poisson process, then mn

: N (A) > 0} = µ(A), Pr{Nni (A) > 0} → Q{N

i=1

and

mn

: N (A) ≥ 2} = 0, Pr{Nni (A) ≥ 2} → Q{N

i=1

so the conditions (11.2.7) and (11.2.8) are necessary. Conversely, if (11.2.7) holds for a sequence of sets An ↑ X , we must then : N (X ) > 1} = 0, so that the limit process must be Poisson, and have Q{N (11.2.8) identifies the parameter measure as µ. We remark that as in other applications of weak convergence, it is sufficient, in checking the conditions of the theorem, to let A run through the sets of any covering semiring of continuity sets of µ [see Proposition 11.1.VIII(iv)]. The following special case was the first to be studied and can be regarded as the prototype limit theorem for point processes. Proposition 11.2.VI. Let N be a simple stationary point process on X = R with finite intensity λ, and let Nn denote the point process obtained by superposing n independent replicates of N and dilating the scale of X by a factor n. Then as n → ∞, Nn converges weakly to a Poisson process with parameter measure λ(·), where (·) denotes Lebesgue measure on R.

11.2.


151

Proof. Here we can envisage a triangular array situation in which each Nni (i = 1, . . . , n) has the same distribution as the original process but on a dilated scale. Hence, using Propositions 3.3.I and 3.3.IV, Pr{Nni (0, t] > 0} = Pr{N (0, t/n] > 0} = (λt/n) 1 + o(1) . Summing on i = 1, . . . , n leads to (11.2.8) with µ(·) = λ(·). Similarly, from Proposition 3.3.V, Pr{Nni (0, t] > 1} = Pr{N (0, t/n] > 1} = o(1/n), and again summing on i leads to (11.2.7). The statement and proof need change when X = Rd ; see Exercise 11.2.4. We conclude this section by briefly reviewing some extensions and further developments. Some of the results that we have handled by generating function arguments can be strengthened to give results concerning bounds in variation norm. In particular, there are elegant bounds that follow via the use of Poisson approximants (see Exercises 11.2.1–2). Extensions to the multivariate, nonorderly, and marked point process cases can generally be handled by applying the preceding results to the case where X has the product form X × K for an appropriate mark space K. Example 11.2(a) Convergence to a multivariate independent Poisson process. Suppose there is given a point process in which each point is identifiable as one of a finite set of types 1, . . . , K say. The process can be described by the (k) multivariate processes with component processes Nni (·) (k = 1, . . . , K). We seek conditions for weak convergence of the superpositions to a limit process in which the different types follow independent Poisson processes with parameter measures µk (k = 1, . . . , K). This last process can thus be regarded as a Poisson process on the space X × {1, . . . , K} with overall measure µ such (k) that µk (·) = µ(· × {k}). Similarly, regard the family {Nni : k = 1, . . . , K} as defining a process Nni on X × {1, . . . , K}. To apply Theorem 11.2.V we have to interpret (11.2.7) and (11.2.8), which apply to the overall processes Nni , in terms of the components. We take A at (11.2.7) and (11.2.8) to be a product set of the form B × {1, . . . , K} for some bounded Borel set B that is a stochastic continuity set for each µ1 , . . . , µk ; that is, µk (∂B) = 0

(k = 1, . . . , K).

(11.2.9)

Then (11.2.7) becomes mn i=1

K (k) Pr Nni (B) ≥ 2 k=1

→ 0

(n → ∞),

(11.2.10)

152


which incorporates the requirement (crucial if the limit process is to have independent components) that there should be zero limiting probability for two distinct components each to contribute points to the same bounded B. Similarly, (11.2.8) takes the form that for all bounded Bk for which µk (∂Bk ) = 0, k = 1, . . . , K, mn K K (k) Pr Nni (B) ≥ 1 → µk (Bk ), i=1

k=1

k=1

which in view of (11.2.10) is satisfied if and only if for each bounded B for which µk (∂B) = 0 (k = 1, . . . , K), mn

(k)

Pr{Nni (B) ≥ 1} → µk (B).

(11.2.11)

i=1

Note also that the u.a.n. condition here becomes K (k) lim sup Pr Nni (B) > 0 = 0, n→∞

i

k=1

again apparently incorporating a constraint on the simultaneous occurrence of points of several types. However, because the mark space here consists only of a finite set of types, the u.a.n. condition is equivalent to a componentwise u.a.n. condition, namely, (k)

lim sup Pr{Nni (B) > 0} = 0

n→∞

(k = 1, . . . , K).

(11.2.12)

i

Thus, from Theorem 11.2.V we have the following corollary. Corollary 11.2.VII. For a K-variate independent triangular array satisfying the u.a.n. condition (11.2.12), the necessary and sufficient conditions for convergence to a K-variate Poisson process with independent components are that (11.2.10) and (11.2.11) hold. Necessary and sufficient conditions for convergence to other types of infinitely divisible point processes, in particular the Poisson cluster process, can be derived by referring back to the general results of Theorem 11.2.III. The procedure is similar to that outlined in Theorem 11.2.V: first identify the KLM measure for the process of interest, and then use (iii) to obtain necessary and sufficient conditions on the component probabilities to ensure convergence to the appropriate limit. The particular case of convergence to a Gauss–Poisson process is outlined in Exercise 11.2.5. There are further complements to Example 11.2(a) in Exercises 11.2.6–7. Finally, we return to the question of convergence of u.a.n. arrays of random measures broached at the beginning of this section. From the general structural form of an infinitely divisible random measure given in Proposition 10.2.IX, we have the following condition for the convergence of a u.a.n. array of random measures (see also Exercise 11.2.10).

11.2.


153

Proposition 11.2.VIII. Let {ξni : i = 1, . . . , mn ; n = 1, 2, . . .} be an indemn ξni , pendent u.a.n. array of random measures on the c.s.m.s. X , ξn = i=1 and ξ an infinitely divisible random measure with Laplace representation (10.2.11). Then necessary and sufficient conditions for ξn → ξ weakly, are mn (0) Pni → Λ Q-weakly on M# (i) 0 (X ); i=1

(ii) for all bounded A ∈ BX , lim lim sup ε→0 n→∞

mn E ξni (A)I[0,ε] ξni (A) i=1

= lim lim inf ε→0 n→∞

(iii)

lim lim sup

mn

r→∞ n→∞

mn E ξni (A)I[0,ε] ξni (A) = α(A); and i=1

P{ξni (A) > r} = 0.

i=1

Just as for point processes, the conditions (i)–(iii) of this proposition can be summarized more succinctly into the single requirement that the Laplace functionals should satisfy mn

1−Lni [f ] →

i=1

f (x) α(dx)− X

M# X

exp − f (x) ξ(dx) −1 Λ(dξ). X

(11.2.13) A sketch proof, completely analogous to that of Theorem 11.2.III, is set out in Exercises 11.2.8–9. For a more detailed treatment see Kallenberg (1975, Chapter 6).

Exercises and Complements to Section 11.2 11.2.1 (a) Let P, P ∗ be the distributions of a totally finite point process on X and its Poisson approximant, respectively. Show that 2

P − P ∗ ≤ 2(P{N (X ) > 0}) . (b) Denoting the convolution of measures Pni by *Pni , show that ∗ ≤2 *Pni − *Pni

2

i

(Pni {N (X ) > 0}) .

(c) Conclude that for independent u.a.n. arrays, the weak convergence of the probability measures in parts (i) and (ii) of Theorem 11.2.III can be replaced by their strong convergence on bounded Borel sets A. 11.2.2 For finite point processes Pj with Poisson approximants Pj∗ (j = 1, 2), show that P1∗ − P2∗ ≤ 2P1 − P2 .

154


11.2.3 Restate Theorem 11.2.III as follows: the necessary and sufficient condition for convergence of an independent u.a.n. array is that

mn

(1 − Gni [h]) → i=1

#

N0 (X )

exp X

log h(x) N (dx) − 1 Q(dN )

for all h ∈ V(X ), and that the right-hand side then equals log G[h], where G is the p.g.fl. of the limit point process. Hence or otherwise deduce that Theorem 11.2.III remains valid in the case mn = ∞ provided only that the resultant superpositions are well defined. 11.2.4 Extensions of the prototype limit result for superpositions. (a) Formulate statements analogous to Proposition 11.2.VI when N is stationary but instead of simple it is (i) nonorderly; or (ii) a marked point process (does the limiting process have independent marks?). (b) Consider the case X = Rd for some d ≥ 2 in which Nn is as stated in Proposition 11.2.VI except that any A ∈ BX is now rescaled by a factor n1/d . What conditions on N suffice for the weak convergence of Nn ? What is needed to obtain nontrivial conclusions in the scenarios of (a)?

is 11.2.5 For a Gauss–Poisson process [see Example 6.3(d)] the KLM measure Q described in Exercise 10.2.3. Use this to deduce that if an independent u.a.n. array converges to a Gauss–Poisson process, then the following hold for bounded A, A1 , A2 in BX : (a) i Pr{Nni (A) ≥ 3} → 0; (b) (c)

1 {N (A) = 1} + 2Q 2 {N (A) = 1}; Pr{Nni (A) = 1} → Q i

i Pr{Nni (A1 ) = 1, Nni (A2 ) = 1} → Q2 {N (A1 ) = 1, N (A2 ) = 1} for disjoint A1 , A2 .

11.2.6 Express the result of Example 11.2(a) in terms of multivariate p.g.fl.s. 11.2.7 Formulate a limit theorem for superpositions of independent marked point processes on the space X × K by regarding the components and the limit as point processes on X × K. [Hint: Compare with Example 11.2(a), and assume first that the marks are independent as in Proposition 6.4.IV.] 11.2.8 Compare the statements of Theorem 11.2.III and Proposition 11.2.VIII. To establish the latter, proceed as below. (a) Introduce Poisson approximants and show that for a u.a.n. array their sum converges if the sum of original summands converges. (b) Prove an analogue of Proposition 11.2.II for the convergence of infinitely divisible random measures in terms of the convergence of their components αn and Λn . (c) Finally, apply part (b) to the Poisson approximants. 11.2.9 (Continuation). Show that the equivalence of (11.2.13) and Proposition 11.2.VIII can be regarded as a continuity theorem for Laplace functionals complicated by the detail of the behaviour near the zero measure. 11.2.10 Interpret the conditions in Proposition 11.2.VIII in terms of the setting of Chapter 4 of Parthasarathy (1967).

11.3.

Thinned Point Processes

155

11.3. Thinned Point Processes The notion of thinning a point process to construct another point process has been described in Example 4.3(a) for renewal processes in the simplest case where the thinning occurs independently for each point. The idea underlying the operation is that, in principle, points of a process may occur at any of a very large number of locations, but it is only at a relatively small proportion of such locations that points are observed. The limit theorems described below formulate sufficient conditions for this process of rarefaction (or thinning or deletion) to lead in the limit to a Poisson process. Let N be a point process on X = Rd and p(·) a measurable function on X with 0 ≤ p(x) ≤ 1 (all x). Np(·) is obtained from N by independent thinning according to p(·) when the following holds. Let the realization N (·, ω) consist of the countable set of points {xi } (cf. Proposition 9.1.V); construct a subset of these points by taking each xi in turn, deleting it with probability 1 − p(xi ) and retaining it with probability p(xi ), independently for each point; and regard the set of points so retained as defining a realization of the thinned point process Np(·) (·, ω). Some form of rescaling is needed before a limit theorem can emerge: the simplest set-up is the following. Take X = R and p(x) = p (all x), and after thinning contract the scale in X by an amount p, so that the point x ∈ R p (·), resulting from is mapped into px; equivalently, the point process, say N both thinning and scale-contraction, has Np (A) = k only if from the original process exactly k of the N (p−1 A) points in the set p−1 A are retained in the thinning process. p (·) denote the sequence of point processes obProposition 11.3.I. Let N tained by independent thinning and contraction at rate p = 1/T from a point process N (·) on X = R, and let N∞ denote a stationary Poisson process at rate λ. Then p (·) → N∞ weakly N if and only if as p → 0, for every bounded A ∈ BX , pN (p−1 A) ≡ (1/T )N (T A) → λ(A)

in probability.

(11.3.1)

Proof. For independent thinnings, the mechanics are most easily described in terms of p.g.fls. Indeed, the thinned process can be regarded as an especially simple form of cluster process, in which each of the points of the original process may be regarded as the centre of a cluster, and the cluster itself is either empty (if the point is deleted) or has just one point at the site of the cluster centre. Suppose first that we are given a general thinning function p(x). Then for h ∈ V(R), the p.g.fl. of the cluster member process, given a centre at y, and in the notation of equations (6.3.6) and (6.3.7), is Gm [h | y] = p(y)h(y) + 1 − p(y). Thus, the p.g.fl. Gp(·) [h] can be written in the abbreviated notation (11.3.2) Gp(·) [h] = G[1 − p + ph],

156


where G, the p.g.fl. of the original process, here plays the role of the p.g.fl. of the cluster centre process. In particular, it follows easily from this representation, that a Poisson process remains Poisson after independent deletions [see Exercise 11.3.1(a)]. p Now suppose that the deletion function p(x) ≡ p (all x), and denote by G the p.g.fl. of the point process Np after deletion and rescaling. From equation (11.3.2) we obtain log h(x) Np (dx) Gp [h] = E exp X log 1 − p[1 − h(x/p)] N (dx) = E exp X log 1 − p[1 − h(x)] N (dx/p) . = E exp X

The logarithmic term here equals −p[1 − h(x)] 1 + O(p) for p ↓ 0, so, using continuity of the generating function with respect to convergence in probability, [1 − h(x)]p 1 + O(p) N (dx/p) Gp [h] = E exp − X λ[1 − h(x)] (dx) → exp − X

if and only if (11.3.1) holds. An equivalent proof in terms of the convergence of one-dimensional distributions and using the characterization result of Theorem 9.2.XII is given in Westcott (1976) [see Exercise 11.3.2(a) and Proposition 11.1.IX]. Proof in the case of a renewal process is simpler [see Example 4.3(a) and Exercise 11.3.1(b)]. Equation (11.3.1) requires the individual realizations to satisfy an almost sure averaging property with a deterministic limit; if instead of (11.3.1) we have (11.3.3) pN (p−1 A, ω) → λ(ω)(A) in probability for some r.v. λ(·) defined on the space (Ω, F, P) on which N is defined [and, implicitly, (Ω, F, P) is assumed to be large enough to embrace the independent thinning process], the conclusion of Proposition 11.3.I is modified as below. This in turn is a special case of Theorem 11.3.III, so the proof is omitted. p (·) converges weakly to a mixed Poisson process, Proposition 11.3.II. N with mixing random variable λ, if and only if (11.3.3) holds. The formulation at (11.3.1) or (11.3.3) specifies a particular form of the measure approximated by λ(A), namely, pN (p−1 A). An alternative approach is simply to postulate the existence of a sequence of point processes {Nn (·)}

11.3.


157

such that, given a sequence of thinning probability functions {pn (x)} satisfying (x ∈ X , n = 1, 2, . . .) (11.3.4a) 0 ≤ pn (x) ≤ 1 and sup pn (x) → 0

(n → ∞),

x∈X

the sequence of random measures Λn , where Λn (A) = satisfies Λn → Λ weakly

(11.3.4b) A

pn (x) Nn (dx), then (11.3.5)

for some limit random measure Λ(·). Here, we may also allow the functions pn (·) to be stochastic, subject to the constraints at (11.3.4) and (11.3.5). Note that the operation of ‘scale-contraction’ needs care when the space X = Rd say: the independent thinning operation implies that the expectation measure M (·) of N becomes pM (·) after thinning, so we should look for convergence of pM (A/p1/d ) in order to obtain a nontrivial d-dimensional analogue of the following basic result (see Exercise 11.3.4). Theorem 11.3.III. Let {pn (x): x ∈ X , n = 1, 2, . . .} be a sequence of measurable stochastic processes satisfying (11.3.4), {Nn : n = 1, 2, . . .} a sequence n the process obtained from Nn by independent of point processes, and N thinning according to pn . Then there exists a point process N for which n → N N

weakly

if and only if (11.3.5) holds for some random measure Λ, in which case N is the Cox process directed by Λ. The statement in this theorem allows for more general limits than Proposition 11.3.II precisely because no construction of the ‘increasingly dense’ processes Nn (·) is specified. In the context of Proposition 11.3.II, it is not possible to have any Cox process other than the mixed Poisson process because when pN (p−1 A, ω) → Λ(A, ω), say, as p → 0, then also p1 p2 N ((p1 p2 )−1 A, ω) → p2 Λ(p−1 2 A, ω) as p1 → 0, and taking (for example) X = R and A = (0, 1], we −1 then have Λ((0, p−1 2 ], ω) = p2 Λ((0, 1], ω) for all 0 < p2 < 1, from which it follows that Λ(·, ω) coincides with λ(ω)(·) for some r.v. λ(·). n , with h ∈ V, n of N Proof. We have for the p.g.fl. G Gn [h] = E exp log 1 − pn (x)[1 − h(x)] Nn (dx) . X

When equations (11.3.4) are satisfied we can write − log 1 − pn (x)[1 − h(x)] = pn (x)[1 − h(x)][1 + Rn (x)],

158


where |Rn (x)| ≤ 12 pn (x) and θn = supx∈X |Rn (x)| → 0 as n → ∞. We can therefore write log 1 − pn (x)[1 − h(x)] Nn (dx) = [1 − h(x)] 1 + Rn (x) Λn (dx). − X

X

If now (11.3.5) holds, then the random variables X [1 − h(x)] Λn (dx) converge in distribution to X [1 − h(x)] Λ(dx), so that their Laplace transforms, and n [h], converge to the Laplace transform of their limit, hence also the p.g.fl.s G namely, n [h] → E exp − G

X

[1 − h(x)] Λ(dx)

.

(11.3.6)

The right-hand side here is just the p.g.fl. of the Cox process directed by Λ, which completes the proof that (11.3.5) is sufficient. n converge. We first establish Suppose conversely that the point processes N that the random measures Λn are weakly compact. Referring to Proposition 11.1.V, let S be a closed sphere in X , and consider the random variables n } implies G n [h] → G∞ [h], say, Λn (S). Weak convergence of the sequence {N for h ∈ V, and in particular for h = hz , where 0 ≤ z ≤ 1 and hz (x) =

z 1

(x ∈ S), (x ∈ S).

But, assuming (11.3.4), this is equivalent to convergence of the Laplace transforms E exp[−(1 − z)Λn (S)] to the limit G∞ [hz ], which is continuous in z as z → 1. Then the continuity theorem for Laplace transforms implies that the limit is the Laplace transform of a proper distribution, and hence that the distributions of the random variables Λn (S) are uniformly tight. Thus, given > 0 we can find M < ∞ such that for all n, P{Λn (S) > M } < ; that is, (11.1.1) holds. n converge weakly then, given η > 0, there exists a As for (11.1.2), if the N compact C such that for n = 1, 2, . . . , n (S − C) > 0} < η P{N [i.e., we use the necessity of (11.1.2) for the point processes]. Now set h(x) = 0 or 1 as x ∈ or ∈ / S − C, and deduce as above that from (11.3.4), n (S − C) > 0} − E exp − Λn (S − C) → 0 1 − P{N

(n → ∞).

11.3.


159

Thus, for sufficiently large n, n (S − C) > 0} + η ≤ 2η. E 1 − exp[−Λn (S − C)] ≤ P{N

(11.3.7)

But by a basic inequality, because 1 − e−x is nonnegative and monotonic, P{Λn (S − C) > δ} ≤

E[1 − e−Λn (S−C) ] 2η ≤ . 1 − e−δ 1 − e−δ

Thus, no matter how small δ and , we can find C such that P{Λn (S −C) > δ} < for all sufficiently large n, and hence (by modifying C if necessary) for all n > 0. Thus, both conditions (11.1.1) and (11.1.2) hold for the sequence Λn . It is n that any limit now a simple matter to deduce from the convergence of the G random measure Λ must satisfy 1 − h(x) Λ(dx) = G∞ [h]. E exp − X

It follows that the limit Λ must be unique and that the Laplace functionals of the Λn converge to that of Λ, so that (11.3.5) holds. One of the important applications of point process methods, and of the concept of thinning in particular, is to the study of high-level crossings of a continuous stochastic process. We do not treat this topic in detail, for which see Leadbetter, Lindgren, and Rootzen (1983) and the earlier text by Cramér and Leadbetter (1967), apart from briefly indicating one possible approach as follows. Consider a nonnegative, discrete time process {Xn } and associate with each n the point (n, Xn ) of a marked point process in R × R+ , where R+ plays the role of the mark space K in Definition 9.1.V. Let the underlying process of time points in R be thinned by rejecting all pairs (n, Xn ), for which (say) Xn ≤ M , and let the time axis be rescaled suitably. We may now seek conditions under which the rescaled process converges to a limit as M → ∞. In general, the thinnings here are not independent and the resulting process may exhibit substantial clustering properties. With suitable precautions, however, and assuming some asymptotic independence or mixing conditions, we may anticipate convergence to a Poisson limit. A richer theory might be expected to result if one could retain the values of the marks accepted, albeit themselves rescaled in an appropriate manner. We now give an example of this kind, in the especially simple case where the marks {Xn } are i.i.d.; yet another approach to dependent thinnings, where the probability of thinning is allowed to depend on the previous history, is outlined in Proposition 14.2.XI.

160


Example 11.3(a). Thinning by the tails of a regularly varying distribution. With the set-up just described, suppose that the initial points {ti } form a stationary, ergodic process N0 with finite mean rate m and that the marks Xi are i.i.d. with regularly varying tails [see, e.g., Feller (1971, Section VIII.8) or Bingham, Goldie, and Teugels (1987)], so that for some α > 0, 1 − F (x) = L(x)x−α

(x → ∞),

where L(x) is slowly varying at infinity; that is, L(cx)/L(x) → 1 for all c > 0. Consider now a sequence of point processes on the space R × R+ obtained in the following manner. For each n = 1, 2, . . . , set Nn (t1 , t2 ] × (u, v] = #{(ti , xi ): nt1 < ti ≤ nt2 and an u < xi ≤ an v}, where the sequence of constants {an : n = 1, 2, . . .} is defined by 1 − F (an ) = 1/n,

equivalently,

an = F −1 (1 − 1/n),

and we assume for convenience that the distribution of F is continuous so the inverse F −1 is well defined. Because the marks are independent, the p.g.fl. of the marked process on R × R+ = X × K can be written, for suitable functions h(·), in the form G[h] = E exp log h(t, y) dF (y) N0 (dt) . R

R+

The function h here must of course lie in V(X × K) (see Proposition 6.4.IV), but, because the limit point process is boundedly finite only in subsets of the mark space bounded away from the origin, h should be equal to unity in a neighbourhood of the origin on the mark space R+ . In effect, the metric in the mark space should be modified so that the origin on the y-axis becomes a point at ∞. With this modification the p.g.fl. theory carries through without change. For the rescaled process, we have to consider t y t , dF (y) = h h , y dF (an y) n an n R+ R+ ∞) t * 1 − h , y dF (an y). =1− n 0 The assumption of regular variation of F is equivalent to the weak convergence of the measures nF (an ·) to the measure ν defined by ν(y, ∞) = y −α , because for any interval (u, v] with 0 < u < v < ∞, v dF (an y) = n 1 − F (an u) − 1 − F (an v) n u

L(an u) −α L(an v) −α 1 − F (an u) 1 − F (an v) − = u − v 1 − F (an ) 1 − F (an ) L(an ) L(an ) → u−α − v −α (n → ∞).

=

11.3.


161

Consequently, the innermost integral in the expression for G[h] is expressible as ∞ t * ∞) t y −1 1 + o(1) dF (y) = 1 − n 1 − h , y dν(y); h , n an n 0 0 thus, the p.g.fl. of the rescaled process, Gn say, is given by t y F (dy) N0 (dt) log h , Gn [h] = E exp n an R R+ ) t * 1 − h , y ν(dy) N0 (dt) = E exp n−1 1 + o(1) n R R+ → E exp (1 − h(u, y)) ν(dy)m du as n → ∞, R

R+

using the ergodicity of N0 . The limit process is thus a Poisson process on R × R+ with intensity measure m(·) × ν. For any c > 0, the limit process restricted to points with marks above c is a compound Poisson process where the marks are distributed on (c, ∞) according to the distribution with d.f. Fc (x) = 1 − (x/c)−α . Strictly speaking, the overall process is not a compound Poisson process as defined in Section 6.4, because the ground process is not boundedly finite. Such extended compound Poisson processes appear also in the discussion of stable random measures (see Section 10.2) and self-similar MPPs in Section 12.8. For further examples and applications see Resnick (1986, 1987). We now indicate how the convergence in Theorem 11.3.III can be strengthened. The theorem is the weak convergence of the fidi distributions: we prove the stronger property that the fidi distributions converge in variation norm, and at the same time provide a bound on the rate of convergence. For probability measures P1 , P2 on the measurable space (Ω, E) the variation metric d(P1 , P2 ) can be defined by d(P1 , P2 ) = supB∈E |P1 (B) − P2 (B)|. This metric has the probabilistic interpretation that d(P1 , P2 ) = inf Pr{ω: X(ω) = Y (ω)}, where the infimum is taken over all pairs of measurable functions X, Y on (Ω, E, Pr) inducing the measures P1 , P2 , respectively. A pair (X, Y ) for which equality holds constitutes a maximal coupling for the probability measures P1 , P2 . The metric d(·, ·) differs by a factor of 2 from the variation distance because, in the notation of Appendix A1.3 where it is defined, |P1 (dω) − P2 (dω)| VP1 −P2 = P1 − P2 = Ω

n(T )

= sup T (Ω)

1

|P1 (Ai ) − P2 (Ai )| = 2d(P1 , P2 ).

162


This notation · is as in MKM (1978) where Var(·) = · is also used. Because the limit r.v. in Theorem 11.3.III is Poisson, our concern is with d(Pn , P∞ ), where the limit probability measure P∞ is Poisson and nonatomic. The Renyi–M¨ onch Theorem 9.2.XII asserts that such measures are characterized by their one-dimensional distributions, and by Proposition 11.1.IX it is then enough here to consider the quantity d Nn (A), N (A) ≡ d Pr{Nn (A) ∈ ·}, Pr{N (A) ∈ ·} for any bounded Borel set A [we abuse the notation in replacing the distributions of r.v.s in d(·, ·) by the r.v.s themselves]. Furthermore, for nonnegative integer-valued r.v.s, X, Y say, with distributions {pk }, {qk } say, we have " " " " " d(X, Y ) = d({pk }, {qk }) = sup " (pk − qk )"" = A⊂Z +

=

∞

k∈A

1 2

∞

|pk − qk |

k=0

(pk − qk )+ .

k=0

Proposition 11.3.IV. In the setting of Theorem 11.3.III, for bounded A ∈ BX , n (A), N (A) ≤ E sup |pn (x)| + 1 − exp − |Λn (A) − Λ(A)| . d N x∈A

Proof. Observe that it is enough to prove the result in the context that the functions pn and measures Λn , Λ are deterministic, for if otherwise, describe these entities as functions of ω with distribution µ(·). Then d Npn (A), N (A) = sup |P1 (B) − P2 (B)| B " " " " P1 (B; ω ) − P2 (B; ω ) µ(dω )"" = sup "" B Ω " " sup "P1 (B; ω ) − P2 (B; ω )" µ(dω ) ≤ B Ω = d Npn (A), N (A) | ω µ(dω ). Ω

The first term in the bound comes from Lemma 11.3.V below, and the second comes from the fact that for Poisson r.v.s X, Y with means λ, µ, d(X, Y ) ≤ d(0, Z) = 1 − e−|λ−µ| , where Z is Poisson with mean |λ − µ|. The second term in the bound can be tightened: see Exercise 11.3.5. Lemma 11.3.V. Let X1 , . . . , Xn be independent Bernoulli r.v.s with n p j = Pr{Xj = 1} = 1 − Pr{Xj = 0}, and Y a Poisson r.v. with mean λ = j=1 pj .

11.3.


163

n Then the distributions of Y and S = j=1 Xj have variation distance d(S, Y ) bounded as in n C j=1 p2j ≤ C max{pj }, (11.3.8) d(S, Y ) ≤ n j j=1 pj where C ≤ 0.71 when maxj {pj } ≤ 0.25. Always, C ≤ 1. Remarks. Although Stein–Chen methods have been much used to prove results like this lemma, we give a direct approach based on Fourier transforms. The method of proof below draws in part on work of Samuels (1965) and Kerstan (1964a). It shows that the value of the constant C depends on the value of max{pj } and can be reduced further by supposing that the maximum is smaller than 0.25. Its smallest value, when the maximum → 0, is equal (by the method below) to 0.409, which is tighter than that quoted by Romanowska (1978) for the more restricted case of a simple binomial approximated by a Poisson. Denoting the middle term in (11.3.7) by C, the computations in the later part of the proof below can be tightened further by retaining ξ in (11.3.8) as a function of θ and integrating numerically; then 0.61 can replace 0.70789 ≈ 0.71 in the theorem. The general result that C ≤ 1 follows from work of Barbour and Hall (1984) and is not discussed here. Proof. The sum S has a Poisson binomial distribution, {bk } say, for which the generating function is n

n

bk z k =

(1 − pj + pj z)

(|z| ≤ 1).

j=1

k=0

! An inequality of Newton used in Samuels (1965) implies that ck = bk nk is a log concave sequence; that is, c2k ≥ ck−1 ck+1 for k = 0, 1, . . . , n; equivalently, b2k ≥ 1 + (n − k)−1 (1 + k −1 )bk−1 bk+1 for k = 1, . . . , n − 1. Write πk = πk (λ) ≡ e−λ λk /k! . Then d(S, Y ) =

n

{bk − πk )+ =

k=0

n

πk (bk /πk − 1)+ ,

k=0

and this summation will involve nonzero terms on a single interval of integers, {k0 + 1, . . . , k1 } say, if the sequence of ratios {bk /πk } is unimodal. For this it suffices that the ratio of ratios (bk /πk )/(bk−1 /πk−1 ), which equals kbk /λbk−1 , be monotonic in k, and this is implied by the corollary to the Newton inequality (see Exercise 11.3.6). Consequently, the sup distance d0 (S, Y ) ≡ sup |Pr{S ≤ k} − Pr{Y ≤ k}| k

= max

k0

(πk − bk ),

k=0

n k=k1 +1

(πk − bk ) .

164


By addition, we thus have 2d0 (S, Y ) ≥ d(S, Y ) ≥ d0 (S, Y ), so to bound the variation distance d, it suffices to bound the sup distance d0 . The Fourier inversion relation gives " " π " 1 E(eiθS ) − E(eiθY ) "" e−iθk dθ d0 (S, Y ) = sup "" " 2πi −π 1 − eiθ k π 1 |E(eiθS ) − E(eiθY )| ≤ dθ. 2π −π |2 sin 12 θ| The characteristic functions here are E(eiθY ) = exp[−λ(1 − eiθ )] and E(eiθS ) =

n

1 − pj (1 − eiθ )

j=1

n

= E(eiθY )

exp[pj (1 − eiθ )] [1 − pj (1 − eiθ )].

j=1

Each term in the product here is of the form [1 − p(1 − eiθ )] exp[p(1 − eiθ )] = 1 −

∞ [p(1 − eiθ )]k (k − 1)

k!

k=2 2

≡ 1 − p (1 − eiθ )2 f (p; θ) so |E(e

iθS

iθY

)− E(e

)| = |E(e

iθY

say,

" " " n " 2 iθ 2 " )|" [1 − pj (1 − e ) f (pj ; θ)] − 1"" j=1

" " n [−(1 − eiθ )2 ]r = e−λ(1−cos θ) "" r=1

−λ(1−cos θ)

≤e

n

1 + 2(1 −

p2jk f (pjk ; θ)

j1 0.

11.3.


165

Write ξ = ξ(π). Then using d ≤ 2d0 and the bounds above gives

e−λ(1−cos θ) [eλξ(1−cos θ) − 1] dθ sin 12 θ 0 ξ 2λ π 1 sin 2 θ dθ exp[−2λ(1 − y) sin2 12 θ] dy = π 0 0 ξ 4λ 1 du exp[−2λ(1 − y)(1 − u2 )] dy = π 0 0 1 4λ ξ dy exp[−2λ(1 − y)(1 − u)] du ≤ π 0 0 2 log(1 − ξ) 2 ξ dy = − , ≤ π 0 1−y π 1 π

d(S, Y ) ≤

π

(11.3.9)

uniformly in λ. For maxj {pj } ≤ 0.25, ξ ≤ 0.3513, and thus d(S, Y ) ≤ = 1.102, where we follow Le Cam (1960) in [(2π −1 log(1/0.6487))/0.25] n n writing = j=1 p2j / j=1 pj . Alternatively, the inequality (11.3.8) can be replaced by the tighter bound based on α>0

1

α exp[−α(1 − u2 )] du = c = 0.642374

sup 0

(this supremum is attained at α = 2.255), leading to 2c d(S, Y ) ≤ π

0

ξ

2c log(1 − ξ) 2cξ dy =− < . 1−y π π(1 − ξ)

For maxj {pj } ≤ 0.25, this yields d(S, Y ) ≤ 0.70789, and for fixed n, lim

maxj {pj }↓0

d(S, Y ) ≤

2cξ = 0.40895ξ. π

Exercises and Complements to Section 11.3 11.3.1 (a) Verify that if a Poisson process in X = Rd with intensity measure Λ(·) is subjected to independent thinning with retention function p(x) for measurable p(·), then the thinned process is Poisson with intensity measure Λp , where Λp (A) =

p(x) Λ(dx). A

(b) When N in Proposition 11.3.I is a renewal process, use inter alia transform techniques to show the following [cf. Rényi (1956)]. (i) The rarefaction of any renewal process is a renewal process. (ii) The only renewal process invariant under the operations is the Poisson. (iii) Starting from any renewal process with finite mean lifetime, the limit under the operations is a Poisson process.

166


11.3.2 (a) Apply Proposition 11.1.IX to furnish an alternative proof of Proposition 11.3.I via avoidance functions. (b) Give a direct proof of Proposition 11.3.II either via p.g.fl.s or by extension of part (a). 11.3.3 Use Theorem 11.3.III to show that a distribution P on NX# is a Cox distribution if and only if for each c in 0 < c < 1 there exists a distribution Pc , which under independent random thinning with constant probability c yields P [see Mecke (1968)]. 11.3.4 Extensions of the prototype limit result for thinnings. (a) Formulate statements analogous to Proposition 11.3.I when N is stationary but instead of simple it is (i) nonorderly and either (i ) batches of points are treated in toto, with all points retained with probability p and otherwise all deleted; (i ) each point of a batch is thinned independently (with retention probability p per point); or (ii) a marked point process. In case (ii) does the limit process have independent marks? (b) Consider the case X = Rd for some d ≥ 2 in which Nn is as stated in Proposition 11.3.I except that any A ∈ BX is now rescaled by a factor n1/d . What conditions on N suffice for the weak convergence of Nn ? How are the results of (a) affected by having X = Rd ? √ 11.3.5 (a) Show that ak ≡ supλ>0 πk (λ) λ occurs for λ = k + 12 and that the se√ quence of ratios {ak+1 /ak } is monotonic in k. Show √ that a0 = 1/ 2eλ ≥ ak > ak+1 for k = 0, 1, . . . , and that πk (λ) ≤ 1/ 2eλ (every λ > 0). (b) Express d({πk (λ)}, {πk (µ)}) as an integral with density πj (·) for some j, and√hence deduce that this variation metric is bounded above by √ 2/e | λ − µ |. Thus, except for small Λn , Λ, the last term in Proposition 11.3.IV can be tightened [see Daley (1987)].

11.3.6 (a) Let bk (n; p) = nk pk (1 − p)n−k , k = 0, 1, . . . , n, 0 < p < 1, denote binomial probabilities, and write πk (λ) for Poisson probabilities as in the proof of Lemma 11.3.V. Show directly that {bk (n; p)/πk (λ)} is a unimodal sequence in k by virtue of the monotonicity of (k + 1)bk+1 /bk . (b) Now let {bk } = {bk (n; p1 , . . . , pn )} denote the Poisson binomial distribution of S of the lemma. Show by induction on n that {(k + 1)bk+1 /bk } is monotonic in k. (c) Deduce that d(S, Y ) ≤ maxk {bk /πk } − 1.

11.4. Random Translations The remaining class of stochastic operations that we consider has as its prototype random translations: each point xi in the realization of some initial point process N0 is shifted independently of its neighbours through a random vector Yi , the Yi forming a sequence of i.i.d. random variables. Exercises 2.3.4(b) and 8.2.7 give simple cases. Generalizations occur if the translations are replaced by more general clustering mechanisms in which the mean number of points per cluster is held equal to one: we defer discussion to Section 13.5 in order to take advantage of some properties of Palm distributions.

11.4.

Random Translations

167

One possible approach to such problems is to view the resultant process as the superposition of its individual clusters, one from each point of the initial realization, and to seek to apply the results of Section 11.2 on triangular arrays. Here the nth row in the array relates to the process derived from n stages of clustering (translation); although the number of terms Nni in the nth row is infinite, this does not affect the validity of the criteria provided their superpositions are well defined (see Exercise 11.2.3), as they must be in this case for the resultant processes themselves to be well defined. In the case of random translations in Rd this leads us to a result of the following kind. Let ν(·) denote the common distribution on B(Rd ) for the translations Yi , and νn (·) the n-fold convolution of ν, corresponding therefore to the effect of n successive random translations. For h ∈ V and X = Rd , the p.g.fl. after n translations takes the form h(· + y) νn (dy) Gn [h] = G0 X) (11.4.1) * log h(x + y) νn (dy) N0 (dx) , = E exp X

X

where G0 [ · ] is the p.g.fl. of the initial process N0 . Each process Nni contains just one point, and the u.a.n. condition (11.2.2) reduces to the requirement sup νn (A + xi ) = sup Pr{Y1 + · · · + Yn ∈ xi + A} → 0 i

i

(n → ∞).

(11.4.2) To use Theorem 11.2.V to prove convergence to a Poisson limit, observe that condition (11.2.7) is trivial here because P{Nni (A) ≥ 2} ≡ 0, and (11.2.8) translates into νn (A + x) N0 (dx) = νn (A + xi ) → µ(A) (n → ∞). (11.4.3) X

i

The problem then is to find conditions on the initial process N0 and the distribution ν that will ensure the truth of (11.4.2) and (11.4.3). The former of these is closely related to the concept of the concentration function QA (F ) of a distribution F on B(Rd ) defined for bounded Borel sets A by QA (F ) = supx∈Rd F (x + A). Clearly the expression in (11.4.2) is bounded above by QA (νn ), so the u.a.n. statement there is a direct consequence of the lemma below [for a non-Fourier analytic proof see Ibragimov and Linnik (1971, Chapter 15, Section 2)]. Lemma 11.4.I. Let F be a distribution on B(Rd ) and F n∗ its nth convolution power. For bounded A, QA (F n∗ ) → 0 as n → ∞ if and only if the support of F contains at least two distinct points.

168


Proof. If F is degenerate, then so is F n∗ and QA (F ) = QA (F n∗ ) = 1 for every n for every nonempty A. Otherwise, we show in fact that QA (F n∗ ) ≤ c(F, A)/n1/2

(11.4.4)

for some constant c(F, A), observing that it is enough to prove this in the case d = 1 because for d ≥ 2, noting that QA (F ) increases for monotonic increasing A, we can embed A in a set corresponding to a marginal distribution. Exercise 11.4.2 shows that the order n−1/2 of the bound in (11.4.4) is tight. Write H(y) = (sin 12 y/ 12 y)2 for the c.f. of the probability measure H(·) with triangular density function H (x) = (1 − |x|)+ . Then the Parseval relation (A2.8.8) yields for any d.f. G, positive a, and real γ, ∞ |ω| 1 ∞ −iωγ 1− dω. (11.4.5) H a(x − γ) G(dx) = G(ω)e a −∞ a + −∞ Substitute G = F n∗ , and recognize that the integral on the right-hand side here is over the interval (−a, a), and the left-hand side is real, so that an upper bound that is uniform in γ is given by 1 a 1 a n |F (ω)| dω ≤ exp − 12 n(1 − |F(ω)|2 ) dω, a −a a −a where we have used the inequality x = 1 + (x − 1) < ex−1 with x = |F(ω)|2 . Now |F|2 is the characteristic function of the d.f. Fs of the symmetrized r.v. X − X , where X and X are i.i.d. like X with d.f. F . Thus, 1 − |F(ω)|2 = 1 − E exp[iω(X − X )] = 1 − E cos ω(X − X ) sin2 12 ωy Fs (dy) = 2E sin2 12 ω(X − X ) ≥ 2 |y|>b

for some positive b, which, because F is nondegenerate, can be and is so chosen that Fs (dy) = Pr{|X − X | > b} ≡ η > 0. |y|>b

Use Jensen’s inequality in the form exp E[f (Y )] ≤ E exp[f (Y )] to write exp[−nη sin2 12 ωy] 1 a 1 a Fs (dy) |F (ω)|n dω ≤ dω a −a a −a η |y|>b Fs (dy) 1 exp[−nη sin2 12 z] dz = a |y|>b η|y| |z| 0, supx |g2 (x) − g1 (x)| <

and

g1 (x) ≤ IA (x) ≤ g2 (x)

[this can be achieved, e.g., by taking for g2 the function IA/2 ∗ t/2 and for g1 the function IA−/2 ∗ t/2 , where tα is the triangular distribution on base (−α, α)]. Thus, (11.4.3) holds whenever ν is nonlattice and A is an interval. Because it is enough in Theorem 11.2.V to let A run through intervals, we can conclude from that theorem that if the initial process is lattice with a point on every integer but the distribution ν is nonlattice, then the processes Nn , obtained by successive random translations according to ν, converge weakly to the stationary Poisson process with unit rate. Obviously, the condition that the initial points lie on a lattice can be relaxed, but, unfortunately, it cannot be relaxed far enough to apply almost surely to the realizations of a typical stationary point process. Indeed, it follows from a theorem of Stone (1968) that the essential requirement on the initial process, if this is regarded as fixed, is that N0 (x + An )/(An ) → const.

uniformly in x,

(11.4.7)

170


where An is the sequence of hypercubes Udn of side n in Rd or, more generally, a convex averaging sequence in the sense of Definition 12.2.I. Such a condition is not satisfied almost surely even by the realizations of a stationary Poisson process. On the other hand, averaged forms of (11.4.5), that is, with convergence in L1 or L2 , follow directly from the ergodic theorems of Section 12.2, and in these the uniformity is trivial when the initial process is stationary. We therefore seek an alternative approach that will bypass the probability one requirements on the initial configuration. For this purpose we return to a direct study of the p.g.fl. at (11.4.1) of the processes Nn . To establish weak convergence of the translated versions Nn to a Poisson limit with rate m, it is enough to show that for h ∈ V, p − log h(x + y) νn (dy) N0 (dx) → m [1 − h(x)] dx, X

X

X

because this implies convergence of the Laplace transforms of the random variables on the left-hand side, and hence of the p.g.fl.s. To ease the notation write u(x) = 1 − h(x), so that u vanishes outside a bounded set and satisfies 0 ≤ u ≤ 1. Then the above requirement becomes p log 1 − u(x + y) νn (dy) N0 (dx) → m u(x) dx. (11.4.8) − X

X

X

From Lemma 11.4.I, if ν has at least two points in its support, we can easily deduce that u(x + y) νn (dy) → 0 (n → ∞). θn ≡ sup x

X

We can therefore approximate the logarithm by its leading term, with remainder, for sufficiently large n, bounded by " "" " " " u(x + y) ν (dy) + log 1 − u(x + y) ν (dy) u(x + y) νn (dy). ≤ θ n n n " " X

X

X

Suppose now that N0 is stationary with finite mean rate m. Then (11.4.8) is implied by the corresponding L1 convergence, which leads us to estimate the expected difference by " " " " " E"m u(x) dx + log 1 − u(x + y) νn (dy) N0 (dx)"" X X X " " " " " u(x) dx − u(x + y) νn (dy) N0 (dx)"" ≤ E"m X X " "X " " " u(x + y) νn (dy) + log[1 − u(x + y)] νn (dy) N0 (dx)"", + E" X

X

X

where the second expectation is bounded by u(x + y) νn (dy) N0 (dx) = mθn u(x) dx, θn E X

X

X

11.4.

Random Translations

171

which tends to zero by Lemma 11.4.I. Thus, for weak convergence to the Poisson limit it is enough to show that for measurable u with bounded support and 0 ≤ u ≤ 1, " " " " u(x) dx − u(x + y) νn (dy) N0 (dx)"" → 0. (11.4.9) E""m X

X

X

A complication arises here if the stationary distribution N0 is nonergodic, for the ergodic theorems then assert convergence not to the constant m but to the random variable (asymptotic density of N0 ) Y = E[N0 (Ud ) | I ],

(11.4.10)

where Ud is the unit cube in Rd , I is the invariant σ-algebra, and E(Y ) = m (see Theorem 12.2.IV). In this case (11.4.7) should be replaced by the more general requirement, justified by a completely analogous argument, that " " " " " u(x) dx − u(x + y) νn (dy) N0 (dx)"" → 0. E" Y (11.4.11) X

X

X

Note that (11.4.9) and (11.4.11) may be regarded as L1 versions of (11.4.3). A full discussion of (11.4.11) involves further delicate analysis of the convolution powers of ν; we content ourselves here with the much easier L2 version, assuming the initial point process has boundedly finite second moment measure. This leads us to the following theorem. Theorem 11.4.II. Let N0 be a second-order stationary point process on X = Rd and ν a distribution on Rd that is nonlattice. Then the sequence of point processes {Nn }, derived from N0 by successive random translations according to ν, converges weakly to the stationary mixed Poisson process with p.g.fl. G[h] = E exp

−Y

X

[1 − h(x)] dx

,

(11.4.12)

where Y is given by (11.4.10). Proof. We again use a Fourier argument, observing that in the ergodic case " "2 " " E""m u(x) dx − u(x + y) νn (dy) N0 (dx)"" X X X = var u(x + y) νn (dy) N0 (dx) X X = |˜ u(ω)|2 |˜ ν (ω)|2n Γ(dω), (11.4.13) X

where Γ(·) is the Bartlett spectrum introduced in Definition 8.2.II. The validity of the above relation follows from the Parseval relation (8.6.10) and Lemma ν (ω)|2n |˜ u(ω)|2 . 8.6.V, ensuring the Γ-integrability of |˜ u(ω)|2 and hence of |˜

172


Because |˜ ν (ω)| < 1 for ω = 0 [this holds for nonlattice distributions in Rd for arbitrary d ≥ 1 : see (11.4.6) for the case d = 1], the right-hand side of the identity above converges to Γ{0}, which being equal to var Y where Y is given by (11.4.10) (see Exercise 12.2.9), vanishes for an ergodic process. In the nonergodic case we have to replace Γ by a modified measure Γ∗ introduced as the Fourier transform of the modified covariance measure C ∗ (A × B) = E N0 (A) − Y (A) N0 (B) − Y (B) . Then Γ∗ differs from Γ precisely by the absence of the atom at zero. We can now argue as in the ergodic case and deduce that "2 " " " " u(x + y) νn (dy) N0 (dx) − Y u(x) dx"" → Γ∗ {0} = 0. E" X

X

X

This result implies (11.4.11) and so completes the proof. It is easily seen that both ordinary and mixed stationary Poisson processes are invariant under the operation of random translation (see Exercise 11.4.1). As a corollary to Theorem 11.4.II we now have the following converse. Corollary 11.4.III. Suppose that N is a stationary, second-order point process that is invariant under the operation of random translation according to a nonlattice distribution ν. Then N is a stationary mixed Poisson process. Proof. Take N as the initial distribution in the theorem, and observe that if N is invariant the weak limit of the Nn must coincide with N . A second case of interest arises when the random translations νn are derived from the movements over time n of particles with fixed but random and independently chosen velocities as in Example 8.3(g). If these velocities have a common distribution ν, we can then write νn (dx) = ν(n−1 dx) and observe that, from Exercise 11.4.5(d), QA (νn ) = QA/n (ν) → 0. Moreover, the integral at (11.4.13) becomes |˜ u(ω)|2 |ν(nω)|2 Γ(dω). X

Now if ν is absolutely continuous, |˜ ν (nω)| → 0 as n → ∞ for every ω = 0 by the Riemann–Lebesgue lemma, so that the proof of (11.4.9) and its extension (11.4.11) to the ergodic case can be completed as in the previous discussion. The restriction to integer values n is immaterial here, and we therefore obtain the following further result. Theorem 11.4.IV. Let N0 be as in Theorem 11.4.II, and for all t ≥ 0 let the point processes Nt be derived from N0 by random translations through time t by fixed but random velocities with common distribution ν. If ν is absolutely continuous with respect to Lebesgue measure on Rd , the processes Nt converge weakly to a mixed Poisson process as in (11.4.12).

11.4.

Random Translations

173

Corollary 11.4.V. Let the point process Nt represent the position at time t of a system of particles moving in Rd with fixed velocities chosen independently and randomly according to a distribution ν that is absolutely continuous with respect to Lebesgue measure in Rd . If the distribution of Nt is independent of t, spatially homogeneous, and of second order, then Nt is a mixed Poisson process as in (11.4.12). A more general type of location-dependent random translation is illustrated in the following example. Example 11.4(a) Markov shifts (random translations). Suppose given a point process on X with p.g.fl. G[h] (h ∈ V(X )) and that any particle of this process initially at x is shifted into any A ∈ BX with probability p(A | x), where p(X | x) =

X

p(dy | x) ≤ 1

(all x),

the shortfall q(x) = 1 − p(X | x) being the probability of deletion of the particle. Arguing as for Exercise 11.3.1 yields Gm [h | x] = q(x) + h(y) p(dy | x) = 1 − [1 − h(y)] p(dy | x) X

X

for the p.g.fl. of the (zero- or one-point) cluster associated with x, from which the resultant p.g.fl. for the translated process equals G Gm [h | ·] . The kth tr for the shifted process is given in terms of the correfactorial moment M[k] sponding moment of the initial process by ··· p(dy1 | x1 ) . . . p(dyk | xk ) M[k] (dx1 × · · · × dxk ). X

X

When the initial process is Poisson with parameter measure µ(·) so that log G[h] = − X [1 − h(x)] µ(dx), the p.g.fl. of the shifted process equals exp

−

X

X

[1 − h(y)]p(dy | x) µ(dx) ,

so the shifted process is Poisson also, with parameter measure tr µ (A) = p(A | x) µ(dx) (bounded A ∈ BX ). X

A situation of particular interest arises if µtr = µ; that is, µ is an invariant measure (not necessarily totally finite) for the Markov transition kernel p(· | ·). It follows from the last relation that a Poisson process with this parameter measure is invariant under the Markov shift operation, a result due to Derman (1955).

174


Consider finally the case of a pure shift (so that q(x) = 0 for all x ∈ X ). Suppose that X = Rd and µ(dx) = µ(dx) where on the right-hand side, µ is a constant and denotes Lebesguemeasure on BRd . Then the initial process is stationary and p(dy | x) = F d(y − x) , meaning that the shifts are identically distributed about the positions of the initial points; that is, we have random translations of the points. Then µtr = µ and consequently a stationary Poisson process is invariant under a process of i.i.d. shifts. Before leaving this topic we make a few remarks concerning the L1 theory referred to briefly before Theorem 11.4.II. A key step here is to show that in both situations considered, the distributions νn satisfy the condition, for all bounded Borel sets A ∈ Rd , |νn (y + A) − νn (x + y + A)| dy → 0 uniformly in x. (11.4.14) Rd

This condition, or the apparently stronger but in fact equivalent condition νn ∗ γ1 − νn ∗ γ2 → 0

(11.4.15)

for all pairs γ1 , γ2 of distributions absolutely continuous with respect to Lebesgue measure in Rd , is referred to in MKM (1978) as weak asymptotic uniformity of the sequence {νn }. A particular example of such a sequence is the sequence of uniform distributions on the sets {An } of a convex averaging sequence. The major technical difficulty is then to show that the standard form of conclusion of the mean ergodic theorem, which can be written as " " " " " Hn (x + A) N0 (dx) − Y (A)"" → 0, (11.4.16) E" d R

where Hn is this special case of a weakly asymptotically uniform sequence, can be extended to the general case and therefore implies (11.4.11) in each of the two situations under consideration. The definitive treatment of the L1 case was given by Stone (1968), after earlier work by Dobrushin (1956) and Maruyama (1955) in the context of iterated random translation, and by Breiman (1963) and Thedéen (1964) for the random velocities scheme. Further extensions and generalizations occur in a series of papers by Matthes and co-workers; for details we refer to MKM (1978) especially Chapter 11 and the further references there. An algebraic treatment of (11.4.14) and related properties, when νn are convolution powers, is contained in the papers by Stam (1967a, b). The second-order treatment used to prove Theorem 11.4.II is an extension of the discussion in Vere-Jones (1968). Some partial results concerning (11.4.14) and related topics are covered in Exercises 11.4.4–5.

Exercises and Complements to Section 11.4 11.4.1 Show that in the stationary case, both ordinary and mixed Poisson processes are invariant under the operation of random translation. [Hint: Use the p.g.fl. representations at (11.4.1) and (11.4.12).]

11.4.

Random Translations

175

11.4.2 The binomial distribution {bk (n; p)} = { nk pk (1 − p)n−k } with 0 < p < 1 is the n-fold convolution of the simplest nondegenerate d.f. F that can arise √ with Lemma 11.4.I. The order 1/ n of the bound at (11.4.4) is tight because 1 1 ≤ Q{0} ({bk (n; p)}) ≤ 2π(n + 1)p(1 − p) 4(n + 1)p(1 − p)

[see, e.g., MKM (1978, pp. 476–477) or else Daley (1987)]. 11.4.3 Let P (x, A) denote a stochastic or substochastic kernel defined for all x ∈ Rd and A ∈ B(Rd ) such that it has an infinite invariant measure ν. Consider the operation of random translation according to the kernel P [i.e., a point initially at x is translated to a new point y according to the distribution P (x, ·)]. (a) The Poisson process with intensity measure ν is invariant under this operation. (b) If P is continuous, then the initial process N0 is invariant under this operation if and only if it is a Cox process directed by Y ν, where Y is a nonnegative random variable. (c) Investigate conditions under which the sequence of point processes {Nn } obtained from an initial process N0 by successive iteration of this operation will converge to a limit of the form described in (b). [Hint: See Kerstan and Debes (1969) and Debes et al. (1971). Part (a) goes back to Derman (1955). The case where ν is totally finite is discussed in MKM (1978, Section 4.8).] 11.4.4 For a given distribution F on Rd , let S denote the set of points a in Rd such that for all intervals A, supx |F n∗ (x + a + A) − F n∗ (x + A)| → 0

as n → ∞.

(11.4.17)

Prove the following. (a) S is an algebra. (b) If a ∈ supp(F ), then a ∈ S. (c) If supp(F ) is contained in no proper subalgebra of Rd , then S = Rd . 11.4.5 (Continuation). A sequence of measures {νn } is weakly asymptotically uniformly distributed if for all absolutely continuous distributions σ on B(Rd ) and all x ∈ Rd , (n → ∞). (11.4.18) σ ∗ νn ∗ δx − σ ∗ νn → 0 (a) Show that (11.4.17) implies (11.4.18) in the special case that σ is the uniform distribution on the interval A and {νn } = {F n∗ }. (b) Extend this result and deduce that if F is nonlattice, the sequence of convolution powers of F is weakly asymptotically uniformly distributed. (c) Prove that (11.4.18) is equivalent to νn ∗ σ1 − νn ∗ σ2 → 0

(n → ∞)

for all pairs of absolutely continuous distributions σ1 and σ2 . (d) If (11.4.18) holds then QA (νn ) → 0 (cf. Lemma 11.4.I). [Hint: For further details and applications see MKM (1978, Chapter 11).]

CHAPTER 12

Stationary Point Processes and Random Measures

12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8

Stationarity: Basic Concepts Ergodic Theorems Mixing Conditions Stationary Infinitely Divisible Point Processes Asymptotic Stationarity and Convergence to Equilibrium Moment Stationarity and Higher-order Ergodic Theorems Long-range Dependence Scale-invariance and Self-similarity

177 194 206 216 222 236 249 255

Stationary point processes play an exceptionally important role in applications and lead also to a rich theory. They have appeared already in Chapters 3, 6 and 8, where some basic properties and applications were outlined, and they are central to the discussion not only in the present chapter, but also in Chapters 13 and 15, and to a lesser extent Chapter 14 also. Our main purpose in this chapter is to develop a systematic study of the theory of point processes and random measures that are invariant under shifts in d-dimensional Euclidean space Rd . Much of the theory is likely to appear familiar: some parts are merely variants of the corresponding theory of stationary continuous processes; the greater part can be deduced from the theory of stationary random distributions, or, in R1 , from the theory of processes with stationary increments, but some parts, especially applications, are peculiar to point processes and random measures. Although we have chosen to develop the basic theory for shifts Su acting d on the canonical space M# X , with X = R , the underlying ideas are capable of extension in several directions. In the first instance this refers to point processes and random measures invariant under more general forms of group action, for example, to processes in two- or three-dimensional Euclidean space that are invariant under rotations (i.e., isotropy) as well as shifts (i.e., homogeneity), and to processes on other types of manifold, such as the surface of a sphere or cylinder. In fact, many of the topics included in this chapter can 176

12.1.

Stationarity: Basic Concepts

177

be developed with almost equal facility (i.e., requiring nothing or little more than changes of wording or interpretation) for point processes on a locally compact metric group, and are so developed in the Russian edition of MKM (1982). We examine some such extensions in the present chapter, especially in connection with scale-invariance and self-similarity, whereas others, including isotropy, are taken up in Chapter 15 on spatial point processes. At the same time, the shifts studied in this chapter are examples of a flow on a probability space (Ω, E, P), meaning in general a group {θg } of measurable one-to-one transformations of (Ω, F) onto itself. The probability measure P is invariant under the flow if for all E ∈ E and all θg , P(θg E) = P(E). This more general concept [see, e.g., Baccelli and Brémaud (2003)] is useful in unifying the treatment of processes, including marked point processes (MPPs) and Cox processes, where the canonical space M# X needs extending to accommodate information about the outcomes of auxiliary random variables or processes. An important technical role in establishing the form of both probability and moment structures for stationary processes is played by the factorization theorems summarized in Appendix A2 as Lemma A2.7.II and Theorem A2.7.III. In their basic form they assert that if a measure µ on a product space Rd × K is invariant under shifts in the first component, then µ reduces to a product of Lebesgue measure in Rd and a fixed measure κ on K. These factorization results extend to more general contexts with Rd replaced by a σ-group H, and Lebesgue measure replaced by Haar measure on H. They underlie the structure not only of stationary MPPs, where they apply most obviously, but also of stationary Poisson and Poisson cluster processes, of the moment measures of stationary processes, and of the Palm theory which is the subject of Chapter 13. After a first section on basic concepts and examples, the chapter covers ergodic theorems, moment and mixing properties (Sections 12.2–4), stationary infinitely divisible point processes (Section 12.5), convergence to equilibrium (Section 12.6), long-range dependence (Section 12.7), and scale-invariance and self-similarity (Section 12.8).

12.1. Stationarity: Basic Concepts We consider first X = Rd and invariance properties with respect to translations (or shifts). For arbitrary u, x ∈ X , and A ∈ BX , write Tu A = A + u = {x + u: x ∈ A}.

Tu x = x + u,

M# X

NX# )

(and also of Then Tu induces a transformation Su of equation1 (Su µ)(A) = µ(Tu A) (µ ∈ M# X , A ∈ BX ). M# X

(12.1.1) through the (12.1.2)

It is clear that Su µ ∈ whenever µ ∈ that is, Su maps M# X into (indeed, onto) itself. Moreover, Su is continuous: to see this, let {µn } be 1

M# X;

With the operators T· and S· as defined, the Dirac measure δ(·) has the property that

178

12. Stationary Point Processes and Random Measures

a sequence of measures on BX converging in the w# -topology to a limit µ, and let f be a bounded continuous function vanishing outside a bounded set. Then its translate f (x − u) has similar properties, so from properties of w# -convergence (Proposition A2.6.II), f (x) (Su µn )(dx) = f (x − u) µn (dx) X X →w # f (x − u) µ(dx) = f (x) (Su µ)(dx). X

X

An application of the sufficiency half of Proposition A2.6.II shows that Su µn →w# Su µ, and hence that Su is continuous. Because a shifted counting measure is again a counting measure, and NX# is closed in M# X , the same conclusion holds for the effects of shifts Tu on counting measures. This establishes the following simple but important result. Lemma 12.1.I. For X = Rd and u ∈ Rd , both the mappings Su : M# X → # # M# and S : N →

N defined at (12.1.2) via the shift operator T are u u X X X continuous (and hence measurable) and one-to-one. It now follows that if ξ is a random measure or point process, then so is Su ξ for every u ∈ Rd because Su ξ is then the composition of two measurable mappings. This remark enables us to make the following definition. Definition 12.1.II. A random measure or point process ξ with state space X = Rd is stationary if, for all u ∈ Rd , the fidi distributions of the random measures ξ and Su ξ coincide. If extra emphasis is needed, we call such random measures strictly stationary or stationary as a whole to distinguish them from random measures that are stationary in weaker senses such as second-order stationarity following Proposition 8.1.I. Note also that this definition is the natural extension to Rd and M# X of Definition 3.2.I, and lies behind the summary of stationarity properties in Proposition 6.1.I. Definition 12.1.II can be stated in a compact form by defining a ‘lifted’ operator or transformation S+u that functions at a third level of abstraction, # on measures P on the Borel sets of M# X . For B ∈ B(MX ) set S+u P(B) = P(Su B),

(12.1.3)

where Su B = {Su µ: µ ∈ B}. The remark following Lemma 12.1.I implies that # S+u maps M# X (or NX ) into itself, and an argument similar to the proof of (Su δx )(·) = δx−u (·) and f (x) (Su µ)(dx) =

f (x) µ(d(x + u)) =

f (x − u) µ(dx).

Sometimes, an operator S· for which Su = S−u is used instead. Then the + and − signs in the above equations are interchanged. The operator Su we use is the same as Tu in MKM (1978, p. 258) and as T−u in Kallenberg (1975 or 1983a, Exercise 10.10).

12.1.


179

that lemma shows that the mapping is even continuous (see Exercise 12.1.1). Then Definition 12.1.II is equivalent to stating that a random measure on Rd is stationary if its distribution on M# is invariant under shifts S+u . Rd These concepts and results can be extended to the more general context of a flow referred to in the introduction. In this case the probability space (Ω, E, P) is not given an explicit structure, but it is supposed to be capable of supporting a family of one-to-one measurable transformations of Ω onto itself, {θu : u ∈ G} say, where G has a group structure, θu+v = θu ◦ θv , θ0 is the identity, and (θu )−1 = θ−u . The link to a transformation acting directly on the random measure ξ: (Ω, E) → M# X , BM# is then provided by taking G = X = Rd and requiring that

X

ξ(A, θu ω) = ξ(A + u, ω) or, more briefly, ξ(θu ω) = Su ξ(ω). Most of the examples below illustrate specific cases where the flow is defined for u ∈ G = Rd , but of course shifts in Rd are not the only actions which can be defined by flows. In the case of a marked point process, it is enough to take for Ω the space of counting measures on the product space Rd × K (i.e., NR#d ×K ), and to consider for the flow the family of translations Su acting on the first component only, so that Su N (A × K) = N (Tu A × K). The MPP is stationary if its fidi distributions are invariant under the translations {Su } (equivalently, its probability distribution on M# is invariant under the lifted operators {S+u }). Rd ×K In discussing a stationary MPP N say, we mostly write Su N rather than Su N , the restriction of the shift to Rd being understood. We proceed to a detailed study of stationarity of random measures in Rd , illustrating the constructions in terms of shifts S+u on the canonical probability space. The results of Chapter 9 imply that the distribution of a random measure is completely determined either by its fidi distributions or by its Laplace functional (see Propositions 9.2.III and 9.4.II). Similarly, the distribution of a point process is completely determined by its fidi distributions or by its p.g.fl. (Theorem 9.4.V) or, if the point process is simple, by its avoidance function (Theorem 9.2.XII). Applying these criteria to the definition of stationarity, we deduce that a random measure is stationary if and only if its fidi distributions are stationary, or, equivalently, if and only if its Laplace functional is stationary. Similarly, a point process is stationary if and only if its p.g.fl. is stationary, or, if it is simple, if and only if its avoidance function is stationary. Spelling out the details of these remarks yields the following theorem. Theorem 12.1.III. Let ξ be a random measure on state space X = Rd . Each of the following conditions is necessary and sufficient for ξ to be stationary. (i) For each u ∈ Rd and k = 1, 2, . . . , the fidi distributions satisfy Fk (A1 . . . , Ak ; x1 , . . . , xk ) = Fk (A1 + u, . . . , Ak + u; x1 , . . . , xk ). (12.1.4)

180


(ii) For each u ∈ Rd and f ∈ BM(Rd ) the characteristic functional satisfies Φ[f (·)] = Φ[f (· − u)]. (12.1.5) When ξ is a point process N the conditions are equivalent to the following. (iii) For each u ∈ Rd and h ∈ V(Rd ), the p.g.fl. G satisfies G[h(·)] = G[h(· − u)].

(12.1.6)

If also N is simple, the conditions are equivalent to the following. (iv) For each u ∈ Rd and all bounded Borel sets A ∈ B(Rd ), the avoidance function P0 (·) of N satisfies P0 (A) = P0 (A + u).

(12.1.7)

Furthermore, in (i) and (iv) it is sufficient for the results to hold for disjoint sets Ai and A from a dissecting semiring generating B(Rd ). The final statement of the theorem implies that it is enough in (i) and (iv) to have the statements holding for disjoint sets that can be represented as finite unions of half-open rectangles. It is not possible to relax this condition significantly: in R1 Lee’s counterexample quoted in Exercise 2.3.1 exhibits two processes with the same distributions for N (I) whenever I is an interval, one of these processes being stationary (indeed, a stationary Poisson process) and the other not. See also Exercise 12.1.2. Analogues of this proposition hold for other examples of flows. The details for MPPs are spelled out in Exercise 12.1.3. The next few examples illustrate some applications of Theorem 12.1.III and its extensions. Example 12.1(a) Stationarity of Poisson and compound Poisson processes [continued from Example 9.4(c); see also Lemma 6.4.VI]. From the representation of the Poisson process p.g.fl. at (9.4.17) we have, for X = Rd , log G[h(· − u)] = [h(x − u) − 1] µ(dx) = [h(y) − 1] (Su µ)(dy), (12.1.8) X

X

which under the assumption of stationarity at (12.1.6) is to be equal to [h(y) − 1] µ(dy). X

Because the measure µ is completely determined by its integrals of functions of the form h(y) − 1 for h ∈ V(Rd ), it follows that a Poisson process is stationary if and only if its parameter measure is invariant under translation. Now the only measure on Rd invariant under translations is Lebesgue measure, so µ(·) must be a multiple of Lebesgue measure on Rd ; that is, for some µ ≥ 0, µ(·) = µ(·). Thus, a Poisson process on Rd is stationary if and only if it has a constant intensity with respect to Lebesgue measure on Rd .

12.1.


181

Alternatively, we may first observe that a stationary random measure can have no fixed atoms, then use the fact (Theorem 2.4.II) that a Poisson process has no fixed atoms if and only if its parameter measure is nonatomic, in which case the process is simple, and finally appeal to (12.1.7), which yields e−µ(A) = e−µ(A+u) , implying the same result even more directly. The compound Poisson process, in the general sense of Section 6.4, is an example of a marked point process, but essentially similar techniques can be applied. Using the p.g.fl. approach from Exercise 12.1.3(iii), we have to check, for a constant rate Poisson ground process and i.i.d. marks, that (12.1.8) holds for a function h(u, κ) of the two variables. In fact we have [h(x − u, κ) − 1] µ dx π(dκ) log G[h(· − u, ·)] = X K = [h(y, κ) − 1] µ dy π(dκ) = log G[h(·, ·)]. X

K

Example 12.1(b) Stationarity is preserved by simple random thinnings and translations [continued from Sections 11.3 and 11.4]. Let N be a stationary point process and assume that each point xi of a realization of N is independently and randomly shifted through a random distance Xi , where the {Xi } are identically distributed with common d.f. F (·); to accommodate deletions, we allow the distribution to be defective, and set q = 1 − F (Rd ). Then from equation (11.4.1) the respective p.g.fl.s G and G0 of the shifted process and N are related by G[h(·)] = G0 q + X h(y) F (dy − ·) . Much as in the previous example, when G0 is itself stationary, G0 [h(·)] = G0 [h(· − u)] for all u ∈ Rd and h ∈ V(Rd ). The right-hand side of the expression for G[ · ] then equals G0 q + X h(y − u) F (dy − ·) = G[h(· − u)], so by (iii) the transformed process is again stationary. Pure translations occur when q = 0, else random deletions when F is concentrated at 0. The stationarity of Cox processes and some cluster processes can be verified by similar techniques. Cox processes are important as examples where the flow needs to be defined initially on an extension of the canonical space M# X. Example 12.1(c) Mixed Poisson and Cox processes. For the case of a mixed Poisson process, we may take Ω = NR#d ×R+ . Then the pair (N, λ) corresponds to the choice of a counting measure N ∈ NR#d and a rate λ ∈ R+ . The distribution P can be generated by conditioning as outlined in Section 6.1: Poi(V | λ) Π(dλ) (A ∈ BR+ ), P(V × A) = A

182


where V is a set of realizations from NR#d , Poi(· | λ) is the probability distribution on NR#d of a Poisson process at rate λ, and Π is the distribution of λ on the Borel sets of R+ . As in Example 12.1(a), the flow is the family of shifts Su on the first component. Stationarity is guaranteed, because invariance of the mixture P(·) is implied by invariance of each of the conditional distributions Poi(· | λ). The case of a Cox process is only a little more complicated. Here we may take Ω = NX# × M# X , where the first component refers to the realizations of the point process and the second to the realizations of the directing random measure. The flow must now act simultaneously on both components, so that (12.1.9) N (A, θu ω), ξ(B, θu ω) = N (A + u, ω), ξ(B + u, ω) or in more compact notation θu (N, ξ) = (Su N, Su ξ).

(12.1.9 )

We have then for the distribution P of the Cox process on Ω as above, Poi(V | ξ) Q(dξ) (W ∈ B(M# )), (12.1.10) P(V × W ) = Rd W

where V a set of counting measures, W is a set of directing measures ξ, Poi(· | ξ) is now the distribution of the inhomogeneous Poisson process with parameter measure ξ, and Q is the distribution of the directing random measure ξ. Exercise 12.1.4 indicates how to show that the resultant process is stationary if and only if Q is stationary. In this case the bivariate process N (·), ξ(·) is also stationary, its distribution being invariant under the same shift acting on both components. Similar constructions are possible in other cases where the evolution of the random measure under study is associated with the evolution of some auxiliary process. For stationarity of a general cluster process see Exercise 12.1.6; the important example of a Poisson cluster process is summarized shortly in Proposition 12.1.V where for the first time we meet a measure in (k) Rd invariant under the group of diagonal shifts Dx defined for k ∈ Z+ and d x ∈ R by (12.1.11) Dx(k) (y1 , . . . , yk ) = (x + y1 , . . . , x + yk ), where y = (y1 , . . . , yk ) and yi ∈ Rd for i = 1, . . . , k, so first we examine the structure of such measures. As in Appendix A2.7, the cosets under this group of transformations are images of the main diagonal y1 = · · · = yk . The (k) action of Dx along any such coset is just a shift through the vector x. Thus, we should anticipate that any measure invariant under the diagonal shifts should reduce to a multiple of Lebesgue measure in each such coset. The next lemma makes this idea precise: by the diagonal subspace we mean the space {(y1 , . . . , yk ): y1 = · · · = yk ∈ Rd }.

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Lemma 12.1.IV (Diagonal Shifts Lemma). Let µ be a boundedly finite Borel measure on X (k) with X = Rd . Then µ is invariant under the diagonal (k) shifts Dx of (12.1.12) if and only if it can be represented as a product of Lebesgue measure on the diagonal subspace and a reduced measure µ ˘ on X (k−1) such that, for any function f ∈ BM(X (k) ), and k > 0, f (x1 , . . . , xk ) µ(dx1 × · · · × dxk ) X (k) = dx f (x, x + y1 , . . . , x + yk−1 ) µ ˘(dy1 × · · · × dyk−1 ), (12.1.12) X

X (k−1)

where in the case k = 1, µ ˘(·) = mδ0 (·) in which δ0 denotes Dirac measure, m = µ(Ud ), and Ud is the unit d-dimensional hypercube. Proof. Consider the mapping from X × X (k−1) into X (k) defined by x, (y1 , . . . , yk−1 ) → (x, x + y1 , . . . , x + yk−1 ). (12.1.13) Given any (x1 , . . . , xk ) ∈ X (k) , we have uniquely x = x1 and yi = xi+1 − x1 (i = 1, . . . , k − 1), so the mapping is one-to-one and onto; it is clearly continuous and hence measurable. Under the mapping, the action of the diagonal (k) shifts Dx on X (k) is reduced to the ordinary shift Tx on the X component of the product X × X (k−1) . We therefore have a representation of the original space X (k) to which we can apply Lemma A2.7.II and assert that the image, µ∗ say, of µ induced by the mapping (12.1.13) reduces to a product of ddimensional Lebesgue measure along X and some measure µ ˘ on the other ˘. Then µ ˘ and µ are related as at factor space X (k−1) ; that is, µ∗ = × µ (12.1.12). For an alternative approach see Exercises 12.1.8–9 and 12.6.1–2. Proposition 12.1.V. A Poisson cluster process with a.s. finite clusters, and both cluster centres and cluster members in X = Rd , is stationary if and only if it can be represented in such a way that (i) the cluster centres form a stationary Poisson process in Rd ; and (ii) the cluster members depend only on their positions relative to the cluster centre, and not on the location of the cluster centre itself. In particular, the p.g.fl. of a stationary Poisson cluster process with a.s. finite clusters has a unique representation (its regular representation) of the form ∞ πk dx [h(x)h(x + y1 ) . . . h(x + yk−1 ) − 1] log G[h] = µc k! X X (k−1) k=1

Pk−1 (dy1 × · · · × dyk−1 ),

(12.1.14)

where µc is the intensity of the cluster centre process, {πk : k ≥ 1} is a proper probability distribution of cluster sizes, P0 (·) = δ0 (·), and for k ≥ 2, Pk−1 (·) is a symmetric probability distribution describing the locations of the remaining k−1 cluster members relative to an arbitrary cluster member chosen as origin.

184


Remark. As already noted around Proposition 6.3.V, the representation of a cluster process in terms of cluster centre and cluster member processes is not unique. In the present context, it is even possible to construct a stationary Poisson cluster process from nonstationary components; that is, they do not satisfy conditions (i) and (ii) (see Exercise 12.1.5). What the proposition asserts is that, even in such cases, the process will have an alternative representation where the above conditions do hold, and that when there is more than one representation, the regular representation is always available as an option. Proof. Recall from Proposition 6.3.V that a Poisson cluster process with a.s. finite clusters has a unique representation with p.g.fl. of the form ∞ 1 h(x1 ) . . . h(xk ) − 1 Kk (dx1 × · · · × dxk ), (12.1.15) log G[h] = k! X (k) k=1

where the Khinchin measure Kk has the representation Kk (B) = Jk (B | y) µc (dy)

(12.1.16)

X

in terms of the Janossy density Jk (· | y) of the cluster member process and the intensity measure µc of the Poisson cluster centre process. (Note that we here assume that both cluster centres and cluster members have points in the same space Y = X = Rd .) If the process is stationary, uniqueness of the representation (12.1.14) implies that each of the measures Kk must be invariant under diagonal shifts (x1 . . . , xk ) → (u + x1 , . . . , u + xk ), or equivalently Kk (Tu A1 × · · · × Tu Ak ) = Kk (A1 × · · · × Ak ). The diagonal shifts Lemma 12.1.IV now implies that Kk (·) must reduce to a product of Lebesgue measure along the diagonal, and a boundedly finite ˘ k−1 (·) on B(X (k−1) ) such that (12.1.15) holds. measure K These ingredients can be used to construct a candidate process for the regular representation. We introduce Janossy measures J0 = 0, J1 (dx | y) = δy (dx), and, for k > 1, ˘ k−1 (dy2 × · · · × dyk ), Jk (dx1 × dx2 × · · · × dxk | x) = (1/µc ) δ0 (dy1 ) K ∞ ˘ (k−1) where yi = xi − x for i = 1, . . . , k and µc = k=1 K )/k! is the k−1 (X candidate intensity of a stationary Poisson cluster centre process. We interpret πk = Jk (X (k) )/k! as the probability that a cluster has k members, and ˘ k−1 (·) Pk−1 (·) = (k! µc πk )−1 K

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as the symmetric probability distribution describing the location of the remaining cluster members relative to a given cluster member as centre; from symmetry the cluster member selected as centre may be regarded as being chosen uniformly at random. We leave the reader to verify that back-substitution of these candidate elements results both in a Poisson cluster process with the required properties and in (12.1.14) being satisfied. This establishes the necessity of a representation satisfying the conditions (i) and (ii) of the proposition, as well as the form of the regular representation. That the two conditions are sufficient to guarantee stationarity is a matter of verification. In p.g.fl. terms, condition (ii) of the proposition implies that for every x, Gm [h(·) | x] = Gm [h(· + x) | 0]. It is then straightforward to check that condition (12.1.6) in Theorem 12.1.III is met. Exercise 12.1.6 extends this argument to more general cluster processes; an alternative approach using Radon–Nikodym derivatives and the disintegration of measures is sketched in Exercises 12.1.8–9. The next example examines a particular case of this representation in detail, and shows that it is not always the most natural or convenient for further manipulations. Example 12.1(d) The regular representation of a stationary Neyman–Scott process [continued from Example 6.3(a)]. In the Neyman–Scott model the cluster members have a common distribution F (·) about the cluster centre. To obtain the regular representation we should refer the distribution of the cluster members to an arbitrarily chosen member of the cluster itself as origin. For clusters with just one element we have evidently 0 (A) = δ0 (A); P that is, the cluster is necessarily centred at its sole representative. For k = 2 we obtain 1 (A) = Pr{Y − X ∈ A} = F (x + A) F (dx), P X

where X and Y are independent r.v.s with the distribution F . Similarly for general k ≥ 2, k−1 (A2 × · · · × Ak ) = F (dx) F (x + A2 ) . . . F (x + Ak ). P X

k−1 gives In the branching process interpretation of the cluster members, P the distribution of the locations of the other siblings given that the arbitrarily chosen member comes from a family of size k. The use of (i) or (iv) rather than (ii) or (iii) of Theorem 12.1.III is indicated in the next example. It shows that a stationary measure of renewal type can be defined on NR# irrespective of whether the interval distribution has a finite or infinite first moment. Without a finite mean, this measure is not totally finite and so cannot be used directly to define a stationary point process; it is used in Exercise 12.4.6 to exhibit an example of a weakly singular infinitely divisible point process.

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Example 12.1(e) Stationary regenerative measure and renewal process [see also Exercises 9.1.13–14]. Let µ be a measure, not necessarily totally finite, defined on the space of sequences {Yn : n ∈ Z+ } satisfying Y0 = 0 ≤ Y1 ≤ · · · ≤ Yn → ∞ (n → ∞), and put τn = Yn − Yn−1 (n = 1, 2, . . .). We call µ regenerative when for any positive integer r and τi ∈ R+ (i = 1, . . . , r), r

dF (xi ), µ {τi ∈ (xi , xi + dxi ], i = 1, . . . , r} = µ1 (dx1 ) i=2

where µ1 is a boundedly finite measure on R+ and F is a d.f. on R+ . Then it follows as in Exercise 9.1.14 that µ defines a measure on B(NR#+ ). Our first aim is to show that when µ1 (dx1 ) = [1 − F (x1 )] dx1 , the measure µ is invariant under shifts Su for u > 0, even if it is not a probability measure as in Definition 12.1.II. When the counting measure N on R+ consists of unit atoms at Y1 , Y2 , . . . , with N (0, Yr ] = r for r = 1, 2, . . . as in Exercise 9.1.12, the successive atoms {Yn } for the counting measure Su N for u > 0 are given by Yn = Yn+ν − u, where the index ν = 0 if Y1 > u, = sup{n: Yn ≤ u} otherwise. Consequently, (Y0 ≡ 0), the measure Su µ on {Yn } is related to µ by writing τn = Yn − Yn−1 (Su µ) {τi ∈ (xi , xi + dxi ], i = 1, . . . , r} = µ {τ1 ∈ (u + x1 , u + x1 + dx1 ], τi ∈ (xi , xi + dxi ], i = 2, . . . , r} ∞ µ {τ1 + · · · + τj ≤ u, τ1 + · · · + τj+1 ∈ (u + x1 , u + x1 + dx1 ], + j=1

τj+i ∈ (xi , xi + dxi ], i = 2, . . . , r} ∞ µ {τ1 + · · · + τj ≤ u, = µ {τ1 ∈ (u + x1 , u + x1 + dx1 ]} + j=1

τ1 + · · · + τj+1 ∈ (u + x1 , u + x1 + dx1 ]}

×

r

dF (xi ).

i=2

For convenience, integrate x1 over (0, y] say, so that on the right-hand side, y when µ1 (0, y] = 0 [1 − F (x1 )] dx1 , the (r − 1)-fold product of terms dF (·) has coefficient

u+y

[1 − F (x1 )] dx1 + u

∞ j=1

· · · [1 − F (t1 )] dt1 dF (t2 ) . . . dF (tj+1 ),

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187

where the multiple integral ∞is over the set {t1 + · · · + tj ≤ u < t1 + · · · + tj+1 ≤ u+y}. Writing U0 (x) = j=1 F j∗ (x), so that U0 satisfies the renewal equation

x

F (x − y) dU0 (y),

U0 (x) = F (x) + 0

the multiple integral can be expressed as u−x1 +y y [1 − F (x1 )] dx1 dF (x2 ) 0

u−x1

= 0

u

0

u

u−v

[1 − F (x1 )] dx1

U0 (dv)

+

0

u−x1 −v+y

x+y

[1 − F (u − x)] dx dF (z) x u u−v + U0 (dv) [1 − F (u − v − x)] dx 0

dF (x2 )

u−x1 −v

0

x+y

dF (z). x

Here, the second term equals u u−x x+y dx [1 − F (u − x − v)] dU0 (v) dF (z) 0 0 x u x+y = F (u − x) dx dF (z), 0

x

so the coefficient of the (r − 1)-fold product of terms dF (·) equals

u+y

[1 − F (x)] dx + u

u

[F (y + x) − F (x)] dx = 0

y

[1 − F (x)] dx, 0

showing that ∞ µ is invariant as required. When 0 [1 − F (x)] dx ≡ λ−1 < ∞, λµ(·) is a probability measure, and so also is the measure it induces on B(NR#+ ). We can then identify the counting measure N (·) with such a stationary distribution as a stationary renewal process. We use the following proposition in discussing stationary infinitely divisible point processes; the result is of wider importance (see, e.g., Section 3.4 and the discussion of parallel lines in a stationary line process in Section 15.4). An analogue for MPPs is at Exercise 12.2.10. Proposition 12.1.VI (Zero–infinity Dichotomy). For a stationary random measure ξ on X = Rd , P{ξ(X ) = 0 or ∞} = 1. (12.1.17) Proof. The assertion is equivalent to showing that P{0 < ξ(X ) < ∞} = 0. Supposing the contrary, it necessarily follows that there exist some positive

188


constants a and γ such that for the hypercube Udγ with vertices {± 12 γ, . . . , ± 12 γ} and its complement (Udγ )c ,

P ξ: ξ(Udγ ) > a, ξ (Udγ )c < a = α > 0. Write Tγr Udγ for the shift of Udγ through the vector γr = (γr1 , . . . , γrd ), where r has integer-valued components so r ∈ Zd , and consider the events

Vr = ξ: ξ(Tγr Udγ ) > a, ξ Tγr (Udγ )c < a . By stationarity, P(Vr ) = P(V0 ) = α for all such r, and because the events Vr are disjoint for distinct r, P Vr = P(Vr ) = ∞ · α, r∈Zd

r∈Zd

which is impossible when P is a probability measure unless α = 0. Equation (12.1.17) prompts the following definition. Definition 12.1.VII. A random measure ξ is nonnull when P{ξ = ∅} = 0. It follows from (12.1.17) that a nonnull stationary random measure on X = Rd satisfies P{ξ(X ) = ∞} = 1. The discussion so far has centred on invariance with respect to shifts in Rd , but, as mentioned earlier, the ideas can be carried over with only nominal changes to processes invariant under other types of transformation, such as rotations, permutations of coordinates, or changes of scale. To conclude this section we examine one such example where, as in Rd , the state space itself is the group. In such cases we should anticipate that a basic role will be played by Haar measure which, analogous to the uniform distribution on the circle, or Lebesgue measure on the line, is the unique measure on the group invariant under the group actions. In the case of a Poisson process, for example, the properties of the process are determined by the parameter measure, which inherits the property of being invariant under the group actions from invariance under the corresponding flow. But the only measures invariant under the group actions are multiples of the Haar measure, and so the parameter measure itself must be a multiple of Haar measure [recall Examples 12.1(a) and (c)]. Even when there is no obvious governing measure, Haar measure will reappear in the moment measures, and lurks in the background behind the finitedimensional distributions. Its role in the latter context can be seen most clearly when the state space is compact as in the next example. Example 12.1(f) Stationary point process on the circle S. For a point process with state space the circle S, which we identify with angles θ modulo 2π, the compactness of S implies that the process necessarily has a.s. finite

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189

realizations, so explicit constructions in terms of Janossy measures are possible. Thus, supposing that the realization consists of exactly n points, defined by angles {θ1 , . . . , θn }, its distribution can be described by the symmetrized probability measure (conditional on n) Πn (dθ1 × · · · × dθn ) =

Jn (dθ1 × · · · × dθn ) . Jn (S(n) )

Stationarity (invariance under rotations) implies that for all θ ∈ S and A1 , . . . , An ∈ B(S), Πn (Tθ A1 × · · · × Tθ An ) = Πn (A1 × · · · × An ) so that we again have invariance under diagonal shifts. Here Lemma 12.1.IV implies that Πn can be written in terms of a product of the uniform measure ˘ n on a space of n − 1 arguments on S and a reduced probability measure Π (n) φ1 , . . . , φn−1 : for g ∈ BM(S ) we have g(θ1 , . . . , θn ) Πn (dθ1 × · · · × dθn ) S(n) dθ ˘ n (dφ1 × · · · × dφn−1 ). = g(θ, θ + φ1 , . . . , θ + φn−1 ) Π S 2π S(n−1) (12.1.18) The interpretation of this result is quite simple. If the distribution Πn of n points is rotationally invariant, it can be described by locating one point uniformly around the circle and the other n − 1 points relative to it according ˘ n . The symmetry properties of Πn imply that it to the reduced distribution Π is immaterial which point is designated as the one to be uniformly distributed, and stationarity (i.e., rotational invariance) implies that it is immaterial which point of the circle is chosen to play the role of origin. For example, if n = 2 and densities exist, the distribution of the two points is completely described by a symmetrical density function f (·) such that Π2 (dθ1 × dθ2 ) = (2π)−1 f (θ2 − θ1 ) dθ1 dθ2 . Note that here, as in general, it is a necessary consequence of stationarity that any one-dimensional marginal distribution such as Π2 (· × S) must be uniform. As a more specific example of such a process, consider first any symmetric distribution g(θ) about the origin (pole) θ = 0 [see, e.g., Mardia and Jupp (2000) for examples]. Take any fixed or random number of points independently distributed about the origin, to form the circular analogue of a Neyman–Scott cluster, N (· | 0) say. Then shift the origin to an angle uniformly distributed over S. This is already a single-cluster, stationary process on S. Finally, consider the superposition of N such processes, where N is Poisson distributed with mean ν. The result is a cluster process on S analogous to a Neyman–Scott process in time or space.

190


˘ n (·) defined by In this case, unfortunately, the reduction of Πn (·) to Π (12.1.18) is of little direct value in computing its properties. Even if the realization consists of only two points, the density function f for the angular separation of the two points will be an awkward mixture of densities that arise from pairs of points coming from either single or different clusters. For stationary point processes in general, the locations of points relative to a given point of the process as origin are independent of where that point itself is located. This is the theme of the Palm theory for stationary point processes discussed in Section 13.3. Often the main difficulty in applying the group concepts relates to the fact that the group G of transformations may split the space into equivalence classes in quite a complex manner. By contrast, the shifts act transitively on the whole space (any point can be transformed by a member of G into any other point) so that the equivalence classes are trivial, the whole space forming the unique equivalence class. Marked point processes on Rd form the canonical example of the sort of structure to be expected in more general cases. Here the state space has the representation X = Rd × K, in which the first factor is the group and the second can be regarded as a representation of the space of equivalence classes. This product form is the desired endpoint of analyses based on Lemma A2.7.II and Proposition A2.7.III. Any measure on X that is invariant under the group actions can then be expressed as the product of Haar measure on the group and a measure on the other component K. Of course the probability measure defining the point process does not live on X itself, but on NX# , but once again the underlying factorization of measures on X generally carries with it some corresponding simplifications of the probability distributions and the moment measures. To illustrate, we consider an extension of the previous example to the marked case. Example 12.1(g) A stationary MPP on S. To extend Example 12.1(f) to an MPP, start by supposing that the number n of points in the ground process is fixed, where the ground process is again specified by locating an initial point uniformly at random around the circle, and then locating the other points relative to it according to a reduced (n − 1)-dimensional symmetric ˘ n (φ1 , . . . , φn−1 ) say. To take the specific case when n = 2 as an distribution Π example, the process can be specified by two components: (i) a distribution F (φ) for the angular separation φ (in a given direction, clockwise say); and (ii) a family of symmetric bivariate distributions, G2 (K1 , K2 | φ) say, for the marks (with Ki ∈ BK for a mark space K that is a c.s.m.s.), given the angular separation φ. More generally, the associated distribution of marks can be specified by a family of n-dimensional symmetric distributions on K, Gn (K1 , . . . , Kn | θ, θ + φ1 , . . . , θ + φn−1 ) say, indexed by the angular locations; in the stationary case each Gn is independent of θ ∈ S. The simplest case is that of independent

12.1.


191

marks, but in general the joint distribution of the set of marks may depend on both the number and relative angles of the points in the ground process. Symmetry implies that the marginal distributions of the multivariate mark distribution are equal, and for a fixed number of points this can be taken as defining what is meant by the stationary mark distribution. However, in the general case of a random number of points, the marginal distributions may also depend on the value of n, so that the stationary mark distribution appears as a weighted average of the stationary mark distributions for realizations with different numbers of points (see Exercise 12.1.10).

Exercises and Complements to Section 12.1 12.1.1 (a) Modify the argument leading to Lemma 12.1.I to show that when X = Rd , # Su µ is jointly continuous as a mapping from M# X × X into MX . (b) Show that Su defined at (12.1.3) is a continuous and hence measurable # mapping of the space M# (M# X ) of boundedly finite measures on MX into itself, and that Su preserves measure and hence maps the set of all probability measures on M# X into itself. Verify that Su acts on the space M# (NX# ) of all boundedly finite measures on NX# in a similar way. [Hint: Su inherits the properties of Su in much the same way as Su inherits the properties of Tu . Continuity depends ultimately on the upper continuity of a measure at the empty set (Proposition A1.3.II).] 12.1.2 Give examples of nonstationary point processes for which (a) the avoidance function is stationary; (b) the one-dimensional distributions are stationary. [Hint: For integer-valued r.v.s X, Y, and X + Y, with X, Y ≥ 0 a.s., find a bivariate distribution for dependent X and Y with the same marginal distribution for X + Y as though X, Y are independent. Take X = Z and define the fidi distributions of a point process by using the dependent and independent bivariate distributions for alternate pairs of integers (see Ripley, 1976).] 12.1.3 Stationarity conditions for marked point processes. Verify that the following conditions for MPPs on Rd (i.e., for point processes on state space Rd × K), are equivalent, each corresponding to stationarity. (i) For each u ∈ Rd , k = 1, 2, . . . , and families A1 , . . . , Ak ∈ BRd and K1 , . . . , Kk ∈ BK , the fidi distributions satisfy P ({N (Ai × Ki ) = ni (i = 1, . . . , k)}) ≡ Pk (A1 × K1 , . . . , Ak × Kk ; n1 , . . . , nk ) = Pk ((A1 + u) × K1 , . . . , (Ak + u) × Kk ; n1 , . . . , nk ). (ii) For each u ∈ Rd and f ∈ BM(Rd × K), with Su f (x, κ) = f (x − u, κ), the characteristic functional satisfies Φ[Su f ] = Φ[f ]. (iii) For each u ∈ R and h ∈ V(Rd × K), the p.g.fl. G satisfies d

G[Su h] = G[h].

192


12.1.4 Starting from a Cox process N which with its directing measure ξ satisfies (12.1.9), use the p.g.fl. of N and the Laplace functional Lξ of ξ, which are related by (6.2.3) of Proposition 6.2.II, to verify that a Cox process N on Rd directed by ξ is stationary if and only if ξ is stationary. Using a similar approach show that the joint process (N, ξ) is stationary. [Hint: Stationarity means invariance as indicated at (12.1.4). Check that this holds if and only if ξ is invariant.] 12.1.5 The following two examples show that a stationary cluster process can be realized from nonstationary components. (a) Random thinning with deletion probability µ(x)/[1 + µ(x)] at x ∈ R1 of an inhomogeneous Poisson process at rate [1 + µ(x)] dx, where µ(x) ≥ 0 (all x), yields a stationary Poisson process (cf. Exercise 11.3.1). (b) Take a simple point process on R with points at {2n + U : n = 0, ±1, . . .}, where the r.v. U is uniformly distributed on (0, 1), to be a (nonstationary) cluster centre process. Let clusters be independent and let them consist of precisely two points at distances X1 and 1+X2 from the cluster centre, where for each cluster X1 and X2 are i.i.d. r.v.s. Then the cluster process so constructed is the same as the random translation of a stationary deterministic process at unit rate. 12.1.6 Let N be a cluster process (see Definition 6.3.I) with stationary centre process Nc on Rd and independent component processes Nm (· | y) (y ∈ Rd ) for which the fidi distributions of Nm (· | y), relative to y, are independent of y. Denote the p.g.fl. of Nm by Gm [· | y]. Referring to Lemma 6.3.II and Exercise 6.3.2, show that a stationary cluster process N is well defined if and only if

Rd

(1 − Gm [h | y]) Nc (dy) < ∞

a.s.

(h ∈ V(Rd )).

(12.1.19)

[Hint: Homogeneity of the Nm means that Gm [h(·) | y] = Gm [h(· + y) | 0].] 12.1.7 Stationary deterministic lattice processes [see Example 8.2(e) for the case d = 1]. Let the r.v. Y be uniformly distributed over the unit cube Ud in Rd , and let Zd denote the set of all integer-valued lattice points in Rd . Show that the point process N with sample realizations {n + Y : n ∈ Zd } is stationary. (Call N the stationary cubic lattice process at unit rate in Rd .) If the span of the lattice in the direction of the xi -axis, i = 1, . . . , d, is changed from 1 to ai , where the positive reals ai satisfy di=1 ai = 1, verify that stationarity at unit rate is retained.

É

12.1.8 (a) Let f (·) be a nonnegative measurable function on R satisfying for each fixed u ∈ R f (x + u) = f (x) (a.e. x). (12.1.20)

Ê

Show that there exists a finite constant α such that f (x) = α a.e. [Hint: y F (y) = 0 f (x) dx satisfies the Hamel equation F (x + y) = F (x) + F (y).] (b) Extend the result of (a) to Rd . [Hint: Apply (a) in a coordinatewise manner, deducing at the first step, for example, that in place of the constant α is a measurable function α(xd−1 ) (xd−1 ∈ Rd−1 ) satisfying (12.1.20).]

12.1.


193

12.1.9 Radon–Nikodym approach to construction of stationary cluster elements. (a) Check that each of the measures Kk1 (·) ≡ Kk ( · × X (k−1) ) in Proposition 12.1.V reduces to a multiple of Lebesgue measure. (b) Define the Radon–Nikodym derivatives Pk−1 (· | x) as in the discussion under (6.3.34) by Pk−1 (B | x) Kk (dx×X (k−1) ) = Kk (A×B)

(A ∈ BX , B ∈ B(X (k−1) ))

A

and observe that, for each fixed u, Pk−1 (Tu A2 × · · · × Tu Ak | x + u) and Pk−1 (A2 × · · · × Ak | x) are versions of the same density and hence equal a.e. (c) For fixed A2 , . . . , Ak , show that the function Pk−1 (Tu A2 × · · · × Tu Ak | u) in part (b) is a measurable function of u, implying by Exercise 12.1.8(b) that it reduces a.e. to a constant Pk−1 (A2 × · · · × Ak | 0). [Hint: For fixed u, the Radon–Nikodym theorem shows that Pk−1 (Tu A2 × · · · × Tu Ak | x + u) = Pk−1 (A2 × · · · × Ak | x) Kk1 -a.e. x. Integrate Kk (Tu A1 × · · · × Tu Ak ) over u and use Fubini’s theorem to express the result as an integral whose density with respect to the product measure du × dx is Pk−1 (Tu A2 × · · · × Tu Ak | x + u), thereby showing via the Radon–Nikodym theorem its joint measurability in x and u. Hence, by putting x = 0, deduce that Pk−1 (Tu A2 × · · · × Tu Ak | u) is a measurable function of u that is a.e. equal to Pk−1 (A2 × · · · × Ak | 0).] (d) Take a countable semiring A generating B(Rd ) and show that Pk−1 is countably additive on product sets of the form A2 × · · · × Ak for Ai ∈ A and so can be extended uniquely to a measure Pk−1 on B(R(k−1)d ) such that for all product sets with Ai ∈ B(Rd ), Pk−1 (Tu A2 × · · · × Tu Ak | u) = Pk−1 (A2 × · · · × Ak )

a.e.

12.1.10 In the setting and notation of Examples 12.1(f)–(g), put πn = Pr{N (S×K)}. Verify that the stationary mark distribution, for K ∈ BK , equals

∞ 1 n=1

∞

nπn

n=1

nπn S

dθ 2π

S(n−1)

Gn (K, K, . . . , K | θ, θ + φ1 , . . . , θ + φn−1 ) ˘ n (dφ1 × · · · × dφn−1 ). Π

12.1.11 Renewal process and random walk on S. Suppose given a probability distribution G(dθ) on (0, 2π], interpreted as the length of a step in the clockwise direction around the circumference of a circle, and for n = 1, 2, . . . let Un (A) denote the expected number of visits within the first n steps to the set A ⊆ (0, 2π]. Find conditions on G such that

2π

¯ Un (A)/n → 2π(A)/θ,

where θ¯ = 0 θ G(dθ) is necessarily finite and bounded by 2π. Investigate the behaviour when the conditions fail. [Hint: Formulate versions of the direct Riemann integrability and spread-out conditions of Section 4.4 for G, and apply the results for the real line, then wrap around the circle.]

194


12.2. Ergodic Theorems In this section we review some basic ergodic theorems and develop them for random measures and point processes. There are diverse examples of their application through the rest of this chapter and the next, and significant extensions of the theory in Sections 13.4–5. Let (Ω, E, µ) be a measure space and S a measure-preserving operator on this space; that is, µ(S −1 E) = µ(E) for E ∈ E. The classical ergodic theorems assert the convergence, in some sense and under appropriate conditions, of the n averages n−1 r=1 f (S r ω) to a limit function f¯(ω), which is invariant under the action of S [i.e., f¯(Sω) = f¯(ω)] for a measurable function f . When f is µintegrable, the limit function f¯ is also µ-integrable and the individual ergodic theorem asserts convergence µ-a.e. When f ∈ Lp (µ) for some 1 ≤ p < ∞, f¯ ∈ Lp (µ) also and the statistical ergodic theorem asserts convergence in the Lp norm. When µ is a probability measure, the limit function f¯(ω) is a random variable that can be identified with the conditional expectation of f with respect to the σ-algebra I of invariant events under S, that is, of those sets E ∈ E for which µ(S −1 E E) = 0. Writing X for f (ω), Xn for f (S n ω), and YX = E(X | I) for f¯, the individual ergodic theorem in the probability case can be written more graphically in the form 1 a.s. Xr → E(X | I) ≡ YX . n r=1 n

(12.2.1)

An important special case arises when the probability measure is such that the events in I all have probability measure either 0 or 1. In this case the transformation S is said to be metrically transitive with respect to the measure µ, and the process {Xn }, or its distribution, is said to be ergodic. In such circumstances the only invariant functions are constants, the conditional expectation in (12.2.1) reduces to the ordinary expectation, and (12.2.1) takes the familiar form n 1 a.s. Xr → m ≡ EX. n r=1 For a fuller discussion of these results with proofs and references, see, for example, Billingsley (1965). One other prefatory remark is in order. Given a stationary process {X(t): t ∈ R}, define the two σ-fields I1 and I of sets E ∈ E that are invariant under the shift transformations {Sn : n = 0, ±1, . . .} and {St : t ∈ R}, respectively. In general I = I1 , with I ⊆ I1 ; of course, if I1 is trivial, then so is I. This and the consequences of the sandwich relation below cover our main concerns. We consider first the implications of these results for stationary random measures on R. Here we take Ω = M# R and S as the shift through the unit distance. The measure-preserving character of S is then a corollary of

12.2.

Ergodic Theorems

195

1 stationarity. The simplest choice for X is the random variable X = 0 ξ(dx), which has finite expectation whenever ξ has finite mean intensity. Then Xn =

1

n+1

ξ(n + dx) = 0

ξ(dx) n

and the assertion in (12.2.1) becomes " 1 " ξ(0, n] a.s. → E ξ(dx) "" I . n 0

(12.2.2)

If, in particular, ξ is ergodic then ξ(0, n] a.s. → m. n

(12.2.3)

The results (12.2.2) and (12.2.3) seem simple, but they can be applied to many more general situations of which the simplest is to a continuous-time process. Observe first that from the simple sandwich relation ξ(0, T ] [T + 1] ξ 0, [T + 1] [T ] ξ 0, [T ] · ≤ ≤ · T [T ] T T [T + 1] we easily extend (12.2.2) to arbitrary intervals as for Proposition 3.5.I, so that " 1 " ξ(0, T ] a.s. → E ξ(dx) "" I . T 0

(12.2.4)

Because the limit is invariant under all shifts {St : t ∈ R}, it is I-measurable rather than just I1 -measurable, so that the conditional expectation can and will be taken with respect to I. As a corollary, consider the behaviour of a nonnegative measurable function f (·) on R, applied to a stationary measurable stochastic process X(·) on R. If E f X(t) < ∞, we can define a random measure ξ with finite mean intensity by setting f X(t) dt. ξ(A) = A

Applying (12.2.4) to such ξ yields the result 1 T

T

" a.s. f X(t) dt → E f X(t) " I .

0

The only restrictive feature is the limitation to nonnegative functions f : this is not inherent in the ergodic problem but arises from our concern with random measures rather than random signed measures.

196


Similar results hold in higher-dimensional spaces and in the more general context of metric groups considered at the end of Section 12.1. The main point of difficulty concerns the choice of averaging sets to replace the intervals (0, n] in (12.2.2). Even in the plane it is not difficult to find sequences {An } with An ⊂ An+1 and (An ) → ∞ such that the analogue of (12.2.2) fails in some cases (see Exercise 12.2.1). To consider this question further, let (Ω, E, µ) be a measure space acted on measurably by the group of measurable transformations {Sg : g ∈ G}, meaning that (g, ω) → Sg ω is jointly measurable, where G is a σ-group with unique right-invariant Haar measure χ. Note the most important fact that the averaging in ergodic theorems takes place over sets in G and not the state space X . For example, the individual ergodic theorem takes the form that, for suitable sequences {An }, An

f (Sg ω) χ(dg) χ(An )

→ f¯(ω)

µ-a.e.,

(12.2.5)

where, in the probability case, f¯(ω) is the conditional expectation E(f | I) with respect to the σ-algebra of events invariant under the whole family {Sg : g ∈ G}. A thorough discussion of extensions of the classical ergodic theorems in this context is given by Tempel’man (1972) [see also Tempel’man (1986) and Sinai (2000, Chapter 4, Section 3.3)] who sets out a range of conditions on the sequence {An }—some necessary, others sufficient—for the validity both of (12.2.5) and of corresponding statistical ergodic theorems. For the present discussion we adopt only the simplest of the conditions he describes. Definition 12.2.I. Let X = Rd . The sequence {An } of bounded Borel sets in Rd is a convex averaging sequence if (i) each An is convex; (ii) An ⊆ An+1 for n = 1, 2, . . . ; and (iii) r(An ) → ∞ (n → ∞), where r(A) = sup{r: A contains a ball of radius r}. Using this terminology, we set out versions of the individual and statistical ergodic theorems, referring to Tempel’man (1972) for proofs and further extensions. Proposition 12.2.II. (a) (Individual Ergodic Theorem for d-dimensional Shifts). Let (Ω, E, P) be a probability space, {Sx : x ∈ Rd } a group of measurepreserving transformations acting measurably on (Ω, E, P) and indexed by the points of Rd , {An : n = 1, 2, . . .} a convex averaging sequence in Rd , and I the σ-algebra of events in E that are invariant under the transformations {Sx }. Then for all measurable functions (random variables) f on (Ω, E, P) with E |f | < ∞, f (Sx ω) dx a.s. An → E(f | I). (12.2.6) (An )

12.2.

Ergodic Theorems

197

(b) (Statistical Ergodic Theorem for d-dimensional Shifts). Under the same conditions as in (a) and for p ≥ 1, "p " " " f (Sx ω) dx − E(f | I)"" → 0 E"" An (An )

all f ∈ Lp (P) .

(12.2.7)

Remark. In general the statistical ergodic theorem holds under weaker conditions on the sequence {An } than the individual ergodic theorem. Versions of the theorem remain true when the probability measure P is replaced by a σfinite measure µ, subject of course to the condition that |f (ω)| µ(dω) < ∞; see Proposition 12.4.V for an application. Our task is to apply these theorems to stationary random measures on the c.s.m.s. X : we consider two cases, X = Rd and X = Rd × K, for unmarked and marked point processes, respectively, where the c.s.m.s. K is a space of marks. # When X = Rd , we identify (Ω, E) with the space M# X , B(MX ) of boundedly finite measures ξ defined on BX , and Sx with the shift taking ξ(·) into ξ(· + x). If ξ has finite first moment measure, stationarity requires that this should reduce to a constant multiple m(·) of Lebesgue measure on Rd . # More generally, if X = Rd × K, we still take (Ω, E) = M# X , B(MX ) but identify {Sx } with shifts in the first coordinate only. Under stationarity, the first moment measure becomes a measure on the product space and it is invariant under shifts in the first component. The factorization Lemma A2.7.II then implies that the first moment measure has the product form ×ν, where ν is a boundedly finite measure on K. If the ground process has finite first moment measure, then it must be a multiple mg (·) of Lebesgue measure on Rd . In this case, ν(K) < ∞ and ν can be normalized to a probability measure π(·) on (K, BK ), the stationary mark distribution. The first moment measure is then of the form mg × π. We proceed to an extension of these remarks to the conditional expectation of ξ with respect to the appropriate invariant σ-algebra I. Lemma 12.2.III. Let ξ be a random measure on the product space X = Rd × K and I the σ-algebra of invariant events with respect to the shifts Sx in Rd . When ξ is stationary with respect to these shifts and such that its expectation measure exists, there exists an I-measurable random measure ψ(·) on K such that for all nonnegative measurable functions f on X , E Rd ×K

" " " f (x, κ) ξ(dx × dκ) " I = ψ(dκ) K

f (x, κ) (dx) Rd

P-a.s. (12.2.8)

In particular, for bounded B ∈ B(Rd ) and K ∈ BK , " E ξ(B × K) " I = (B) ψ(K)

P-a.s.

(12.2.9)

198


Proof. Let X be any r.v. on (Ω, E, P) with finite expectation and G ∈ I be an invariant set, so that P(G Sx G) = 0. Then X(ω) P(dω) = X(ω) P(dω) = X(S−x ω) P d(S−x ω) G S G G x X(S−x ω) P(dω), = G

so for all x ∈ R , d

E(X | I) = E(Sx X | I)

P-a.s.

(12.2.10)

Take X = ξ(A), and recall from 9.1.XV that there is a version of Proposition the conditional expectation, E ξ(·) | I ≡ η(·) say, which is again a random measure. Then (12.2.10) asserts that η(Sx A) = η(A) P-a.s. Take A of the form B × K as at (12.2.9), and let B and K run through the members of countable rings generating B(Rd ) and BK , respectively, and x through a countable dense set in Rd . Because only a countable family of null sets is involved, we can assume that (12.2.10) holds simultaneously for all such B, K, x, and for ω outside a single set V with P(V ) = 0. For ω ∈ /V it now follows from Lemma A2.7.II that η(B × K, ω) = (B) ψ(K, ω) for some kernel ψ on K × Ω. But η(·) was chosen to be an I-measurable random measure, so for each K the left-hand side is an I-measurable r.v. (more precisely, it can be extended to all ω ∈ Ω, in such a way as to form such a r.v.). Also, for a fixed ω ∈ / V , ψ(K, ω) is countably additive and its extension to V can be constructed so as to retain this property. Thus, ψ(·) is a random measure on K, from Proposition 9.1.VIII. This establishes (12.2.9), and (12.2.8) follows by standard extension arguments. Applying the definition in this lemma to a stationary MPP leads to an analogue of Proposition 12.1.VI (see Exercise 12.2.10). When X = Rd in Lemma 12.2.III, K reduces to a single point, and thus the random measure ψ is then an I-measurable random variable Y = E ξ(Ud ) | I , where Ud is the unit cube in Rd . Then (12.2.10) becomes the more familiar assertion that E ξ(A) | I = Y (A) P-a.s. (12.2.10 ) We can now state the main theorem of this section. It treats both marked and unmarked processes, and combines simple versions of both the individual and the statistical ergodic theorems. For more extensive results see MKM (1978, Section 6.2), Nguyen and Zessin (1979a), and the further discussion in Sections 13.4–5.

12.2.

Ergodic Theorems

199

Theorem 12.2.IV. Let the random measure ξ on X = Rd × K for some c.s.m.s. K be stationary with respect to shifts on Rd and have boundedly finite expectation measure × ν. Let ψ be the invariant random measure defined as in Lemma 12.2.III. Then for any convex averaging sequence {An } on Rd and any ν-integrable function h on K, 1 h(κ) ξ(An × dκ) → h(κ) ψ(dκ) (n → ∞) (12.2.11) (An ) K K a.s. and in L1 norm. If the second moment measure exists and 2 d E < ∞, h(κ) ξ(U × dκ) K

then convergence at (12.2.11) also holds in mean square. For an unmarked process (K reduces to a single point) the statements are equivalent to ξ(An ) → Y = E[ξ(Ud ) | I] (n → ∞) (12.2.12) (An ) a.s. and in L1 mean, and also in mean square if the second moment measure of ξ exists. Proof. We give details mainly for the unmarked case, and consider the proof of (12.2.12). d For some fixed ε > 0, let gε be a continuous function in R such that (i) gε (x) ≥ 0, Rd gε (x) dx = 1; and (ii) the support of gε (·) ⊆ Sε (0), the ball in Rd with centre at 0 and radius ε. Now define a function f on M# X by f (ξ) = gε (y) ξ(dy). Rd

It is clear that f is measurable and, because ξ has finite expectation measure, f is a P-integrable function with gε (x) dx = m. E(f ) = m Rd

Observe that when {An } is a convex averaging sequence, so are the related sequences {Aεn } and {A−ε n } with elements defined by A−ε and Aεn = Sε (x). n = {x: Sε (x) ⊆ An } x∈An

Also,

f (Sx ξ) =

Rd

gε (y) ξ(x + dy) =

Rd

gε (u − x) ξ(du).

200


This leads to the sandwich relation f (Sx ξ) dx = ξ(du) gε (u − x) dx A−ε Rd A−ε n n ξ(du) gε (u − x) dx = ≤ ξ(An ) ≤ Rd

Aεn

f (Sx ξ) dx,

Aεn

where the inequalities are consequences of properties (i) and (ii) of gε (·), which further imply that gε (u − x) dx = 0 (u ∈ / An ) and gε (u − x) dx = 1 (u ∈ An ). A−ε n

Aεn

Invoking Proposition 12.2.II(a), we obtain Y lim inf

ξ(An ) ξ(An ) (Aεn ) (A−ε n ) ≤ lim inf ≤ lim sup ≤ Y lim sup (An ) (An ) (An ) (An )

P-a.s.

Because r(An ) → ∞ and An is convex, (Aεn )/(An ) → 1 and (A−ε n )/(An ) → 1 as n → ∞. This establishes the a.s. assertion in (12.2.12). Also from Proposition 12.2.II(b), with p = 1, we have, with f as defined above and for n → ∞, " " " " " " " ε f (Sx ξ) dx " −ε f (Sx ξ) dx " " " An " An − Y − Y → 0 and E E" " → 0. " " ε) " " " " (A (A−ε ) n n Denoting the first terms in these differences by Ln and Un , respectively, these equations imply that E|Un − Ln | → 0 as n → ∞. Furthermore, (Aεn ) ξ(An ) (A−ε n ) Ln ≤ ≤ Un , (An ) (An ) (An ) where the coefficients of Ln and Un converge to 1, so E|ξ(An )/(An ) − Y | → 0 as n → ∞. This establishes the L1 convergence in (12.2.12). When the second moment measure exists, a similar argument with the L2 norm replacing the L1 norm establishes the L2 convergence. Turning now to the general marked case, let h(·): K → R be measurable and ν-integrable. Define a function f on M# X by gε (y)h(κ) ξ(dy × dκ), f (ξ) = Rd ×K

and observe that f (Sx ξ) is the same integral with gε (y) replaced by gε (y − x). Then form the integrals f (Sx ξ) dx and f (Sx ξ) dx, A−ε n

Aεn

12.2.

Ergodic Theorems

201

and invoke the general forms of Proposition 12.2.II and Lemma 12.2.III to assert that f (Sx ξ) dx Aεn → h(κ) ψ(dκ), (Aεn ) K with a similar statement holding with Aεn replaced by A−ε n ; here ψ(·) is the invariant random measure defined by Lemma 12.2.III. Similar inequalities and arguments now apply as in the unmarked case, and yield (12.2.11) in its a.s., L1 and L2 forms. As simple special cases of (12.2.12) and (12.2.11), the theorem yields the following corollaries. Corollary 12.2.V. (a) When ξ is stationary and metrically transitive with finite mean density m, ξ(An ) → m a.s. and in L1 norm. (An )

(12.2.13)

(b) If {(xi , κi )} is the realization of a stationary ergodic MPP on Rd × K, then with ν as in Theorem 12.2.IV, 1 (An )

a.s.

h(κi ) →

i:xi ∈An

h(κ) ν(dκ).

(12.2.14)

K

For versions of the L2 norm results, see Exercises 12.2.7–8. Numerous other special cases and corollaries follow from Theorem 12.2.IV such as the following (see also the exercises to this section). Proposition 12.2.VI. Under the conditions of Theorem 12.2.IV, for any measurable integrable function h(·) on Rd and with Y as at (12.2.12), Rd

h(y)ξ(An + y) dy = (An )

An

ξ(dx)

Rd

h(u − x) dx

(An )

→Y

h(y) dy Rd

a.s. and in L1 norm.

(12.2.15)

In all of these results, the convex averaging sequence {An } can of course be specialized to sequences of balls about the origin or nested hyper-rectangles whose smallest dimension → ∞. Higher-order ergodic theorems, requiring the existence of higher-order moment measures, are discussed in Section 12.6 and reappear in Chapter 13 in connection with higher-order Palm distributions. A different type of extension is outlined briefly in the proposition below. Proposition 12.2.VII (Weighted Averages). Let {an (·)} be a monotonic increasing sequence of nonnegative functions, convex upward, {An } a convex

202


averaging sequence in Rd , and ξ a stationary random measure on Rd with finite intensity m. Then as n → ∞, a (x) ξ(dx) a.s. An n → Y ≡ E ξ(Ud ) | I . (12.2.16) a (x) dx An n Proof. Define an associated random measure ξ on Rd × R by ξ (A × B) = ξ(A)(B). Then ξ is a stationary random measure in Rd+1 , and the sets An = {(x, u): x ∈ An , 0 ≤ u ≤ an (x)} are those of a convex averaging sequence in Rd+1 . Equation (12.2.16) follows by applying Theorem 12.2.IV to ξ . Some other classes of weighting functions can be handled by using the more general averaging sequences considered by Tempel’man (1972). In particular, this includes the class an (x) = a(x/tn ), where the nonnegative measurable function a(·) has bounded support (e.g., the unit cube) and {tn } is a sequence of nonnegative reals → ∞; a(·) need not be convex upward. Also, the assumptions of Proposition 12.2.VII can be trivially extended to the case where there exist positive constants bn such that an (x) ≤ bn an+1 (x). Ergodic theorems are important in nearly all branches of point process theory, whether in establishing properties of point process models, or in developing estimation and testing procedures in statistics, or in analyzing the behaviour of simulation routines for point process models. We conclude this section with an application to the frequency of occurrence of special configurations of points in a Poisson process. Discussions of particular configurations of points are closely related to one method of introducing Palm probabilities as ergodic limits (see in particular Theorem 13.2.VI). Such results can often be reduced to a direct application of Theorem 12.2.IV itself by introducing a suitable auxiliary random measure. We illustrate the procedure in a case where the expectation in the limit can be evaluated explicitly. Example 12.2(a) Configurations in a Poisson process. Let N be a stationary Poisson process in Rd at rate µ. Consider first the configuration consisting of a single point of the process with no neighbours within a distance a. The general (estimation) procedure is to take a convex region A, which we suppose to be a member of a convex averaging sequence, and count the number of points in A satisfying the required condition. Write this as the sum IB (Sxi N ), Y (A) = i:xi ∈A

where B = {N : N ({0}) = 1 = N Sa (0) }. Evidently, the sum can also be written as the counting process integral IB (Sx N ) N (dx), Y (A) = A

12.2.

Ergodic Theorems

203

and can be regarded as the value of a further point process Y (·) if A is allowed to range more generally over bounded sets of B(Rd ). In fact, Y (·) here is just a dependent thinning of the original process (see Section 11.3). Applying Theorem 12.2.IV to Y (·) yields the result that for increasing A, Y (A) →E IB (Sx N ) N (dx) = µpB , (12.2.17) (A) Ud where pB may be regarded as the probability that a given point will be retained in the thinning process: later, we show that pB can be interpreted as the Palm probability of the event B. In the special case considered here, we can evaluate the expectation by a simple approximation argument using the independence properties of the Poisson process as follows. The probability that there is a point in the small region (x, x+ δx) and none in the remainder of a ball Sa (x) centred at x is µ(δx) exp −µ Sa (x) −(δx) , and because the process is simple this is also the expected number of such configurations associated with the element (x, x + δx). Integration over Ud gives the limit as µe−µV (a) , where V (a) = Sa (x) is the volume of a sphere of radius a. Theorem 12.2.IV here asserts that the average density of points in A that have no neighbours closer than a, approaches this value as a limit when (A) → ∞ through a convex averaging sequence. Similarly, the average density of points in A that have at least k neighbours within a distance a approaches the limit µ 1 − e−µV (a) 1 + µV (a) + · · · + [µV (a)]k−1 /(k − 1)! . Finally, consider the numbers of pairs of points that lie within a distance a of one another. Taking one point of any such pair as a reference origin, at x say, the number of such pairs to which it belongs is just N Sa (x) − 1. Summing over all points in A leads to the integral 1 N Sa (x) − 1 N (dx), Y2 (A) = 2 A

the factor 12 arising (asymptotically when the edge effects from points near the boundary of A become negligible) from the fact that each point of each pair is counted twice. Dividing by (A) and noting that Y2 (·) defines a random measure to which the theorems can be applied, we obtain the quantity E[Y2 (Ud )] as the limiting value of such an average density of pairs; again this can be evaluated by an approximation argument that uses the independence properties of the Poisson process as d 1 E N Sa (x) \ Sδx (x) N (dx) E[Y2 (U )] = lim 2 δx→0 d U 1 µ V (a) − V (δx) µ dx = 12 µ2 V (a). = lim 2 δx→0

Ud

Of course, the expected numbers of pairs is related to the second factorial moment measure of the process, and the general version of the above argument leads to a higher-order ergodic theorem in which the reduced factorial moment measures appear as ergodic limits.

204


Exercises and Complements to Section 12.2 12.2.1 Suppose the stationary point process N (·) in R2 has sample realizations {Y + (n1 , 2n2 ): n1 , n2 = 0, ±1, . . .}, where Y is uniformly distributed on the rectangle (0, 1] × (0, 2]. Contrast the a.s. limits of N (An )/(An ) (n → ∞) when (2◦ ) An = n (1◦ ) An = (0, n] × (0, 1]; j=1 (j − 1, j] × (j − 1, j]; (4◦ ) An = (0, n] × (0, n]; (3◦ ) An = (0, 1] × (0, n]; (5◦ ) An = {(x, y): x ∈ (0, n], 0 ≤ y ≤ 2x/n}; and (6◦ ) An = {(x, y): x ∈ (0, n], x ≤ y ≤ x + 1}. 12.2.2 Show that Theorem 12.2.IV, which in the text is deduced from Proposition 12.2.II, implies the latter in the sense that, if ξ is a stationary random measure on Rd satisfying the assumptions of Theorem 12.2.IV and (12.2.12) holds, then (12.2.6) holds also. [Hint: For such ξ and f (ξ(·)) a functional with finite expectation, define a new random measure ξf by

ξf (A) =

(bounded A ∈ B(Rd )).

f (Sx ξ(·)) dx

A

Apply Theorem 12.2.IV to ξf to deduce Proposition 12.2.II(a) as applied to the original random measure ξ.] 12.2.3 Show formally that (a) if ξ has a trivial invariant σ-field, then the only invariant functions are the constant functions; and (b) metric transitivity implies ergodicity [i.e., (12.2.13) holds]. 12.2.4 Let F be a σ-algebra in B(M# X ) and ξ(·) a random measure on X with finite expectation measure. (a) Use Theorem 9.1.XIV to show that there exists a random measure ψ such that (bounded A ∈ BX ). ψ(A) = E[ξ(A) | F ] a.s. (b) Show that if F is chosen to be the σ-algebra of events invariant under shifts Sx , then ψ(·) is invariant in the sense that Sx ψ(A) = ψ(A)

a.s.


[Hint: Show first that the indicator function of any event in F is invariant, and ultimately that any F -measurable r.v. is invariant.] 12.2.5 Interpret the strong law result at Exercise 4.1.1 in the setting of Theorem 12.2.IV. 12.2.6 Establish statistical ergodic theorem versions of the individual ergodic theorems at Theorem 12.2.IV, Corollaries 12.2.V–VI and Proposition 12.2.VII. [Hint: Use Proposition 12.2.II(b).] 12.2.7 L2 convergence of a stationary MPP in Rd . Show that if the stationary ergodic MPP ξ in Rd satisfies the conditions of Theorem 12.2.IV, including the existence of finite second moments, then for any convex averaging sequence {An }, 2 ξ(An × K) lim E = 0, − ψ(K) n→∞ (An )

where ψ(·) is as in Lemma 12.2.III.

12.2.

Ergodic Theorems

205

12.2.8 Let the simple point process N on R have stationary second-order distributions and finite second-order moment. Then the expectation function U (·) of (3.5.2) satisfies U (x)/x → λ (x → ∞) for some λ ≥ λ [Exercise 8.1.3(e) or Lemma 9 in Daley (1971)]. Examine the implications of Proposition 12.2.II for an L2 norm result. 12.2.9 (a) Use Theorem 12.2.IV, the inversion result at (8.6.8), and the identification of the Bartlett spectrum Γ(·) as in Definition 8.2.II to show that for a nonergodic second-order stationary random measure ξ on Rd , Γ({0}) = var Y, where Y is the I-measurable r.v. as in (12.2.10 ) and (12.2.12). (b) For a second-order stationary random measure ξ on R and with V (x) = var ξ(0, x], recall from Exercise 8.1.3(b) that limx→∞ x−2 V (x) exists and ∞ is finite. Defining v(s) = 0 e−sx dV (x), use an Abelian theorem for Laplace–Stieltjes transforms [e.g., Widder (1941, p. 181)] to show that lims→0 s2 v(s) = limx→∞ 2x−2 V (x). Then show from (8.2.3) that Γ(dω) 1 2 s v(s) = Γ({0}) + , 2 2 2 R\{0} 1 + ω /s and conclude that var ξ(0, x] ∼ x2 Γ({0}) (x → ∞). 12.2.10 Let N be a marked point process on Rd ×K, stationary as in Lemma 12.2.III. Extend Proposition 12.1.VI to the statement that for each K ∈ B(K), P{N (Rd × K) = ∞} + P{N (Rd × K) = 0} = 1. 12.2.11 A stationary nonisotropic point process in R2 . For i ∈ Z let {Ni (·)} be independent copies of a simple stationary point process on R for which var Ni (0, 1] < ∞ and with generic realization {xij : j ∈ Z} at rate λ1 ; let {yi : i ∈ Z} be a realization of some other simple stationary point process N 0 (·) on R at rate λ0 and with finite second moment measure, independent of the Ni . Write N (·) for the counting measure of the point process in R2 with realizations {(xij , yi ): i, j ∈ Z}. (a) Show that N is a stationary point process in R2 . (b) Using the convex averaging sets {A1n } and {A2n } specified by A1n = (0, n] × (0, n2 ] and A2n = (0, n2 ] × (0, n], show that both N (A1n )/(A1n ) a.s. and N (A2n )/(A2n ) → E[N (A11 )] = λ0 λ1 . 0 (0,y] Ni (0, x] for the rect(c) Use the representation N ((0, x] × (0, y]) = N i=1 angle (0, x] × (0, y] to show that var N ((0, x] × (0, y]) equals x y [1 + 2U (u)] du + λ0 λ21 x2 2[U 0 (v) − λ0 v] dv, λ0 yλ1 0

0

where U (·) and U 0 (·) are the expectation functions for the Ni and N 0 , respectively. Deduce that, when var N1 (0, x] = O(x) and var N 0 (0, y] = O(y) for x, y → ∞, and for the same convex averaging sets as in (b), var N (A1n ) = O(n4 ) but var N (A2n ) = O(n5 ) and (A1n ) = n3 = (A2n ).

206

12. Stationary Point Processes and Random Measures (d) Suppose that both Ni (·) and N (·) have covariance density functions, c(·) and c0 (·) say. Show that Pr{N ((x, x + dx) × (y, y + dy)) > 0 | N ({0, 0}) = 1} =

λ[c(x) + λ] dx

if y = 0,

λλ0 [c0 (y) + λ0 ] dx dy

if y = 0.

Does N have a reduced covariance density, c((x, y)) say?

12.3. Mixing Conditions In practice, the useful applications of the ergodic theorem are to those situations where the ergodic limit is constant or, in other words, where the process is metrically transitive (the invariant σ-algebra is trivial). It is therefore important to characterize as fully as possible the various classes of processes that have this property. Now the absence of nontrivial invariant events is closely related to the absence of long-term dependence, and thus, checking for metric transitivity is generally accomplished by verifying that some kind of asymptotic independence or mixing condition is satisfied. This section contains a review of such conditions and outlines some of their applications. As in the previous section we suppose that either X = Rd or X = Rd × K, and write Sx for the operator on M# X defined by shifts as in (12.1.2) or, for X = Rd × K, by shifts in the first coordinate Sx ξ(·, K) = ξ(· + x, K). We also write Ud2a for the hypercube in Rd with sides of length 2a and vertices (±a, . . . , ±a), and P for the probability measure of a random measure on M# X or a point process on NX# . Definition 12.3.I. A stationary random measure (respectively, point process) on state space X = Rd or Rd × K is # (i) ergodic if, for all V , W in B(M# X ) [respectively, B(NX )] 1 P(Sx V ∩ W ) − P(V )P(W ) dx → 0 (a → ∞); (12.3.1) (Uda ) Uda (ii) weakly mixing if for all such V , W , " " 1 "P(Sx V ∩ W ) − P(V )P(W )" dx → 0 d (Ua ) Uda

(a → ∞); (12.3.2)

(iii) mixing if for all such V , W , P(Sx V ∩ W ) − P(V )P(W ) → 0

(x → ∞);

(12.3.3)

(iv) ψ-mixing (on R1 ) if for u > 0, t ∈ R, and a function ψ(u) with ψ(u) ↓ 0 as u → ∞, |P(V ∩ W ) − P(V )P(W )| ≤ ψ(u) (12.3.4) whenever V ∈ σ{ξ(A): A ⊆ (−∞, t]} and W ∈ σ{ξ(B): B ⊆ (t+u, ∞)}.

12.3.

Mixing Conditions

207

The conditions are written in order of increasing strength: it is clear that mixing implies weak mixing, which in turn implies ergodicity. Furthermore, any completely random measure, such as the Poisson process, clearly satisfies all four conditions. The first three conditions apply to point processes and random measures generally; the fourth is introduced specifically to illustrate the central limit theorem at Proposition 12.3.X. Before examining the conditions in more detail, we show that in general it is enough to check the properties # on any semiring of events generating the Borel sets in M# X : replacing MX by # NX throughout leads to the same statement for point processes. Lemma 12.3.II. For a stationary random measure the limits in (12.3.1–4) hold for all V , W ∈ B(M# X ), if and only if they hold for V , W in a semiring S generating B(M# ). X Proof. We establish the truth of the assertion for (12.3.3) (mixing); the other cases are proved similarly. Let F ⊆ B(M# X ) denote the class of sets for which (12.3.3) holds. It is clear that if (12.3.3) holds for finite families of disjoint sets V1 , . . . , Vj and W1 , j k . . . , Wk , then it holds also for V = i=1 Vi and W = i=1 Wi . So, if (12.3.3) holds for sets in a semiring S, it holds for sets in the ring R generated by S. Suppose that W ∈ F and Vn ∈ F for n = 1, 2, . . . with Vn ↑ V . Now " " "P(Sx V ∩ W ) − P(V )P(W )" " " " " ≤ "P(Sx V ∩ W ) − P(Sx Vn ∩ W )" + "P(Sx Vn ∩ W ) − P(Vn )P(W )" " " + "P(Vn )P(W ) − P(V )P(W )", in which the first term on the right-hand side is bounded above by P Sx (V Vn ) ∩ W 0, we can fix n large enough such that each term < ε, uniformly in x. For the middle term, having fixed n, (12.3.3) holds for the pair Vn , W , so for x > some x0 , this term < ε also. Thus, " " "P(Sx V ∩ W ) − P(V )P(W )" < 3ε (all x > x0 ). Similarly, we may also replace W by a sequence {Wn } ⊆ R with Wn → W , showing that F is closed under monotone limits. Thus, F is a monotone class which, because it includes R, includes σ{R} = B(M# X ). Our aim now is to establish links with the theorems of the previous section. The next proposition establishes the equivalence of metric transitivity (trivial invariant σ-algebra) with the ergodicity condition at (i) of Definition 12.3.I above. It implies that in talking of an ergodic point process we may use the two criteria indifferently.

208


Proposition 12.3.III. A stationary random measure or point process is ergodic if and only if it is metrically transitive; that is, the invariant σ-algebra I is trivial. Proof. Let ξ be ergodic as at (12.3.1) above and let A be an invariant event. Putting V = W = A in (12.3.1), observe from invariance that P(Sx A ∩ A) = P(A) and hence, using ergodicity, that 1 P(A) − [P(A)]2 dx → 0 (a → ∞), d (Ua ) Uda which is possible only if P(A) = 0 or 1. Conversely, suppose that I is trivial, so that (12.2.6) takes the form, in the notation as there, 1 f (Sx ξ) dx → E[f (ξ)] a.s. (An ) An Let V , W be as in Definition 12.3.I and take f (ξ) = IV (ξ), so that Ef (ξ) = P(V ) and (12.2.6) yields 1 IV (Sx ξ) dx → P(V ) a.s. (An ) An Writing gn (ξ) for the left-hand side of this equation, observe that 0 ≤ gn (ξ)≤1 # and that gn (ξ) is a measurable function of ξ in M# X , B(MX ) . Integrating over W and using dominated convergence and Fubini’s theorem, we obtain 1 IV (Sx ξ) dx P(dξ) → P(V )P(W ), (An ) An W which reduces to (12.3.1) in the special case An = Udn . Just as ergodicity is related to the invariant σ-algebra I being trivial, so mixing is related to the σ-algebra of tail events defined on the process ξ being trivial (this σ-algebra being, in general, larger than I). To define these events, denote by Ta , for each a > 0, the σ-algebra of events defined by the behaviour of ξ outside Uda ; that is, Ta is the smallest σ-algebra in B(M# X ) with respect to which the ξ(A) are measurable for A ∈ B(X \Uda ). Definition 12.3.IV. ∞ The tail σ-algebra of the process ξ is the intersection T∞ ≡ a>0 Ta = n=1 Tn . An element of T∞ is a tail event. Thus, T∞ defines the class of events that are determined by the behaviour of ξ outside any bounded subset of X . It is not difficult to show that, modulo sets of P-measure zero, any invariant event is in the tail σ-algebra (see Exercise 12.3.2). The converse is not true, however: periodic processes provide typical examples of processes that are ergodic but for which the tail σ-algebra is non-trivial (see Exercise 12.3.1).

12.3.

Mixing Conditions

209

The triviality result referred to above is set out below. Triviality of the tail σ-algebra is also closely related to the concept of short-range correlation; see Exercise 12.3.4 for a definition and details of the relationship. Note that the term short-range correlation is well established in the physics literature, although it relates to stochastic dependence rather than any second-order product-moment property; paradoxically, the terms long- and short-range dependence (see Section 12.7 for the point process setting) are established in the statistical literature as pertaining to a second-order or correlational (!) property. Proposition 12.3.V. If the tail σ-algebra is trivial, then the random measure ξ is mixing. Proof. Let V be any set in B(M# X ). Because Tn ↓ T∞ , we have for any random variable with expectation, and hence in particular for the indicator function IV (·), E(IV | Tn ) → E(IV | T∞ ) a.s. [this is a standard result for backward martingales; see, e.g., Chung (1974, Theorem 9.4.7)]. When T∞ is trivial, the right-hand side here reduces to E(IV ) = P(V ) a.s. Consequently, given ε > 0, we can choose n0 such that for n ≥ n0 , " " "E(IV | Tn ) − P(V )" < ε a.s. (12.3.5) Now let W be a cylinder set belonging to the σ-algebra generated by the family {ξ(A): A ∈ B(Uda )}. For x sufficiently large, namely, x > d1/2 n, Tx Uda lies in the complement of Udn and hence Sx W ∈ Tn . For fixed n and x large enough, the indicator function ISx W is therefore Tn -measurable and so satisfies E(ISx W IV | Tn ) = ISx W E(IV | Tn ) a.s. Taking expectations and using (12.3.5), we obtain P(Sx W ∩ V ) = P(Sx W )P(V ) + E(ISx W Y ), where the r.v. Y has |Y | < ε a.s. This establishes the mixing property for arbitrary V and any cylinder set W . Because the cylinder sets generate B(M# X ), the proposition follows from Lemma 12.3.II. Mixing is defined above in terms of probabilities of events; the conditions can equally be stated in terms of expectations of random variables defined on the process: see Exercises 12.3.5–6 for details in the context of convergence to equilibrium, which topic is discussed more fully in Section 12.5. We already observed in the proof of the last proposition that it is enough to verify the mixing or weak mixing or ergodicity conditions for cylinder sets, that is, for sets of the type that occur in the definition of the fidi distributions. Although this may sometimes be convenient, it is generally easier to check

210


the conditions in a form that relates to the generating functionals rather than directly to the fidi distributions. The next proposition provides such conditions: the discussion here, as in the applications that follow, is based on Westcott (1972). In the proposition below we use the shift operator S. defined on functions h(·) by (Sx h)(y) = h(y + x) [compare with (12.1.1)]. Proposition 12.3.VI. (a) Let ξ be a random measure, L[·] its Laplace functional, and h1 , h2 functions in BM+ (X ). (i) ξ is ergodic if and only if for all such h1 , h2 , 1 L[h1 + Sx h2 ] − L[h1 ]L[h2 ] dx → 0 (n → ∞). (12.3.6) d (Un ) Udn (ii) ξ is weakly mixing if and only if for all such h1 , h2 , " " 1 "L[h1 + Sx h2 ] − L[h1 ]L[h2 ]" dx → 0 (n → ∞). d (Un ) Udn

(12.3.7)

(iii) ξ is mixing if and only if for all such h1 , h2 , L[h1 + Sx h2 ] → L[h1 ]L[h2 ]

(x → ∞).

(12.3.8)

(b) Let N be a point process, G[·] its p.g.fl., and h1 , h2 functions in V(X ). (i) N is ergodic if and only if for all such h1 , h2 , 1 G[h1 Sx h2 ] − G[h1 ]G[h2 ] dx → 0 (n → ∞). (12.3.9) (Udn ) Udn (ii) N is weakly mixing if and only if for all such h1 , h2 , " " 1 "G[h1 Sx h2 ] − G[h1 ]G[h2 ]" dx → 0 (n → ∞). d (Un ) Udn

(12.3.10)

(iii) N is mixing if and only if for all such h1 , h2 , G[h1 Sx h2 ] → G[h1 ]G[h2 ]

(x → ∞).

(12.3.11)

Proof. (a) Let {Aj : j = 1, . . . , J} and {Bk : k = 1, . . . , K} be bounded on X , and consider the family of random variables {ξ(Aj )} ∪ {ξ(Tx Bk )}. If ξ is mixing then (12.3.3) implies that the joint distribution of this family converges to the product of the two joint distributions of the families {ξ(Aj )} and {ξ(Bk )}. It follows that the multivariate Laplace transforms of these joint distributions satisfy for real nonnegative {αj } and {βk } J K αj ξ(Aj ) − βk ξ(Tx Bk ) E exp − j=1

k=1

J K → E exp − αj ξ(Aj ) E exp − βk ξ(Bk ) . j=1

k=1

12.3.

Mixing Conditions

211

But this is just the statement (12.3.8) for the special case that the hi are the simple functions h1 (x) =

J

αj IAj (x),

h2 (x) =

j=1

K

βk IBk (x).

(12.3.12)

k=1

Now any h ∈ BM+ (X ) can be monotonically and uniformly approximated by simple functions of this form, so an argument similar to that of Lemma 12.3.II shows that (12.3.8) holds as stated. Conversely, when (12.3.8) holds, take h1 , h2 to be simple functions as at (12.3.12). Then it follows from the continuity theorem for Laplace transforms that the joint distributions of {ξ(Aj )} and {ξ(Tx Bk )} converge for all families {Aj } and {Bk }, and hence that (12.3.3) holds for the corresponding cylinder sets, that is, a semiring generating B(M# X ). Then by Lemma 12.3.II, (12.3.3) holds generally and so ξ is mixing. Analogous statements hold in the other cases; we omit the details. It is important for our proof below of Proposition 12.3.IX to observe that the convergence properties in equations (12.3.6–11) hold for wider classes of functions than those with bounded support. For example, when each hi is the monotone limit of a sequence {hin } ⊂ BM+ (X ), (12.3.8) holds provided we interpret the functionals as extended Laplace functionals [see (9.4.11)]. To see this, recall that any function in BM+ (X ) is the monotone limit of simple functions, and first assume that the functions h1 and {h2n } are simple functions. By the argument leading to (12.3.12) we then have, when ξ is stationary and mixing, L[h1 + Sx h2n ] → L[h1 ]L[h2n ] as x → ∞. Thus, 0 ≤ |L[h1 ]L[h2 ] − L[h1 + Sx h2 ]| " " " " ≤ "L[h1 ](L[h2 ] − L[h2n ])" + "L[h1 ]L[h2n ] − L[h1 + Sx h2n ]" " " + "L[h1 + Sx h2n ] − L[h1 + Sx h2 ]" ≡ δ1n + δ2n (x) + δ3n (x),

say.

Now " " " δ3n (x) = "E exp − h1 dξ exp − Sx h2n dξ − exp − Sx h2 dξ by nonnegativity and monotonicity, ≤ |L[Sx h2n ] − L[Sx h2 ]| by stationarity, = |L[h2n ] − L[h2 ]| and by monotone convergence, L[h2n ] ↓ L[h2 ] as n → ∞. Similarly, δ1n ≤ |L[h2n ]−L[h2 ]| and so, given ε > 0, we can make both δ1n < ε and δ3n (x) < ε, uniformly in x, by choosing n sufficiently large. Fixing such n, (12.3.12) now implies that we can make δ2n (x) < ε by taking x sufficiently large. Thus, (12.3.8) holds for simple h1 ∈ BM+ (X ) and h2 ∈ BM+ (X ), and a similar argument establishes it for h1 ∈ BM+ (X ) as well.

212


Extensions of the rest of (12.3.6–11) are established in a similar manner. It is also pertinent to note that in part (b), it is enough to restrict the functions hi to the subspace V0 (X ) ⊂ V(X ). This follows directly from part (a) and the remark following (9.4.14). Using these results, we can investigate the mixing and ergodicity properties of some classes of point processes, namely Cox and cluster processes in Propositions 12.3.VII–IX, and interval properties (e.g., renewal processes) in Exercise 12.4.1 and Section 13.4. Proposition 12.3.VII. A stationary Cox process N on X = Rd is mixing, weakly mixing, or ergodic if and only if the random measure Λ directing N has the same property. Proof. Recall from Proposition 6.2.II that the p.g.fl. G[·] of a Cox process N is related to the Laplace functional L[·] of the random measure Λ directing N by G[h] = L[1 − h] for h ∈ V(X ). To verify the mixing property we start from the relation G[h1 Sx h2 ] = L[1 − h1 Sx h2 ] = L[1 − h1 + Sx (1 − h2 ) − (1 − h1 )Sx (1 − h2 )]. Because each 1 − hi (i = 1, 2) has bounded support, the last term vanishes for sufficiently large x, and for such x, appealing to (12.3.8) for x → ∞, G[h1 Sx h2 ] = L[(1 − h1 ) + Sx (1 − h2 )] → L[1 − h1 ]L[1 − h2 ] = G[h1 ]G[h2 ]. This argument is reversible and proves the result concerning the mixing property. Proofs for the weakly mixing and ergodicity properties are similar. Corollary 12.3.VIII. A stationary mixed Poisson process N is mixing if and only if it is a simple Poisson process. Proof. From Example 9.4(d) and Exercise 12.1.4, the directing measure ξ must be a random multiple of Lebesgue measure, Λ(·) say, so for h ∈ V(Rd ) the p.g.fl. G[h] of N equals φΛ

Rd

[1 − h(x)] dx ,

where φΛ (·) denotes the Laplace–Stieltjes transform of Λ. Now as x → ∞, [1 − h1 (x)] dx + [1 − h2 (x)] dx , L[(1 − h1 ) + Sx (1 − h2 )] → φΛ Rd

Rd

which can equal the product of the φΛ Rd [1 − hi (x)] dx for all hi ∈ V(Rd ) if and only if φΛ is an exponential function, and hence the distribution of Λ is concentrated at a single point. By a stationary cluster process N on Rd , we mean a process as in Exercise 12.1.6 (or Proposition 12.1.V for a stationary Poisson cluster process).

12.3.

Mixing Conditions

213

Proposition 12.3.IX. A stationary cluster process is mixing, weakly mixing, or ergodic, whenever the cluster centre process has the same property. Proof. We give details for the mixing case only; the other cases can be treated in a similar fashion. Also, although we generally follow the p.g.fl. proof of Westcott (1971) [for an alternative proof see MKM (1978, Proposition 11.1.4)], some further argument is needed as in Daley and Vere-Jones (1987). In particular, we use the idea of extended p.g.fl.s and use Proposition 12.3.VI(b) with functions hi ∈ V0 (X ) (see the last remark following that proposition). Then in view of (12.1.18) and (12.3.11) with hi ∈ V0 (X ), it is enough to deduce from Gc [h1 Sx h2 ] → Gc [h1 ]Gc [h2 ] as x → ∞ that (12.3.13) Gc Gm [h1 Sx h2 | · ] → Gc Gm [h1 | · ] Gc Gm [h2 | · ] . Formally, the mixing property implies that the right-hand side here is the limit at x → ∞ of ˜ 1 Sx h ˜ 2 ], (12.3.14) Gc Gm [h | · ] Sx Gm [h2 | · ] ≡ Gc [h ˜ i (y) = Gm [hi | y]; so, formally, it is enough to show that as x → ∞, where h ˜ 1 Sx h ˜ 2 ] − Gc Gm [h1 Sx h2 | · ] → 0. (12.3.15) Gc (h We have said ‘formally’ because, although hi ∈ V0 (Rd ), the same need not ˜ i . However, by replacing the generic cluster Nm (· | 0) necessarily be true of h ˜ i is expressed as by Nmn (· | 0) = Nm (· ∩ Udn | 0) and letting n → ∞, each h ˜ the limit of the monotonic sequence {hin } for which hin ∈ V0 (Rd ). Consequently, appealing to the convergence properties of extended p.g.fl.s established in Exercise 9.4.6(b) and the extended form of the mixing property (12.3.11) noted below Proposition 12.3.VI, it follows that when Nc (·) is mix˜ 1 Sx h ˜ 1 ] Gc [h ˜ 2 ] as x → ∞. ˜ 2 ] → Gc [h ing, Gc [h To complete the proof, define x (u) = Gm [h1 Sx h2 | u] − Gm [h1 | u] Sx Gm [h2 | u]. As upper bounds on x (u) we have x (u) ≤ Gm [Sx h2 | u](1 − Gm [h1 | u]) ≤ 1 − Gm [h1 | u], x (u) ≤ Gm [h1 | u](1 − Gm [Sx h2 | u]) ≤ 1 − Gm [Sx h2 | u]; as lower bounds we have x (u) ≥ Gm [h1 Sx h2 | u] − Gm [Sx h2 | u] = −E exp X log[Sx h2 (y)] Nm (dy | u) × 1 − exp X log h1 (y) Nm (dy | u) ≥ −(1 − Gm [h1 | u])

214


and, similarly, x (u) ≥ −(1 − Gm [Sx h2 | u]). Because h2 ∈ V0 (Rd ) and Nm (Rd | u) < ∞ a.s., Gm [Sx h2 | u] → 1 as x → ∞ and thus x (u) → 0 as x → ∞. Also, because the cluster process exists, (12.1.18) holds, and therefore |x (u)| is bounded above by 1 − Gm [h1 | u], which is Nc -integrable a.s. Indeed, again because h1 ∈ V0 (Rd ) and Nm (Rd | u) < ∞ a.s., it also holds that 1 ≥ Gm [h1 | u] ≥ c1 , uniformly in u, for some positive constant c1 , and similarly for Gm [Sx h2 | u]. Thus, χx (u) ≡ Gm [h1 | u]Sx Gm [h2 | u] ≥ c > 0 for some constant c, uniformly in x and u. Now " " "Gc Gm [h1 Sx h2 | ·] − Gc Gm [h1 | ·]Sx Gm [h2 | ·] " = |Gc [χx + x ] − Gc [χx ]| " " = "E exp Rd log[χx (u) + x (u)] Nc (du) − exp Rd log χx (u) Nc (du) " |x (u)| Nc (du) . ≤ 1 − E exp − log 1 + χx (u) Rd This expression → 0 as x → ∞ because 1 ≥ χx (u) ≥ c > 0 (all x and u), x (u) → 0 pointwise, and x (u), and hence c−1 x (u) also, is Nc -integrable a.s. uniformly in x (see Exercise 9.4.6). Thus, (12.3.13) is proved. One of the classical applications of mixing conditions is in establishing conditions for a central limit theorem. Here, the point process or random measure character of the realizations plays an entirely minor role; it is not the local behaviour but the behaviour over large time spans that is important. Results for point processes and random measures can, indeed, be written down directly from the results in texts such as Billingsley (1968) for stochastic processes in general. Because rescaling is involved, the limits need no longer correspond to random measures, and convergence needs to be expressed in terms of convergence in a function space such as D(0, 1). For example, adapting Billingsley’s Theorem 20.1 to random measures yields the following. Proposition 12.3.X. Let the stationary random measure ξ on X = R be ψ-mixing for ∞some continuous, monotonic, nonnegative function ψ(·) on R+ for which 0 [ψ(t)]1/2 dt < ∞, and have boundedly finite first and second moment measures, with mean rate m and reduced covariance measure C(·) satisfying 0 < σ 2 = C(R) < ∞. Then the sequence of random processes {Yn } defined by (0 ≤ t ≤ 1) Yn (t) = {ξ(0, nt] − mnt}/σn1/2 converges weakly in D(0, 1) to the Wiener process on (0, 1).

Exercises and Complements to Section 12.3 12.3.1 Prove that a stationary renewal process can exist if and only if the lifetime distribution has a finite mean and that such a process is ergodic but need

12.3.

Mixing Conditions

215

not be mixing. Show that if the lifetime distribution is nonlattice then the process is mixing. [Hint: Use the renewal theorem in different forms. A periodic renewal process can be made into a stationary process by suitably distributing the initial point (see e.g. the stationary deterministic process at Exercise 12.1.7), but such a process is not mixing because, for example, the events V = {N (0, 12 ] > 0} and W = {N ( 21 , 1] > 0} do not satisfy the mixing property (12.3.1) when x is an integer and the process has period 1.] 12.3.2 Show that any event in I is equal (modulo sets of measure zero) to an event in T∞ , but not conversely. [Hint: See Exercise 12.3.1 for the converse. For W ∈ I, consider W = ∞ n=1 Sxn V , where xn → ∞ as n → ∞.] 12.3.3 As an example of a cluster process that is mixing but for which the cluster centre process is not mixing, take the cluster centre process to be a mixture of two Neyman–Scott cluster processes with member distributions F1 and F2 about the centre, and the cluster member process again of Neyman–Scott type with distribution F3 such that F1 ∗ F3 = F2 ∗ F3 . 12.3.4 Say that a process ξ has short-range correlation if for W ∈ B(M# X ) with P(W ) > 0, and arbitrary ε > 0, there exists a bounded set A ∈ BX such that on the sub-σ-algebra of B(M# X ) determined by {ξ(B): B ∈ X \A}, the variation norm of the difference as below satisfies PX \A (· | W ) − PX \A (·) < ε. Show that ξ has short-range correlation if and only if the tail σ-algebra T∞ is trivial. [Hint: Consider the sequence of Radon–Nikodym derivatives pn (·) ≡ dPX \An (· | W )/dPX \An (·) for An = Udn (for example). Show that these functions {pn (·)} constitute a martingale that converges to a limit p∞ (·), which is T∞ -measurable, and that h∞ = constant a.s. if PX \A (· | W ) − PX \A (·) > c > 0 for some real c and every bounded A ∈ BX . The result is attributed to Lanford and Ruelle (1969) in MKM (1982, Theorem 1.10.1).] 12.3.5 Let r.v.s X(ξ), Y (ξ) be defined on the stationary random measure ξ and have finite expectations. Show that ξ is mixing if and only if for all such r.v.s X, Y ,

#

MX

X(Sx ξ)Y (ξ) P(dξ) → E[X(ξ)] E[Y (ξ)]

(x → ∞).

12.3.6 Stability condition for mixing process. Let P, P0 be measures on M# X with P0 P. Show that if P is mixing then for Px defined by Px (V ) =

#

MX

IV (Sx ξ) P0 (dξ)

(V ∈ B(M# X )),

Px → P weakly as x → ∞. [Hint: Let p(ξ) be a measurable version of the Radon–Nikodym derivative; apply Exercise 12.3.5.] 12.3.7 A process is mixing of order k if for all V0 , . . . , Vk−1 ∈ B(M# X ), P(V0 ∩ Sx1 V1 ∩ · · · ∩ Sxk−1 Vk−1 ) → P(V0 )P(V1 ) . . . P(Vk−1 ) as xi → ∞ (i = 1, . . . , k − 1) in such a way that xi − xj → ∞ also for all i = j. Show that when T∞ is trivial the process is mixing of all orders. [Hint: Use the method of proof of Proposition 12.3.V. See also MKM (1978, Theorem 6.3.6) and MKM (1982, Theorem 6.2.9).]

216


12.3.8 Verify the assertions of Proposition 12.3.IX concerning conditions for a stationary cluster process to be weakly mixing or ergodic.

12.4. Stationary Infinitely Divisible Point Processes This shorter section discusses stationarity and mixing conditions for infinitely divisible point processes, resuming the notation and terminology of Section 10.2. It can be viewed as an extended example concerning mixing conditions, at the same time yielding a more refined classification of such point processes. denote the KLM measure of a stationary Consider first stationarity. Let Q infinitely divisible process so that from (10.2.9), (dx) − 1 Q(d N ) exp log h(x − u) N log G[h(· − u)] = =

N0# (X )

N0# (X )

exp X

X

u dN ). log h(y) N (dy) − 1 Q(S

(12.4.1)

When N is stationary, this must coincide with the original form of (10.2.9), (dy) − 1 Q(d N ). exp log h(y) N log G[h] = N0# (X )

X

by As in (12.1.3), we can define a new measure S+u Q N : Su N ∈ B} = Q{ S+u Q(B) so that (12.4.1) could equally well be written in the form (10.2.9) with S+u Q in place of Q. But by Theorem 10.2.V the KLM measure is unique, so Q and must coincide, and we have established the following result (see Exercise S+u Q 12.4.1 for a generalization). Proposition 12.4.I. An infinitely divisible point process on Rd is stationary if and only if its KLM measure is stationary (i.e., invariant under the shifts S+u ). In Section 10.2 we established some relationships between total finiteness of the KLM measure and the representation of an infinitely divisible point process as a Poisson cluster process. This relationship can be sharpened when the point process is stationary. Proposition 12.4.II. Let N be a stationary infinitely divisible point process on Rd . (a) If N has a representation as a Poisson randomization then it is singular. (b) N is regular if and only if it can be represented as a Poisson cluster process with a stationary Poisson process of cluster centres and a cluster structure that depends only on the relative locations of the points in a cluster and not on the location of the cluster itself.

12.4.

Stationary Infinitely Divisible Point Processes

217

is totally finite, so that Proof. Suppose first that the KLM measure Q # P(·) ≡ Q(·)/Q(N0 (X )) is the probability measure of a stationary point . The special property (10.2.8) of the KLM measure now implies process N N (X ) = 0} = 0, so coupled with Proposition 12.1.VI we must have that P{ P{N (X ) = ∞} = 1, which with (10.2.10b) proves (a). From the decomposition at Proposition 10.2.VII of an infinitely divisible point process into its regular and singular components, and from the fact that N (X ) = ∞} into itself, it follows that we may discuss the effects S+u maps P{ of stationarity separately for each type of process. From the discussion around Proposition 10.2.VIII, we know already that a regular infinitely divisible point process has a representation as a Poisson cluster process. Statement (b) then follows from Proposition 12.1.V. Observe that, from (a) of this proposition, the stationary singular infinitely divisible distributions can be classified into those with totally finite KLM measures, namely the Poisson randomizations, and those with unbounded KLM measure. An alternative and more interesting classification can be based on the Q-measures of the sets of trajectories with zero or positive asymptotic densities, as indicated below and in Example 13.2(c). First, however, note that Proposition 12.3.IX implies that a stationary Poisson cluster process, and hence any regular stationary infinitely divisible process, is necessarily mixing and hence ergodic. Interest centres therefore around the mixing properties of the singular stationary infinitely divisible processes. We follow essentially Kerstan and Matthes (1967) and MKM (1978, Chapter 6), starting from a simple general property. Proposition 12.4.III. If the stationary random measure ξ on X = Rd is ergodic, it cannot be represented as a mixture of two distinct stationary processes. Proof. Suppose the contrary, so that P = αP1 + (1 − α)P2 say, where 0 < α < 1 and the other three terms are stationary probability measures on M# X . Evidently, P1 P and the Radon–Nikodym derivative dP1 /dP is invariant under shifts Sx . When P is ergodic the only invariant functions are constants a.s., so P1 = cP; hence, c = 1 because both P and P1 are probability measures. Thus, P1 = P, and similarly P2 = P, showing that the decomposition is trivial. As a converse to this result, by noting that a Poisson randomization is by definition a nontrivial discrete mixture of distinct components, we conclude as follows. Corollary 12.4.IV. No stationary Poisson randomization, nor more generally any process that can be represented as the superposition of a stationary process and a Poisson randomization, is ergodic. From this result it may seem plausible that no singular stationary infinitely divisible process could be ergodic. Such is not the case: the next result, due

218


to Kerstan and Matthes (1967), spells out in terms of the KLM measure those properties that lead to ergodicity or mixing, and we show that such properties include examples of singular infinitely divisible processes. Proposition 12.4.V. Let N be a stationary infinitely divisible point process on X = Rd with KLM measure Q. (a) N is ergodic if and only if for all bounded A, B ∈ BX , 1 (Udn )

Ud n

(B) > 0} dx → 0 N : N (Tx A) > 0 and N Q{

(n → ∞), (12.4.2)

in which case N is also weakly mixing. (b) N is mixing if and only if for all such A, B, (B) > 0} → 0 N : N (Tx A) > 0 and N Q{

(x → ∞).

(12.4.3)

Proof. We consider first part (b). We use the p.g.fl. formulation of the mixing condition, writing now = log G[h] = G[h]

# NX

exp X

N ) (h ∈ V(Rd )). log h(x) N (dx) −1 Q(d

Then the mixing condition (12.3.11) is expressible as 1 ] + G[h 2] 1 Sx h2 ] → G[h G[h

(x → ∞).

(12.4.4)

To show that this condition is the same as (12.4.3), take h1 = 1 − IB , h2 = 1 − IA , and consider the difference 1 ] − G[h 2] = 1 Sx h2 ] − G[h G[h

# NX

i

ai bi −

i

ai −

bi + 1 Q(dN ),

i

(12.4.5) where ai = h1 (xi ), bi = h2 (xi − x), and the xi are the points of the particular . For the particular h1 , h2 , the integrand on the right vanishes realization N for which both N (B) > 0 and N (Tx A) > 0, when except for realizations of N it reduces to unity. Thus, the left-hand side of (12.4.5) coincides with the expression at (12.4.1) for such h1 , h2 , and thus (12.4.4) implies(12.4.3). Conversely, when (12.4.3) holds, (12.4.4) holds whenever 1 − hi are indicator functions as above. For more general hi ∈ V(Rd ), the difference at (12.4.5) is dominated by the corresponding difference when h1 and h2 are replaced by 1 − IB and 1 − IA , where B and A are the supports of 1 − h1 and 1 − h2 . Thus, (12.4.3) also implies (12.4.4) for these more general hi , and part (b) is established. To prove part (a) we need to develop some auxiliary results. Because is stationary, although perhaps only σ-finite, an extension of the ergodic Q

12.4.


219

theorem (see comment following Proposition 12.2.II) can be applied to deduce the limit, for any convex averaging sequence {An }, 1 n→∞ (An )

lim

) dx = f¯(N ) f (Sx N

Q-a.e.

(12.4.6)

An

) Q(dN ) < ∞. whenever the B(NX# )-measurable function f satisfies N # f (N X Let IV be the indicator function of the set V = {N : N (A) > 0} for bounded ) < ∞ by (10.2.8), so the limit in (12.4.6), I¯V (N ) say, A ∈ B(Rd ). Then Q(V exists Q-a.e. We assert that if the infinitely divisible process is ergodic, then this limit must be zero Q-a.e. To see this, consider for any fixed positive c : I¯V (N ) > c > 0}, which is measurable and invariant because the set Jc = {N I¯V (·) is. Furthermore, c) ≤ cQ(J

) Q(d N ) ≤ I¯V (N

Jc

≤ lim sup n→∞

1 (An )

An

# NX

# NX

) Q(d N ) I¯V (N

) Q(d N ) dx = Q(V ) < ∞. IV (Sx N

measure and is invariant. If in fact Q(J c ) > 0, Consequently, Jc has finite Q # we can construct a stationary probability measure PV on NX by setting c ∩ ·)/Q(J c ), PV (·) = Q(J and it then follows that the original process has the Poisson randomization of PV as a convolution factor. But for an ergodic process this is impossible ) = 0 Q-a.e. c ) = 0 for every c > 0, and I¯V (N as by Corollary 12.4.IV, so Q(J asserted. : N (B) > 0}, and Now let B be any bounded set in B(Rd ), write W = {N consider the relations 1 N : N (A) > 0 and N (B + x) > 0} dx Q{ (An ) An 1 )IS W (N ) Q(d N ) dx IV (N = x (An ) An NX# 1 ) ) dx Q(d N ). IV (N ISx W (N = # (An ) An NX We have just shown that the inner integral here → 0 as n → ∞ Q-a.e., and because this integral ≤ 1 and Q(V ) < ∞, we can apply the dominated convergence theorem and conclude that the entire expression → 0 as n → ∞; that is, (12.4.2) holds.

220


The converse implication depends on the inequality [see under (12.4.5)] 1 Sx h2 ] − G[h 1 ] − G[h 2] 0 ≤ G[h N : N (A) > 0 and N (Tx B) > 0} = q(x) ≤ Q{

say,

where A, B are the supports of 1 − h1 and 1 − h2 . Using the elementary inequality eα − 1 ≤ αeα (α > 0) and taking exponentials, we obtain 0 ≤ G[h1 Sx h2 ] − G[h1 ]G[h2 ] ≤ G[h1 ]G[h2 ]q(x)eq(x) .

(12.4.7)

N : N (A) > 0} < ∞ uniformly in x, it follows that the Because q(x) < Q{ difference in (12.4.7) is bounded by Kq(x) for some finite positive constant K. Then the integral at (12.3.9) is bounded by K q(x) dx, (Udn ) Udn which → 0 as n → ∞ when (12.4.3) holds. Proposition 12.3.VI(b) part (i) now shows that the process must be ergodic, and because the difference at (12.4.7) is already nonnegative, the stronger convergence statement at (12.3.10) must hold, and the process is in fact weakly mixing as asserted. The arguments used in the preceding proof can be taken further. Consider in particular the invariant function I¯V (·) introduced below (12.4.6), putting can be classified Q-a.e. A = Ud say for definiteness. The trajectories N by defining : I¯V (N ) = 0}, : IV (N ) > 0}. Ss = {N Sw = {N Both these subsets of NX# are measurable, and their complement has zero Q s , where can be decomposed as Q =Q w + Q measure, so Q w ∩ ·), w (·) = Q(S Q

s = Q(S s ∩ ·). Q

The notation here comes from the definition of a singular stationary infinitely s vanishes and strongly singular divisible point process as weakly singular if Q if Qw vanishes. Just as a general infinitely divisible point process can be represented as the superposition of regular and singular components, so in the stationary case the singular component can be further represented as the superposition of weakly singular and strongly singular components. Evidently, a Poisson randomization is strongly singular, and any stationary singular process that is ergodic must be weakly singular (in fact, the condition is necessary and sufficient: see Exercises 12.4.2–4). Examples of weakly singular processes are not easy to construct: one such construction that uses a modified randomization procedure starting from a renewal process with infinite mean interval length is indicated in Exercise 12.4.6. Other examples arise from the so-called stable cluster processes described in Section 13.5. See also Example 13.3(b). For additional results see MKM (1978, Sections 6.3 and 9.6), and MKM (1982, Section 6.2).

12.4.


221

Exercises and Complements to Section 12.4 12.4.1 Show that if an infinitely divisible point process on the c.s.m.s. X is invariant under a σ-group {Tg : g ∈ G} of transformations (Definition A2.7.I), then its KLM measure is also invariant under the transformations Sg induced by {Tg }. [Hint: See Proposition 12.4.I.] 12.4.2 Show that a singular infinitely divisible process is strongly singular if and only if it can be represented as a countable superposition of Poisson randomizations. [Hint: Consider the restriction Qn of Q to the set Jn = {N : I¯V (N ) ∈ ((n + 1)−1 , n−1 ]}, where V = {N : N (A) > 0} for some bounded A ∈ B(Rd ) (cf. the proof of Proposition 12.4.II). Then Qn is totally finite, so it can be the KLM measure of a Poisson randomization. The original process is equivalent to the convolution of these randomizations.] 12.4.3 Show that for a singular infinitely divisible point process the conditions below are equivalent. (a) The process is weakly singular; (b) The process is ergodic; (c) If V ∈ I then Q(V ) = 0 or ∞. [Hint: To show (a) ⇔ (b), use the characterization of weak singularity in terms of I¯V together with relevant parts of the proof of Proposition 12.4.II. Modify the argument of Exercise 12.4.2 to show that (a) ⇔ (c).] 12.4.4 Show that T∞ is trivial for every regular infinitely divisible point process. [Hint: Use the short-range correlation inequality of Exercise 12.3.4 and equation (12.4.7) to show that for some positive c, |P(V ∩ W ) − P(V )P(W )| ≤ c Q{N (B) > 0 and N (X \A) > 0}, where for bounded A, B ∈ BX , V and W are in the sub-σ-algebras determined by {ξ(B ): B ∈ BX , B ⊆ B} and {ξ(A ): A ∈ BX , A ∈ X \A}. Now use the regularity of ξ to show that the right-hand side → 0 as A ↑ X , and hence that events in T∞ have probability 0 or 1. MKM (1982, p. 97) attributes the result to K. Hermann.] 12.4.5 For n = 1, 2, . . . let Nn be independent stationary infinitely divisible point processes on R that are Poisson randomizations (with mean number = 1) ∞ ∗ = of Poisson processes at rate λn , where n=1 λn < ∞. Verify that N ∞ N is a well-defined stationary point process that is singular infinitely n n=1 divisible with infinite KLM measure. 12.4.6 Let a stationary infinitely divisible point process on R+ have as its (stationary) KLM measure a finite positive multiple αµ of the shift-invariant regenerative measure of Example 12.1(e), so that (using the notation from there) y

Q{N (0, y] > 0}) = α 0

[1 − F (u)] du,

222

12. Stationary Point Processes and Random Measures which has a finite or infinite limit according as the d.f. F (·) has finite or infinite mean. Show that for x > y > 0 and z > 0, Q({N (0, y] > 0}, {N (x, x + z] > 0})

=α 0

y

x−u+z

[1 − F (u)] du

[1 − F (x − u + z − v)] U0 (dv), x−u

and that this quantity has limit zero when F has infinite mean. Hence, conclude that the point process is then weakly singular. [See MKM (1978, Section 9.6) for some other details.] 12.4.7 Show that if an infinitely divisible process is mixing then it is mixing of all orders as in Exercise 12.3.7.

12.5. Asymptotic Stationarity and Convergence to Equilibrium The issue of convergence to equilibrium, or stability, for point process models has already surfaced in Chapter 4, where we illustrated the use of coupling arguments to obtain classical results for renewal and Wold processes, and in Exercise 12.3.6, where convergence was established assuming both mixing and an absolute continuity condition. This section treats convergence to equilibrium in the more general context of simple and marked point processes on R. Similar results can be developed for random measures with essentially only changes in terminology, but are left to the exercises. The special problem of convergence to equilibrium from the Palm distribution is taken up in Section 13.4. The tools available to tackle these problems have been extended greatly by the development of coupling and shift-coupling methods, which we illustrate in the present section, and the work of Brémaud and Massoulié on Poisson embedding, which we outline in Section 14.7 and use there to develop conditions for convergence to equilibrium in terms of conditional intensities. Lindvall (1992) is a basic reference on coupling ideas; for shift-coupling see Aldous and Thorisson (1993) and Thorisson (1994, 2000). In the point process context, systematic treatments concentrating on applications to queues appear in Franken et al. (1981), Sigman (1995), and Baccelli and Brémaud (1994); convergence to equilibrium has been discussed, for example, by Lindvall (1988) for finite-memory processes, Sigman (1995) for shift-coupling and queues, Brémaud and Massoulié (1996) and Massoulié (1998) for generalizations of the Hawkes process, and Last (2004) for the stress-release model and its analogues. Additional references are given in these papers and below. Apart from the basic theoretical interest of these issues, their importance has increased in recent years because of the greatly increased role of simulation methods, especially Markov chain Monte Carlo methods, and the associated

12.5.

Asymptotic Stationarity and Convergence to Equilibrium

223

need to estimate the ‘burn-in times’ required for initial system values to approximate the equilibrium values that are usually the endpoint in view. When such problems arise in applications, we are commonly given a ‘law of evolution’ of the process and seek to establish, first, whether such a law is compatible with a stationary form for the process, and second, whether the process will converge to that stationary form when started at t = 0 from some ‘initial distribution’. To try to capture these ideas more precisely, consider first the action of shifts on a point process (simple or marked) defined on the half-line [0, ∞) = R+ . If the process has distribution P say, on B(NR#+ ×K ), then for u > 0, the action of the general shift operator S+u on P is well defined: it corresponds to shifting the time origin to u. Thus S+u P ≡ Pu is a probability measure defined on the Borel sets of N # ([−u, ∞) × K). Any such measure can be projected onto the smaller sub-σ-algebra B(NR#+ ×K ), on which there is then a measure that corresponds to a new (simple or marked) point process on R+ × K. For the rest of this section the notation Pu = S+u P denotes the probability measures of the shifted point processes induced in this way on R+ × K. We call an MPP on the half-line stationary if its distribution is invariant under these positive shifts. Trivially, the restriction to the half-line of an MPP stationary on the whole line is stationary on the half-line. Conversely, any MPP stationary on the half-line can be extended to a process stationary on the full line by shifting the origin forward (which does not change the distributions) and taking the limit of these shifted processes (Exercise 12.5.1). In the discussion below we use the notation (C, 1), as for Cesáro summability in analysis, to denote the behaviour of integral averages. Definition 12.5.I. Let N be an MPP on R+ × K, and let P, Pu be the probability measures on B(NR#+ ×K ) associated as above with N and its shifted versions Su N . Then N is asymptotically stationary u (resp., (C, 1)-asymptotically stationary) if as u → ∞, Pu (resp., P u = u−1 0 Pv dv) converges weakly to a limit P ∗ corresponding to a limit MPP N ∗ . It is strongly asymptotically stationary (resp., strongly (C, 1)-asymptotically stationary) if the convergence holds in variation norm. Brémaud and Massoulié and others refer to asymptotic stationarity as a stability property, having in mind the analogy with the stability of differential equations. Lemma 12.5.II. Any limit measure P ∗ arising in Definition 12.5.I necessarily corresponds to a stationary point process on R+ . Proof. When

S+u P → P ∗

weakly

(u → ∞),

(12.5.1)

then for all finite u, the sequences S+x+u P and S+x P must have the same weak limit P ∗ as x → ∞, implying that P ∗ = S+u P ∗ .

224


A similar argument holds in the (C, 1) case if we apply the shift S+x to the x u+x integrals and note that the contributions from the integrals 0 and u on the end-intervals are asymptotically negligible. The convergence at (12.5.1) is equivalent to requiring that for u > 0 the fidi distributions should satisfy Fk (A1 + u, . . . , Ak + u; x1 , . . . , xk ) → Fk∗ (A1 . . . , Ak ; x1 , . . . , xk ), (12.5.2) either directly or in the (C, 1) sense, where on the right-hand side F ∗ refers to the fidi distributions of P ∗ . Thus, if the shifted distributions of the initial point process converge to limits which constitute a family of fidi distributions, then these limit distributions must be those of a stationary point process. The underlying problem can be phrased alternatively as follows. Given a stationary MPP with associated probability measure P ∗ , find the class of MPP distributions P for which (12.5.1) or (12.5.2) holds. In this sense, the problem of convergence to equilibrium can also be interpreted as a domain of attraction problem. Consider then what more stringent requirements may be needed for convergence to equilibrium without the (C, 1)-averaging. It is convenient here to specify the distribution P of the initial point process on the half-line by treating it as the conjunction of two components: a distribution Π(·) of initial conditions Z defined on some appropriate c.s.m.s. Z, and a kernel P(· | z) that governs the evolution of the process N given the initial condition Z = z, so that P(V | z) Π(dz) (V ∈ B(NR#+ )). (12.5.3) P(V ) = Pr{N ∈ V } = Z

This formulation follows Brémaud and Massoulié (1994, 1996), which in turn has its origins in the fundamental paper of Kerstan (1964b). For u ≥ 0 let Pu (·) denote the distribution of Su N (so that in our earlier notation, Pu = S+u P and P0 = P), and suppose that there exists a family of distributions Πu on the space Z of initial conditions such that for these u, Pu (V ) =

Z

P(V | z) Πu (dz)

(V ∈ B(NR#+ ))

(12.5.4)

with the same kernel P(· | z) as in (12.5.3). We can interpret Πu as the initial conditions that apply when the time origin is shifted to u. When such a representation is available, it is not difficult to show that convergence of the distributions Πu of the initial conditions implies convergence of the shifted distributions Pu . Proposition 12.5.III. Let Z be a c.s.m.s. of initial conditions Z for some point process and {Πu } a family of probability measures on BZ . Let P(V | z)

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225

be a measurable family of point processes on R+ , and suppose that for all u ≥ 0 and V ∈ B(NR#+ ), Pu (V ) ≡ S+u P0 (V ) ≡ S+u

Z

P(V | z) Π0 (dz) =

Z

P(V | z) Πu (dz).

(a) If Πu → Π∗ weakly, and the kernel P(· | z) takes continuous functions on NR#+ into continuous functions on Z, then Pu → P ∗ weakly and P ∗ (V ) = P(V | z) Π∗ (dz) (i.e. the point process N with distribution P = P0 is Z asymptotically stationary with limit distribution P ∗ ). (b) If Πu → Π∗ in variation norm, then the point process N of (a) is strongly asymptotically stationary with the same limit distribution P ∗ , and Pu − P ∗ ≤ Πu − Π∗ .

(12.5.5)

Proof. To establish asymptotic stationarity we have to prove weak conver gence of the Pu . This means showing that N # (R+ ) h(N ) Pu (dN ) converges for any bounded, continuous function h(N ) on NR#+ . But under assumption (a), the integral N # (R+ ) h(N ) P(dN | z) defines a bounded continuous function of z ∈ Z, and then the weak convergence of Πu to Π∗ implies h(N ) Pu (dN ) = h(N ) P(dN | z) Πu (dz) N # (R+ )

Z

→

Z

N # (R

N # (R+ )

h(N ) P(dN | z) Π∗ (dz) = +)

N # (R

h(N ) P ∗ (dN ). +)

Under assumption (b), the same relations hold under the weaker requirement that the integral N # (R+ ) h(N ) P(dN | z) is merely bounded and measurable, which follows from the boundedness of h and the measurability condition on the family P(· | z). The bound in (12.5.5) follows from the fact that P(· | z) is a stochastic kernel, so that, for any totally finite signed measure Ψ on BZ , , , , , , P(dN | z) Ψ(dz), , , # N (R+ )×R+ |h(N )| P(dN | z)[Ψ+ (dz) + Ψ− (dz)] ≤ Ψ, ≤ sup h:|h(N )|≤1

N # (R+ )

R+

where Ψ+ , Ψ− are the positive and negative parts of Ψ from its Jordan– Hahn decomposition, and the supremum is taken over all measurable functions bounded by 1. The major advantage of this form of representation is that the process of ‘initial conditions’ can usually be represented as a Markov process Z(t) say,

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thus reducing the problem of convergence to equilibrium for point processes to the well-developed theory of convergence to equilibrium for Markov processes. Such a representation is always possible in principle because the initial condition can be taken as the history of the process up to time 0, and such a history can itself be regarded as a point in a c.s.m.s. This still leaves us within the framework of Markov processes on a c.s.m.s. (or more generally a Polish space). Consequently, problems for point processes concerning conditions for the existence of and convergence to equilibrium distributions, are reduced to corresponding problems for general Markov chains [see, e.g., Meyn and Tweedie (1993a, b)]. Similarly, the notions of coupling and shift-coupling, which again have their origins in Markov chain theory, can be invoked and applied within the point process context. The treatment becomes greatly simplified if a condensed representation can be found, for example, one in which the space of initial conditions can be reduced to a low-dimensional Euclidean space. The goal in practice, then, is to identify a simple representation for the space Z of initial conditions, and to examine the conditions for convergence of the induced Markov process Z(·). To establish a representation (12.5.4) for a given point process model, the first step is to identify a convenient state space Z for the Markov process of initial conditions. The essential feature that such a representation should possess is that knowledge of the current value Z(u) should be sufficient to fully determine the fidi distributions for the evolution of the process beyond time u, any further information from the past being redundant. The other components needed to complete the representation are the form of the state transition probabilities for the Markov process Z(·) and the mappings P(· | z) determining the evolution of the process from a given value of z. Most commonly, this is done by identifying a mapping, Φ: NR#− → Z say, that condenses the information from the past trajectory {N (t): t < 0} of the point process onto a particular value z = Z(0) ∈ Z. When this transformation is applied to Su N , its value Φ(Su N ) = Z(u) represents the updated value of the initial condition, so that from there on its evolution is governed by P(· | Z(u)). Under these conditions the transition probabilities for the process Z(·) B ∈ BZ are given by Pu (B | z) = P Su Φ−1 (B) | z = S+u P Φ−1 (B) | z (B ∈ BZ ). Integrating over the further evolution from u to u + v yields the Chapman– Kolmogorov equation P Sv Φ−1 (B) | z Pu (dz | z) Pu+v (B | z) = Z = Pv (B | z ) Pu (dz | z). Z

Thus, once the mapping Φ has been identified, the provisions of Proposition 12.5.III, especially the bound (12.5.5), can be invoked to establish asymptotic stationarity for the associated point process.

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We now illustrate the proposition with some well-known examples. Example 12.5(a) Convergence to equilibrium of renewal and Wold processes. Consider first the simpler case of the renewal process, supposing the lifetime distribution to be nonlattice, with finite mean µ = λ−1 . A convenient choice for Z here is the forward recurrence time defined as in (3.4.15). The space Z becomes the half-line R+ and the conditional point process corresponding to P(· | z) is the ordinary renewal process (with initial point at 0) shifted through z. It remains to investigate the conditions in parts (a) and (b) of the proposition. Continuity of the integral N # (R+ ) h(N ) P(dN | z) as a function of Z, here the value z say of the forward recurrence time, follows from the observation (Lemma 12.1.I) that the shifted realization Su N of an element N ∈ NR#+ depends continuously on u, so if h(·) is a bounded and continuous function of N (·), h(Su N ) is a bounded continuous function of u. Continuity of N # (R+ ) h(N ) P(dN | Z = z) then follows by dominated convergence. The other requirement of condition (a) amounts here to the weak convergence of the distribution Πu of the forward recurrence time at time u to its equilibrium form Π∗ . This was established for nonlattice lifetime distributions with finite mean in Example 4.4(a). In fact, the stronger result required for condition (b) holds when the lifetime distribution is spread out, as noted in Exercise 4.4.4. Thus a renewal process having a spread-out lifetime distribution with finite mean is both weakly and strongly asymptotically stationary, from an arbitrary initial condition. Similar results for the Wold process can be obtained from the discussion in Section 4.5. Here it is convenient to describe the initial conditions in terms of the pair (X, Y ) = t0 (N ), t0 (N ) − t−1 (N ) representing the time to the first event and the length L0 (N ) of the (unobserved) initial interval. Let (Xu , Yu ) denote their values after a shift of the time origin through u. Conditions for the convergence of the pair (Xu , Yu ) are obtained in Proposition 4.5.VI. For example, if the recurrence time distribution G in that proposition is nonlattice but spread-out, and the stationary interval distribution has finite mean, then convergence holds in variation norm and the associated Wold process is strongly asymptotically stationary. In particular this is the case when the transition kernel satisfies the simpler Condition 4.5.I . The next example introduces two extensions of principle: the point process under consideration is now an MPP, and the underlying Markov process may not be directly observable. The extension to a marked process causes no essential difficulties; the function h in the proposition now needs to be taken as a function on the space NR#+ ×K rather than on NR#+ , with the shifts defined on the first component only, but no essentially new points arise. To accommodate the non-observable parts of the process, or more generally any random variables in addition to those described by the observed history of the point process, the probability space needs to be taken in the general form (Ω, E) and the explicitly defined shifts Su on NR#+ replaced by a general flow

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θu on (Ω, E) which is compatible with the shifts [see the discussion following (12.1.3) and elsewhere in Section 12.1]. The initial conditions must then carry sufficient information to define the initial state of all components of the process, observable or otherwise, and it is again their evolution with u which governs the convergence of the point process. Example 12.5(b) Convergence to equilibrium of BMAPs and MMPPs [see Examples 10.3(e), (h), (i)]. The BMAPs of Example 10.3(i) are treated there as examples of MPPs. But, any BMAP shares with the simpler MMPP model of Examples 10.3(e),(g) the feature that its evolution is fully determined by the evolution of a hidden or at best partly observed Markov process, X(t) say. Indeed, equations (10.3.23) spell out the form of the conditional intensity as a function of the current state of the process. In such a situation, the space of initial conditions Z reduces to the state space X of the process X(t) (and in these examples X is just a finite set of points, {1, . . . , K} say). The family of distributions P(· | z) of Proposition 12.5.III becomes the family of distributions on (NR# , BN # ) of the BMAP started from the initial condition R X(0) = z. The distribution Π0 of the proposition is the distribution of X(0) on the finite set X, π0 = {πk (0)} say, and Πu , corresponding to the distri u u uQ is the matrix of bution of X(u), is given by π u = π0 P , where P = e transition probabilities pij (u). Because X is finite, the transition probabilities P u converge in variation norm to the limit matrix π∞ 1 , where π∞ is the stationary distribution, as soon as the process X(t) is irreducible. It then follows from Proposition 12.5.III(b), and specifically from (12.5.5), that a BMAP with irreducible governing process X(t) is strongly asymptotically stationary, no matter what the initial condition. In tackling more complex examples, several different approaches have been suggested. In most cases, as illustrated already for the renewal process, the process Zu takes the form of a continuous-time, mixed jump–diffusion process on a continuous state space such as Rd . Conditions for the convergence to equilibrium for such processes, based on extensions of Foster–Lyapunov conditions for stability, have been set out with a considerable degree of generality in Meyn and Tweedie (1993b); their book, Meyn and Tweedie (1993a), contains a compendium of related material on general state-space Markov chains. Application of the Markov process conditions to any particular point process example requires a careful analysis of the properties of the induced process {Zu }, and may be far from a trivial exercise (cf. Exercise 12.5.2). For example, in a comprehensive study, Last (2004) has used this approach to derive stability properties for a class of models that are related to the stress-release model of Example 7.2(g). Further examples include models in reliability for repairable systems [Kijima (1989), Last and Szekli (1998)] and work-load processes [Browne and Sigman (1992)]; see also Exercise 12.5.3. The difficulty revolves around the subtle behaviour of the Markov process itself. Zheng (1991) derived general conditions for irreducibility and positive recurrence of the simple stress-release model, and Last extended the discussion to more

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229

general processes, and gave estimates for the rate of convergence and convergence of moments. For the more complex linked stress-release model and its variants, as in Example 7.3(d), many basic questions remain open. Even sufficient conditions for ergodicity have been established only in very special cases [Bebbington and Borovkov (2003)], and no necessary and sufficient conditions are known. Another general approach is to invoke coupling or shift-coupling results. These have the advantage that they directly yield convergence in variation norm. They may be applied to the process of initial conditions, to the point process itself, or to the process of intervals appearing in the Palm distributions. Early applications of coupling arguments to point processes appear in Hawkes and Oakes (1974), Berbée (1979), and Lindvall (1988). Recent texts such as Lindvall (1992), Sigman (1995), and Thorisson (2000) cover much wider territory. We gave both the underlying definition of coupling and the basic coupling inequality in Lemma 11.1.I. To apply the inequality directly to point processes, we take the stochastic processes X(t), Y (t) in Lemma 11.1.I to be shifted versions St N, St N , of two simple or marked point processes N, N , so that the parameter t refers to the extent of the time shift. It is convenient to treat the processes as defined on a common probability space (Ω, E, Pr), and to denote by θt the flows associated with the shifts St . , N of N, N , and a We say that N and N couple if there exist versions N finite stopping time T , such that θT N = θT N a.s. The last equation means that the trajectories on [0, ∞) of the shifted versions are a.s. equal, so that = θT +t N a.s. for t > 0. Similarly we say that N and N shift-couple θT +t N , N of N, N respectively, and finite stopping times if there exist versions N = θT N ; that is, θt+T N = θt+T N a.s. for t ≥ 0. In this T, T such that θT N context the basic coupling inequality of Lemma 11.1.I extends as follows. Note that because coupling equalities hold between shifts of versions of the original MPPs, rather than between the original MPPs themselves, the inequalities involve only the distributions of the original processes. Lemma 12.5.IV (Coupling Inequalities). Let N, N be jointly distributed by P, P the probability measures induced on MPPs and denote # on R+ × K, B N (R+ × K) by N, N , respectively, and · the total variation norm. (a) Suppose N and N couple, with coupling time T . Then S+t P − S+t P ≤ 2 Pr{T > t}.

(12.5.6a)

(b) Suppose N and N shift-couple, with coupling times T, T . Then , t , t ,1 , +u P du − 1 +u P du , ≤ 2 Pr{max(T, T ) > U t}, , S S ,t , t 0 0

(12.5.6b)

where U is uniformly distributed on (0, 1), independent of T, T , N, N .

230


Proof. The first inequality is proved in Lemma 11.1.I. For the other we follow Thorisson (1994; 2000, Theorem 5.3.1). By definition of shift-coupling, there exist a.s. finite stopping times T , T such that θT N = θT N is a successful coupling of N and N . Let U be a r.v. uniformly distributed on (0, 1) independent of (T, T , N, N ). Then so is 1−U , and hence for any finite t > 0, θU t N and θ(1−U )t N are copies of the same process. For such U and t, (T +U t) mod t = t T /t+U −T /t+U is uniformly distributed on (0, t), independent of (T, T , N, N ) (x denotes the largest integer ≤ x). Therefore θ(T +U t) mod t N is also a copy of θU t N . Similarly, θ(T +U t) mod t N is a copy of the process θU t N . On the set C ≡ {U t + max(T, T ) ≤ t}, (T + U t) mod t = T + U t, and thus, θ(T +U t) mod t N = θU t θT N = θU t θT N = θ(T +U t) mod t N

on C,

the central equality by the assumed shift-coupling. Thus on C we have a shift-coupling of θU t N and θU t N , and therefore S+U t P − S+U t P ≤ 2[1 − Pr(C)]. But 1 − Pr(C) = Pr{U t + max(T, T ) > t} = Pr{max(T, T ) > (1 − U )t}, which is the same as Pr{max(T, T ) > U t}. Taking expectations over U in the displayed inequality yields the left-hand side of (12.5.6b). As a corollary, we obtain the following conditions for asymptotic stationarity. Proposition 12.5.V. Let N, N be jointly distributed MPPs on R+ × K, and suppose that N is stationary. (a) If N couples with N , then N is strongly asymptotically stationary, with limit process N . (b) If N shift-couples with N , then N is strongly (C, 1)-asymptotically stationary, with limit process N . Proof. The proof follows from Lemma 12.5.IV and the observation that when N is stationary, its distribution is invariant under the shifts S+u . As a further corollary of these results it follows that the ergodic theorems from Section 12.2 extend to processes which are asymptotically stationary. Suppose that N is asymptotically stationary and that both N and the limit process N have boundedly finite first moment measures. The limit process N is stationary by assumption, so that the ergodic Theorem 12.2.IV applies to N , as also to any of its a.s. versions. If N shift couples to N there are versions of both processes, the realizations of which coincide a.s. after certain realization-dependent but finite time-shifts. The existence a.s. of the (C, 1) t limits t−1 0 f (Su N ) du for the realizations of the version of the stationary process therefore implies the existence a.s. of the same (C, 1) limits for the version of the approximating process. But the a.s. existence of the (C, 1) limits does not depend on which version is used (the versions are equal outside a set of realizations with zero probability measure), and so the ergodic behaviour applies also to the original version of the approximating process.

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Corollary 12.5.VI. The a.s. ergodic results of Theorem 12.2.IV apply to asymptotically stationary MPPs whenever the original process and its stationary limit have boundedly finite first moment measures. In fact, a great deal more follows from the work of Aldous and Thorisson, who showed in particular that coupling is strongly linked to mixing properties and the behaviour of the tail σ-field T∞ , whereas shift-coupling is linked in a parallel way to ergodicity properties and the behaviour of the invariant σ-field I. The results depend on a deep analysis of the ergodic properties of Markov chains, which we do not develop in detail here, referring to the papers cited for proofs, and to Sigman (1995) for an elaboration of their applications to point processes. The most important results for our purposes are summarized in the proposition below [see in particular Thorisson (1994, Section 4)], where for the sake of completeness Proposition 12.5.V is included. Proposition 12.5.VII. Suppose that N and N are two MPPs on X = R+ × K, and that N is stationary. (A) (Coupling Equivalences) The following statements are equivalent. (i) N and N couple. (ii) N is strongly asymptotically stationary with limit process N . (iii) N and N induce the same probability distribution on the tail σ-algebra T∞ . (B) (Shift-Coupling Equivalences) The following statements are equivalent. (i) N and N shift-couple. (ii) N is strongly (C, 1)-asymptotically stationary with limit process N . (iii) N and N induce the same probability distribution on the invariant σ-algebra I. Let us note at least one of the remarkable consequences of these results. Corollary 12.5.VIII. Weak and strong (C, 1)-asymptotic stationarity coincide. Proof. Suppose N is weakly (C, 1)-asymptotically stationary with limit process N , and denote by P, P the probability measures they induce on (X , BX ). Then for any invariant set E ∈ NX# , P(E) = S+x P(E) → P (E), so P and P coincide on I. Thus condition B(iii) holds and hence N and N shift-couple. The rest of the corollary follows from Proposition 12.5.VII. In applications we are still left with the problem of identifying situations where coupling holds. For point processes, identifying shift-coupling is generally easier for the associated sequence of intervals than for the continuous-time process, because regeneration points, which often lie at the heart of coupling arguments, are more commonly linked to the intervals. Thus a common approach is to establish shift-coupling for the intervals, and then to refer to the

232


Palm equivalence between counting and interval properties to establish shiftcoupling and associated ergodic results for the continuous-time process. See the further discussion in Section 13.5 and Sigman (1995). As an example of coupling (rather than shift-coupling) arguments, we outline in the next example the basic argument used by Hawkes and Oakes (1974) to establish convergence to equilibrium of the simple Hawkes process. The extensive work on linear and nonlinear Hawkes processes by Brémaud, Massoulié, and colleagues is outlined in Section 14.7 in conjunction with their ‘Poisson embedding’ technique. Example 12.5(c) Convergence to equilibrium of the Hawkes process [continued from Examples 6.3(c), 7.2(b)]. We introduced the Hawkes process at Example 6.3(c) as an example of a Poisson cluster process with a special kind of branching process cluster. Then Example 7.2(b) shows that, when started with the zero initial condition at t = 0, the process is characterized by a conditional intensity function of the form

t

µ(t − u) N0 (du),

λ0 (t) = ν +

(12.5.7)

0

where ν is an immigration rate and µ(u) is a density for the intensity measure, assumed to satisfy the condition

∞

µ(x) dx ≡ ρ < 1.

(12.5.8)

0

The existence of a stationary version of the process follows from the Poisson clustering representation, or can be established by letting the origin retreat to −∞ in (12.5.7) (see Exercise 12.5.4). We now compare two versions of the process: N0 starts from 0 at time 0 and follows (12.5.7), whereas N † is a stationary version with the complete conditional intensity t µ(t − u) N † (du) (12.5.9) λ† (t) = ν + −∞

and mean rate m = ν/(1 − ρ). For both versions, we consider the effect of shifting the origin forward to s; equivalently, we consider the shifted versions Ss N0 and Ss N † that bring the origin back to 0. Ss N † can be split into two components: one component has the same structure as Ss N0 , being built up from clusters initiated by † consists of the immigrants arriving after time −s, and the component N−s ‘offspring’ of points that occurred before time −s. On R+ the contributions from the latter form a Poisson process whose intensity, conditional on H−s , is given by −s † λ†−s (t) = µ(t − u) N−s (du). (12.5.10) −∞

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233

Note that from the integrability of µ as at (12.5.8), ∞ µ(x)dx → 0 (s → ∞), E[λ†−s (t)] = m t+s

and more generally, for any T < ∞, T Pr{Ns† (0, T ) > 0} = E 1 − exp − 0 λ†−s (t) dt T ≤ E 0 λ†−s (t)dt ∞ µ(x) dx → 0 (s → ∞). (12.5.11) ≤ mT s

Consider now the corresponding probability measures on N # (R+ ), P0 and P say, but restrict attention to their projections onto N # ([0, S]). Using the property P ∗ Q = P · Q of the variation norm, which implies that the contribution from the common distribution of Ss N0 disappears from the difference below, we have †

† [0, S] > 0} → 0 Sˆs P0 −P † [0,S] = Sˆs P0 − Sˆs P † [0,S] ≤ Pr{N−s

(s → ∞).

This implies convergence of the fidi distributions of the two processes on [0, S], for any S > 0, hence their weak convergence, and hence the weak asymptotic stationarity of N0 . We can extend this result to initial conditions specified by a distribution P J of a point process N J on R− , assuming that for t > 0 the process evolves according to the same dynamics as before. As in the previous discussion, for t > 0 the conditional intensity can then be written as the sum of two components, one corresponding to N0 and deriving from immigrants arriving after t = 0, and the other the residual risk remaining from events occurring J , we before t = 0. Denoting the shifted version of the latter process by N−s can write its contribution to the conditional intensity as in (12.5.10) but with † J replacing N−s . Similar arguments to those used before show that, for a N−s given realization of N J on R− , S −S ∞ J Pr{N−s [0, S] > 0} ≤ Ss N J (du) µ(t − u) dt = ∆(s, i), t=0

−∞

i=1

−s−ui +S

where ∆(s, i) = −s−ui µ(x) dx, and the ui are the points of N J in R− . If the initial distribution P J is concentrated on an individual realization, it follows that theresultant process will be weakly asymptotically stationary provided Σ(s) ≡ i ∆(s, i) < ∞, and Σ(s) → 0 as s → ∞. For under these conditions, comparing the two projections on [0, S], we have Sˆs P J − P † [0,S] ≤ Sˆs P0 − Sˆs P J [0,S] + Sˆs P0 − P † [0,T ] † J ≤ Pr{N−s (0, S) > 0} + Pr{N−s [0, S] > 0} → 0 (s → ∞).

234


If the initial distribution P J is such that N J has mean density m(u) on R− , then a similar conclusion holds provided −s s m(u) µ(t − u) dt du → 0 (s → ∞), t=−∞

0

which is certainly satisfied if, for example, m(u) is bounded. Under slightly stronger conditions we can establish strong asymptotic stationarity as well. Suppose that, in addition to (12.5.8), µ(x) also satisfies the condition ∞ x µ(x) dx < ∞. (12.5.12) 0

This condition implies that, if a parent event has a direct offspring, the mean gestation period, that is to say, the mean time to the appearance of the offspring, is finite, and hence, because (12.5.8) implies that the mean number of offspring is also finite, that the random time T from the appearance of an ancestor (cluster centre) to the last of his descendants (last cluster member) is also finite; that is, E(T ) < ∞. The probability that a cluster is initiated at some time −u ≤ −s, and still produces some members in R+ , is given by 1 − FT (u) = Pr{T > u}. Treating this as a thinning probability, and using the Poisson character of the arrival of ancestors (cluster centres), the probability that (in the stationary process) none of the ancestors arriving before −s produces offspring in R+ is equal to ∞ † [0, ∞) > 0)} = 1 − exp − ν [1 − FT (u)] du . Pr{N−s s

∞

† [0, ∞) > 0)} → 0. Because 0 [1 − FT (u)] du = E(T ) < ∞, we have Pr{N−s The occurrence time Ts for the last point in R+ of a cluster initiated before time −s is therefore a.s. finite, and acts as a coupling time between the processes Ss N0 and N † on R+ . The coupling time inequality (12.5.6a) then yields ≤ 2Pr{Ts > 0} = 2Pr{N † [0, ∞) > 0)} → 0 (s → ∞). Sˆs P0 − P † [0,∞]

−s

Thus, when the density µ satisfies both (12.5.8) and (12.5.13), the process starting from zero is strongly asymptotically stationary. Again the result can be extended to more general initial conditions N J , for example, when ∞ 0 µ(t − u) N J (du) dt = σ(|uj |) < ∞, (12.5.14) t=0

−∞

∞

where σ(u) = u µ(x) dx and {uj } enumerates the points of N J over (−∞, 0). The arguments are similar to those used before, and are outlined in Exercise 12.5.6. Note that the argument used above to derive strong asymptotic stationary is not peculiar to Hawkes processes, but holds for a wide range of Poisson cluster processes. Exercise 12.5.7 gives some details and examples.

12.5.


235

Exercises and Complements to Section 12.5 12.5.1 Prove the statement above Definition 12.5.I, that a process stationary under positive shifts on the half-line can be extended to a process stationary on the whole line. [Hint: Consider {Su P: u > 0}.] 12.5.2 Generalized stress release model. Following Last (2004), consider a piecewise linear Markov process X(t) determined by the following components: a positive constant ν representing the linear increase of X(t) between jumps; a locally bounded risk function Ψ(x): R → R+ representing the instantaneous rate of occurrence of jumps given that X(t−) = x; and a stochastic kernel J(x, B) representing the probability of a jump into B ∈ BR+ given X(t) = x ∈ R. (a) Show that, as for the BMAP processes, the process X(t) uniquely determines the associated MPP, say NX ≡ {(tn , κn )} of jump times and jump sizes, and that conversely knowledge of the MPP on R+ determines X(t) uniquely up to the initial value X(0). (b) Show also that if X(t) is positive recurrent, in the sense that, if X(t) has distribution Πt on R+ , there exists a stationary distribution Π∗ such that Πt − Π∗ → 0 as t → ∞, then NX is strongly asymptotically stationary. (c) Show that if X(t) is ‘geometrically ergodic’ in the strong sense that Πt − Π∗ ≤ C exp(−βt) for positive constants C, β, then also NX is geometrically asymptotically stationary in the sense that in Definition 12.5.I, Pu − P ∗ ≤ C exp(−β t) for some constants C , β . [Hint: Use the norm-preserving property of the stochastic kernel, as in establishing (12.5.5).] 12.5.3 (Continuation). Application to reliability models. For each of the examples listed below, identify the form of the components, and find sufficient conditions to ensure that the resulting process X(t) is (i) well-defined, (ii) positive recurrent, and (iii) geometrically ergodic. (a) The simple stress release model of Example 7.2(g) and Exercises 7.2.9–10. (b) The repairable system model [e.g., Block et al. (1985), Kijima (1989), and Last and Szekli (1998)], in which X(t) denotes the ‘virtual age’ of a system subject to failure, and after every failure an instantaneous repair takes place, restoring the system to some fraction 0 ≤ θ < 1 of its ‘age’ before failure. (c) The workload-dependent queueing process [e.g., Browne and Sigman (1992), Meyn and Tweedie (1993b)], in which W (t) = max{−X(t), 0} decreases linearly between jumps until either another jump (upwards) occurs, or W (t) = 0 in which case it remains zero until the next jump occurs. W (t) here can be interpreted as the workload in a queueing system in which both the arrival and the service rates may depend on the current value of W (t). [Hint: The basic aim is to find variations on condition (7.1.3) that will allow Foster–Lyapunov drift conditions, as developed in Meyn and Tweedie, to be applied to the process in question. Once convergence of the Markov process has been established, Proposition 12.5.III can be invoked to transfer the results to the associated point process. Last (2004) gives a very general discussion, as well as applications to the specific examples mentioned.] 12.5.4 Investigate conditions under which asymptotic stationarity of the point process implies, in an appropriate sense, asymptotic convergence of its first,

236

12. Stationary Point Processes and Random Measures second, and higher moment measures to the moment measures of the stationary version of the process.

12.5.5 Consider the Hawkes process of Example 12.5(c). Show that, for the process started from zero as in (12.5.7), there is an instantaneous mean rate m(t) = E[λ0 (t)] which is always finite and satisfies m(t) ≤ m = ν/(1 − ρ). Imitate the arguments leading to (12.5.11) to show that the sequence Ss P0 defines a strong Cauchy sequence of point processes on [0, S], and so implies the existence of a limit point process on [0, S], which in turn, because S is arbitrary, implies the existence of the distribution P † of an equilibrium process N † on R+ such that Ss P0 → P † weakly as s → ∞. 12.5.6 Let N J be an initial realization on R− satisfying (12.5.14), and suppose that (12.5.12) holds. Show that the process started from N J is strongly asymptotically stationary. [Hint: Extend the arguments below (12.5.12) to show that for both Ss N J and Ss N † , the probability that the residual components on R+ are nonempty converge to zero, so that both couple to Ss N0 .] 12.5.7 Convergence to equilibrium of Poisson cluster processes. (a) Consider a Poisson cluster process as in Proposition 6.3.III with constant intensity µc for the cluster centres, and stationary cluster structure such that the time T from the first to the last member of the cluster satisfies E(T ) < ∞. Imitate the last part of the discussion in Example 12.5(c) to show that the process is strongly asymptotically stationary. (b) Show that the conditions are satisfied for the Neyman–Scott process whenever both the number N of points and the distance X of a satellite point from the cluster centre have finite (absolute) first moments. Similarly for the Bartlett–Lewis model show that the conditions are satisfied whenever the number of points N and the distance X between successive cluster points both have finite means. 12.5.8 (Continuation). Investigate conditions under which the strong asymptotic stationarity for Poisson cluster processes can be strengthened to geometric asymptotic stationarity as in Exercise 12.5.2(c).

12.6. Moment Stationarity and Higher-order Ergodic Theorems The essential simplification of the moment structure implied by stationarity derives from the application of Lemma A2.7.II as in the diagonal shifts Lemma 12.1.IV. It amounts to a diagonal factorization: each moment measure is represented as a product of Lebesgue measure along the main diagonal and a reduced measure in a complementary subspace. These reduced measures determine the moment structure of the process and have long been studied, usually as densities, in applications. They appear in many different guises, notably as the moment measures of the Palm distributions in Chapter 13, and in the higher-order ergodic theorems discussed at the end of this section and

12.6.

Moment Stationarity and Higher-order Ergodic Theorems

237

again in Section 13.4. Their Fourier transforms provide the various spectra of the random measure, the second-order version (Bartlett spectrum) being discussed in detail in Chapter 8. The role of the factorization theorem in the present context emerged in the work of Brillinger (1972) and Vere-Jones (1971), being subsumed by the more general results on the disintegration of moment measures developed in Krickeberg (1974b). We start with some definitions, supposing again that X = Rd , and writing Tx for the shift operator as in (12.1.1). Although we include MPPs in the exposition below, for which stationarity was noted below Definition 12.1.II, the resulting expressions are more involved and are left largely as exercises. Definition 12.6.I. (a) A random measure or point process on X = Rd is kthorder stationary if its kth moment measure exists, and for each j = 1, . . . , k, bounded Borel sets A1 , . . . , Aj , and x ∈ Rd , Mj (Tx A1 × · · · × Tx Aj ) = Mj (A1 × · · · × Aj ).

(12.6.1)

(b) An MPP on Rd × K is kth-order stationary if its kth moment measure exists, and for each j = 1, . . . , k, bounded Borel sets A1 , . . . , Ak and K1 , . . . , Kk ∈ BK , Mj (Tx A1 × K1 × · · · × Tx Aj × Kj ) is independent of x ∈ Rd . If ξ is a stationary random measure, the joint distributions of {ξ(Tx A1 ), . . . , ξ(Tx Aj )} coincide with those of {ξ(A1 ), . . . , ξ(Aj )} (see around Definition 12.1.II), so that a stationary random measure for which the kth-order moment measure exists is kth-order stationary. The converse implication is not true in general (see Exercise 8.1.1), but in particular parametric models, moment stationarity, even of relatively low order, generally requires stationarity of the process as a whole. For example, a Poisson process is stationary if and only if it is first-order stationary. The imposition of conditions on Mj for j < k in Definition 12.6.I is certainly redundant in the case of a simple point process, for the lower-order moment measures appear as diagonal concentrations (see Proposition 9.5.II) and are thereby identified uniquely (see Exercise 9.5.7). It may be redundant more generally, but the question appears to be open. It is relatively easy, however, to find a process for which the second cumulant measure is stationary but the expectation measure is not (see Exercise 8.1.2). The case k = 1 of the condition (12.6.1) simply asserts that the expectation measure M1 (·) is invariant under shifts. It must therefore reduce to a multiple of the unique measure on Rd with this property, namely, Lebesgue measure. We thus have the following proposition that incorporates parts of Propositions 6.1.I, 8.1.I and 8.3.II. Proposition 12.6.II. (a) A random measure on Rd is first-order stationary if and only if its expectation measure is a finite positive multiple m (the mean density) of Lebesgue measure on Rd . (b) A marked random measure or MPP on Rd × K is first-order stationary if and only if its expectation measure is a product × F of Lebesgue measure

238


on Rd and a boundedly finite measure F on BK ; F is totally finite if and only if the expectation measure of the ground process has finite mean density mg , and mg = F (K). The proportionality constant referred to in this proposition is usually denoted m and called either the mean density or the rate of the process. For k > 1, the conditions at (12.6.1) imply, via the generating properties of rectangle sets, that the whole measure Mk is invariant under the group of (k) diagonal shifts Dx defined by (12.1.11). The diagonal shifts Lemma 12.1.IV now implies the following proposition for the unmarked case. The corresponding results for the marked case are outlined in Exercise 12.6.9. Proposition 12.6.III. For any kth-order stationary random measure or ˘ k, M ˘ [k] , C˘k , and C˘[k] repoint process on Rd , there exist reduced measures M lated to the corresponding kth-order measures Mk , M[k] , Ck , and C[k] through equations, valid for any function f ∈ BM(X (k) ), of the type f (x1 , . . . , xk ) Mk (dx1 × · · · × dxk ) X (k) ˘ k (dy1 × · · · × dyk−1 ). (12.6.2) = dx f (x, x + y1 , . . . , x + yk−1 ) M X

X (k−1)

˘ k, M ˘ [k] , C˘k , and C˘[k] the reduced kth-order moment measure, the We call M reduced kth-order factorial moment measure, the reduced kth-order cumulant measure, and the reduced kth-order factorial cumulant measure, respectively; see Proposition 13.2.VI for their interpretation as moment measures of the Palm distribution. For k = 1 these reduced measures all coincide and equal the mean density m = M1 (Ud ). For k = 2 we mostly use C˘2 , which we also call the reduced covariance measure. It is defined on BX , and its properties and applications form the main content of Chapter 8. Note that the disintegration furnished by (12.6.2) is of the form ˘ k, Mk = × M

(12.6.3)

where denotes standard Lebesgue measure on Rd (so, (Ud ) = 1) and thus any scale factors remain in the reduced measure. The same disintegration result can also be obtained via an argument involving Radon–Nikodym derivatives with respect to the first-moment measure, as in Exercises 12.1.8–9. This alternative approach is outlined in Exercises 12.6.1–2 and leads to a decomposition of the form ˘ k) Mk = M1 × (m−1 M with M1 = m. This and its role in Palm theory has led some authors to ˘ k as the definition of the reduced measure; we have preferred not adopt m−1 M to adopt this convention, mainly because of its incompatibility with the usual definition of the stationary form of the covariance function when the measure is absolutely continuous.

12.6.


239

Some of the more accessible properties of the reduced moment measures are given in the next proposition; analogous statements for factorial moment and cumulant measures can be given (see Exercise 12.6.3). A more extended list of properties for the case k = 2 is given in Proposition 8.1.II for the unmarked point processes and in Proposition 8.3.II for MPPs. ˘ k be the kth-order reduced moment measure Proposition 12.6.IV. Let M for the kth-order stationary random measure ξ on X = Rd . ˘ k (·) is a symmetric measure on X (k−1) (invariant under permutations of (i) M the arguments in the product space) and is invariant also under the ‘shift reflection’ transformation mapping (u1 , u2 , . . . , uk−1 ) into (−u1 , u2 − u1 , . . . , uk−1 − u1 ). ˘ k is also absolutely (ii) When Mk is absolutely continuous with density mk , M continuous, and its density m ˘ k is related to mk by ˘ k (x2 − x1 , . . . , xk − x1 ). mk (x1 , x2 , . . . , xk ) = m (iii) For all bounded Borel sets A1 , . . . , Ak−1 ∈ X , ˘ k (A1 ×· · ·×Ak−1 ) = E ξ(x+A1 ) . . . ξ(x+Ak−1 ) ξ(dx) . M

(12.6.4)

(12.6.5)

Ud

Proof. In (12.6.2) set f (x1 , . . . , xk ) = g(x1 )h(x2 − x1 , . . . , xk − x1 ), where g(·) and h(·) are bounded Borel functions of bounded support on X and X (k−1) , respectively, so that g(x1 )h(x2 − x1 , . . . , xk − x1 ) Mk (dx1 × · · · × dxk ) X (k) ˘ k (dy1 × · · · × dyk−1 ). = g(x) dx h(y1 , . . . , yk−1 ) M (12.6.6) X

X (k−1)

Now let the variables x2 , . . . , xk in the argument of h(·) on the left-hand side be permuted. Because of the symmetry properties of Mk (·), this leaves the integral unaltered. Observe also that it corresponds to permuting the variables y1 , . . . , yk−1 in the argument of h(·) on the right-hand side of (12.6.6). Equivalently, it corresponds to leaving the variables in h(·) unaltered and ˘ k . Because a measure on X (k−1) is uniquely permuting the variables in M ˘ k must determined by the integrals of all such functions h(·), it follows that M be invariant under permutations of its arguments; that is, it is symmetric. Alternatively, if we interchange x1 and x2 on the left-hand side of (12.6.6), the integral is unaltered, and from (12.6.3) the right-hand side becomes ˘ k (dy1 × · · · × dyk−1 ). dx g(x + y1 )h(−y1 , y2 − y1 , . . . , yk−1 − y1 ) M X

X (k−1)

240


But

X

g(x + y1 ) dx =

X (k−1)

X

g(x) dx, so we conclude that

˘ k (dy1 × · · · × dyk−1 ) h(y1 , . . . , yk−1 ) M ˘ k (dy1 × · · · × dyk−1 ), = h(−y1 , y2 − y1 , . . . , yk−1 − y1 ) M X (k−1)

from which there follows the shift reflection invariance assertion in (i). If Mk has density mk with respect to Lebesgue measure on Rdk , invariance of Mk implies that m k (x, x + y1 , . . . , x + yk−1 ) is independent of x, so on cancelling the factor X g(x) dx in (12.6.6) we obtain ˘ k (dy1 × · · · × dyk−1 ) h(y1 , . . . , yk−1 ) M X (k−1) = h(y1 , . . . , yk−1 ) mk (x, x + y1 , . . . , x + yk−1 ) dy1 . . . dyk−1 . X (k−1)

˘ k is absolutely continuous with density mk (x, x + y1 , . . . , x + yk−1 ), Thus, M which is equivalent to the assertion (ii). #k−1 Finally, in (12.6.6) set g(x) = IUd (x) and h(y1 , . . . , yk−1 ) = j=1 IAj (yj ). Then because X g(x) dx = (Ud ) = 1, (12.6.5) follows directly. ˘ k are necessarily nonnegative, the same Although the reduced measures M is not true of reduced cumulant measures C˘k . In the simplest nontrivial ˘ 2 (A) − m2 (A) [see (8.1.6)], so that for its Jordan–Hahn case, C˘2 (A) = M decomposition C˘2 = C˘2+ − C˘2− into positive and negative parts, we have C˘2− (A) ≤ m2 (A) < ∞


˘ 2 (A) = (m2 + m)(A) so For the simple (stationary) Poisson process, M + − C˘2 (A) = m(A), C˘2 (A) = 0, but for the stationary deterministic process on R with span a as in Example 8.3(e), C˘2+ consists of atoms of mass 1/a at the points ka (k ∈ Z) and C˘2− (A) = (A)/a2 for A ∈ BR . Thus, although this process has 0 ≤ var N (0, x] ≤ 14 , neither of C˘2+ and C˘2− is totally finite. ˘ k (·) as an expectation suggests the exisThe identification at (12.6.5) of M tence of higher-order ergodic theorems in which the reduced moment measures appear as the ergodic limits. To identify the limits in the nonergodic situation, we use the following application of Lemma 12.2.III, where I again denotes the σ-algebra of invariant events. Lemma 12.6.V. Let ξ be a strictly stationary random measure with finite kth moment measure. Then there exists a symmetric I-measurable random measure ψ˘k on X (k−1) , invariant also under the shift reflections of Proposition 12.6.IV, such that for bounded Borel functions f of bounded support on X (k) , " " f (x1 , . . . , xk ) ξ(dx1 ) . . . ξ(dxk ) "" I E (k) X = dx f (x, x + y1 , . . . , x + yk−1 ) ψ˘k (dy1 × · · · × dyk−1 ). (12.6.7) X

X (k−1)

12.6.


241

In particular, for bounded A1 , . . . , Ak−1 ∈ BX , " " ˘ ψk (A1 × · · · × Ak−1 ) = E ξ(x + A1 ) . . . ξ(x + Ak−1 ) ξ(dx) "" I . (12.6.8) Ud

Proof. Represent X (k) in the product form X × X (k−1) via the mapping (12.1.13). On the space X (k) , ξ induces a new random measure, namely, the k-fold product ξ (k) of ξ with itself, and ξ (k) is stationary with respect to diagonal shifts. Its image under (12.1.13) is therefore stationary with respect to shifts in the first component. We now have a situation to which the general result in Lemma 12.2.III applies, with X (k−1) playing the role of the mark space K. On the product space X × X (k−1) there exists a σ-algebra of sets invariant under shifts in the first component, and the image of ξ (k) under (12.1.13) has a conditional expectation with respect to this σ-algebra, which factorizes into a product of Lebesgue measure on X and an I-measurable random measure ψ˘k on X (k−1) , which is readily checked as having the properties described in the lemma. Before proceeding to the ergodic theorems, we give a further example, albeit somewhat artificial in character, to illustrate some of the types of behaviour that can occur in the nonergodic case. Example 12.6(a) Poisson cluster process with dependent clusters. Suppose that X = R and that cluster centres occur at rate λ. Set up a common pattern for the clusters from a fixed realization {y1 , . . . , yZ } of a finite Poisson process on R with a nonatomic parameter measure µ(·), so that Z is a Poisson r.v. with mean E(Z) = µ(R) = ν and, conditional on Z, the r.v.s y1 , . . . , yZ are i.i.d. r.v.s with distribution µ(·)/ν. Then, given a realization {xi } of the cluster centre process, we associate with the cluster centre xi the cluster (xi + yj : j = 1, . . . , Z}, so that the whole process has as its realization the points {xi + yj : i = 0, ±1, . . . ; j = 1, . . . , Z}. The r.v.s {Z, y1 , . . . , yZ } define a σ-algebra of events, which, in fact, coincides with the invariant σ-algebra I for the whole process. We can then compute moment characteristics of the process as follows. (1◦ ) For k = 1, the mean density given the invariant σ-algebra I, that is, the r.v. Y of (12.2.12), here equals λZ; the mean density of the whole process equals m = E(λZ) = λµ(R) = λν. (2◦ ) For k = 2 and before reduction, the second-order moment measure given I has three components: a multiple λ2 Z 2 of Lebesgue measure in the plane; a line concentration with density λZ along the main diagonal x = y; and line concentrations of density λ along each of the Z(Z − 1) lines y = x + yi − yj , where i = j but both orderings are permitted. Then the reduced moment measure ψ˘2 (·) on BR can be written ψ˘2 (du) = λ2 Z 2 du + λZδ(u) du + λ δ(u − yi + yj ) du, i =j

242


where the δ-function terms represent atoms at 0 and the points ±|yi − yj | (i = j). Taking expectations leads to the reduced second moment measure: ˘ 2 (du) = λ2 ν(ν + 1) du + λνδ(u) du + λ M

µ(x + du) µ(dx). R

(3◦ ) Third- and higher-order moments can be built up in a similar way by considering all possible locations of triples of points {yi } and so on. Observe that ψ˘2 is just the form that the reduced moment measure would take if the cluster structure were fixed for all realizations: the process would then be infinitely divisible and ergodic. A variant of this Poisson cluster process, but having conditionally independent clusters, can be obtained by treating the clusters as a Cox process directed by some a.s. finite random measure ξ replacing the fixed measure µ(·) above: regard ξ as fixed for any given realization with the points in each cluster now being determined mutually independently according to a Poisson ˘ 2 as above equals the random process with parameter measure ξ(·). Then M ˘ 2 is obtained by a further averaging measure ψ˘2 of this process and the new M over the realizations of ξ. Further variants of the model are possible. We are now in a position to state the higher-order version of the ergodic Theorem 12.2.IV [see also Nguyen and Zessin (1976)]. Extensions to the marked case of Lemma 12.6.V and the result below are outlined in Exercise 12.6.10. For further extensions see Sections 13.4–5, especially Propositions 13.4.I and 13.4.III. Theorem 12.6.VI. Let ξ be a strictly stationary random measure for which the kth moment measure exists, ψ˘k the invariant random measure defined by (12.6.8), and B1 , . . . , Bk−1 a family of bounded Borel sets in Rd . Then, for any convex averaging sequence {An } in Rd , as n → ∞, 1 (An )

a.s. ξ(x + B1 ) . . . ξ(x + Bk−1 ) ξ(dx) → ψ˘k (B1 × · · · × Bk−1 ). (12.6.9)

An

In particular, if ξ is ergodic, 1 (An )

a.s. ˘ ξ(x+B1 ) . . . ξ(x+Bk−1 ) ξ(dx) → M k (B1 ×· · ·×Bk−1 ). (12.6.10)

An

Proof. Given a bounded Borel function g of bounded support on X (k) , consider the random function g(x1 , . . . , xk ) ξ(dx1 ) . . . ξ(dxk ), f (ξ) ≡ X (k)

noting that, by assumption, we have E[f (ξ)] < ∞.

12.6.


243

Appealing to Proposition 12.2.II(a) and evaluating the limit in (12.2.6) from (12.6.7), we obtain, for n → ∞, 1 f (Sx ξ) dx (An ) An 1 dx g(u1 − x, . . . , uk − x) ξ(du1 ) . . . ξ(duk ) = (An ) An X (k) a.s. → dx g(x, x + y1 , . . . , x + yk−1 ) ψ˘k (dy1 × · · · × dyk−1 ). (12.6.11) X (k−1)

X

In particular, by taking g(x1 , . . . , xk ) = gε (x1 )h(x2 − x1 , . . . , xk − x1 ), where gε (·) has the same properties as in the proof of Theorem 12.2.IV, it follows, for example, that Aε f (Sx ξ) dx equals

n

gε (u1 − x) dx Aεn

X (k)

gε (u1 − x) dx

= Aεn

≥

h(u2 − u1 , . . . , uk − u1 ) ξ(du1 ) . . . ξ(duk )

ξ(du1 ) An

X (k)

X (k−1)

h(v1 , . . . , vk−1 ) ξ(du1 ) ξ(u1 + dv1 ) . . . ξ(u1 + dvk−1 )

h(v1 , . . . , vk−1 ) ξ(u1 + dv1 ) . . . ξ(u1 + dvk−1 ).

Thus, we can use the approximation argument exploited in Theorem 12.2.IV to deduce that, for nonnegative bounded functions h of bounded support in X (k−1) , as n → ∞, 1 ξ(du) h(v1 , . . . , vk−1 ) ξ(u + dv1 ) . . . ξ(u + dvk−1 ) (An ) An X (k−1) a.s. → h(v1 , . . . , vk−1 ) ψ˘k (dv1 × · · · × dvk−1 ). (12.6.12) X (k−1)

Equations (12.6.9) and (12.6.10) are now easily derived as special cases of (12.6.12). It is, of course, a corollary of (12.6.8) that ˘ k (B1 × · · · × Bk−1 ). E ψ˘k (B1 × · · · × Bk−1 ) = M An L1 version of Theorem 12.6.VI is given in Exercise 12.6.8. For point processes, the left-hand side of (12.6.10) suggests a natural class of nonparametric estimates for the reduced moment measures, as for example the estimate + [2] (B; A) = 1 N ∗ (xi + B), (12.6.13) M (A) i:xi ∈A

∗

where N (B) = N (B) − δ0 (B), introduced at (8.1.25) in discussing secondorder factorial moment measures. In practice, estimates of this kind are

244


subject to serious biases arising from edge effects, as discussed briefly around (8.1.26) and in greater detail in texts such as SKM (1995). In such contexts, Theorem 12.6.VI provides a starting point for proving the consistency of the estimates or of variants in which (A) is replaced by N (A), itself representing an estimate of M1 (A) = m(A). The resulting quantity can be written + [2] (B; A) ≈ M

N (A) m ∗ N (xi + B) N (A) i=1

(12.6.14)

and represents the average, over the points of A, of the counts of points in sets B relative to the points of A as origin. As such, it is a point estimate of the first-order moment measure of the Palm process associated with N , as discussed further in Section 13.4. Again, Theorem 12.6.VI provides the basis for proving the consistency of such estimates, and is so used in the discussion of fractal dimension in Section 13.6. Often, the natural interpretation of the estimates such as in (12.6.13) is in terms of point configurations. For example, in the case k = 3, the third-order factorial moment measure gives information about the occurrence of triplets of points of the realization, taking one point of the triplet as origin. In the discussion of Section 13.6, for example, use is made of sets Bk−1,r = {(u1 , . . . , uk−1 ): max uj < r}, which for k = 3 gives information about the proportion of triplets in the realization with the property that all three points of the triplet lie within a maximum distance r of one of the three points. Kagan (see Exercise 12.6.11) has used estimates of two-, three- and four-point configurations at ‘small’ scale in examining possible relations between shocks and aftershocks in earthquake studies. As for finite processes in Chapter 5, moment densities are often used as an aid to understanding both qualitative and quantitative behaviour of models, as we illustrate in our concluding example. Example 12.6(b) Interacting point processes. Suppose given two stationary simple point processes Nj (j = 0, 1) with mean densities mj . They evolve independently except that each successive point ti of the process N1 is followed by a dead time Zi , during which any point of the process N0 is deleted. Suppose that Zi = min(Yi , ti+1 −ti ), where {Yi } is a sequence of i.i.d. nonnegative r.v.s independent of both N0 and N1 . We observe N1 and the thinned process N2 consisting of those points of N0 that are not deleted. Our aim is to describe the first and second moment measures of the output (N1 , N2 ), particularly as they relate to the same measures for N0 . To this end it is convenient to use the {0, 1}-valued process J(t) for which 0 if 0 < t − ti ≤ Zi for some i, J(t) = 1 otherwise (so, t ∈ i (ti + Zi , ti+1 ]).

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245

Because the marked point process {ti , (Yi , ti+1 − ti )} on R × R2+ is stationary whenever N1 (with realizations the points {ti }) is stationary, it follows that then J(·) is itself a stationary process, with first and second moments α = EJ(t) (all t), β(u) = E J(t)J(t + u) (all t, u). Furthermore, J is determined by N1 and {Yi }, and is thus independent of N0 . Consequently from N2 (dx) = J(x) N0 (dx) it follows that N2 has mean density m2 given by 1 1 J(x) N2 (dx) = E E[J(x)] N2 (dx) = αm0 . m2 = EN2 (0, 1] = E 0

0

In addition, for bounded measurable h of bounded support in R2 , and writing D = {(x, y): x = y} for the diagonal of R2 , the second factorial measure (2) M[2] (·) of N2 satisfies R2 \D

(2) h(x, y) M[2] (dx × dy) = E =E

R2 \D

R2 \D

h(x, y) N2 (dx) N2 (dy) h(x, y)J(x)J(y) N0 (dx) N0 (dy) .

Thus, when N0 has a reduced factorial moment density m ˘ [2] (·), we have h(x, y)J(x)J(y)m ˘ [2] (x − y) dx dy E R2 \D h(x, y)β(x − y)m ˘ [2] (x − y) dx dy, = R2 \D

and N2 has a reduced factorial moment measure which likewise has a density (2) m ˘ [2] (·); it is given by (2)

m ˘ [2] (u) = β(u)m ˘ [2] (u). Finally, for the cross-intensity we find similarly (using differential notation for brevity) that E N1 (dx) N2 (dy) = m0 γ(y − x) dx dy, where γ(u) dx = E[J(x + u) N1 (dx)]. Here, γ(u) can be interpreted as the rate of occurrence of points ti of N1 such that ti +u lies outside any dead-time interval. Any more detailed evaluation of the quantities α, β(·), and γ(·) of the process N1 requires in turn more specific detail about its structure. Ergodicity of N1 is enough to show via the ergodic theorem that n ∞ (X − Zi ) i=1 n i α = lim = m1 E[(Xi − Yi )+ ] = m1 G(v) 1 − F (v) dv, n→∞ 0 i=1 Xi

246


where G is the common d.f. of the Yi , and Xi = ti+1 − ti with common d.f. F given by F (x) = lim Pr{N1 (0, x] ≥ 1 | N1 (−h, 0] ≥ 1} h↓0

(see Chapters 3 and 13). The functions β and γ are necessarily more complex, involving joint distributions, as we now illustrate in the case that N1 is a stationary renewal process with lifetime d.f. F as just given. Writing π(t) = Pr{J(t) = 0 | N1 has a point at the origin} for the conditional dead-time probability function, the regenerative properties of N1 imply that π(·) satisfies the renewal equation t π(t − v) dF (v) π(t) = 1 − G(t) 1 − F (t) + 0 t 1 − G(t − v) 1 − F (t − v) dU (v), = 0

where U (·) is the renewal function generated by the d.f. F [see (4.1.7)]. When F is such that the nonlattice form of the renewal Theorem 4.4.I holds, (4.4.2) yields ∞ 1 − G(v) 1 − F (v) m1 dv (t → ∞). π(t) → 0

Thus, writing B for a stationary backward recurrence time r.v. (see Section 4.2), we also have lim π(t) = Pr{Z > B} = 1 − α. t→∞

For general t, we have γ(t) = m1 [1 − π(t)], so it remains only to identify β(·). When N1 is stationary, write Bt and Tt for the backward and forward recurrence time r.v.s at time t, noting that stationarity implies from (4.2.7) and (4.4.2) that their joint distribution has

∞

Pr{Bt > x, Tt > y} = m1

1 − F (v) dv.

x+y

For u > 0 we have β(u) = E J(t)J(t + u)

u 1 − π(u − v) Pr{Bt > Z, Tt ∈ (v, v + dv)} = Pr{Bt > Z, Tt > u} + ∞ u 0 = α − m1 dG(z) π(u − v) 1 − F (z + v) dv, 0

→ α2

(u → ∞)

0

when π(t) → 1 − α

(t → ∞).

12.6.


247

The properties of β(u) and γ(u) noted for large u reflect asymptotic independence, but it is primarily the local properties, for u closer to zero, that are of interest in practice, because it is these that reflect any causal relation between the two processes N0 and N1 . Parametric models are then called for, but take us further from the theme of this section so we refer, for example, to Lawrance (1971) or Sampath and Srinivasan (1977) for specific details (see also Exercises 12.6.12–13).

Exercises and Complements to Section 12.6 12.6.1 For a random measure (not necessarily stationary) on the c.s.m.s. X for which the kth moment measure Mk (·) (k ≥ 2) is boundedly finite, so M (·) ≡ M1 (·) ˘ k (B | x) with exists, establish the existence of a Radon–Nikodym family M the properties below, including the ‘disintegration’ property in (c). ˘ k (B | x) is a measurable function of (a) For each bounded B ∈ B(X (k−1) ), M x that is integrable with respect to M (·) on bounded sets. ˘ k (· | x) is a boundedly finite measure on B(X (k−1) ). (b) For M -almost all x, M (k−1) ), use the fact that Mk (· × B) M (·) to conclude that (c) For B ∈ B(X ˘ k (B | x) M (dx) M

Mk (A × B) =

[A ∈ B(X )].

A

12.6.2 (Continuation). Arguing as in Exercises (12.1.8–9), deduce that when X = ˘ k (· | x) Rd and the process is kth-order stationary, there exists a version of M (k−1) ˘ that is invariant under simultaneous translations; that is, Mk (Dy B | ˘ k (B | x). Hence, give an alternative proof of Proposition 12.6.III. x + y) = M 12.6.3 Give analogous statements to those of Proposition 12.6.IV for the reduced factorial moment measure and for the reduced ordinary and factorial cumulant measures. Investigate the analogue of (12.6.6) when Mk is replaced by M[k] and g and h are indicator functions as in the proof of (12.6.5). Relate the case k = 2 to the ergodicity result underlying (12.6.13). 12.6.4 Find the reduced moment and cumulant measures for a stationary renewal process. In particular, show that if the renewal function has a density h(·), then reduced kth factorial moment measures exist for all k = 2, 3, . . . and have densities m ˘ [k] (x1 , . . . , xk−1 ) = λh(x1 )h(x2 − x1 ) · · · h(xk−1 − xk−2 ), where {x1 , . . . , xk−1 } is the set {x1 , . . . , xk−1 } arranged in ascending order. [Hint: See Example 5.4(b) and Exercise 7.2.3.] 12.6.5 (a) Show that when the reduced kth factorial cumulant measure of a kthorder stationary point process is totally finite, the kth cumulant of N (A) is asymptotically proportional to (A) as A ↑ X through a convex averaging sequence. (b) Show that a stationary Poisson cluster process for which the cluster size distribution has finite kth moment, satisfies the conditions of (a). (c) Show that the conditions of (a) are not satisfied for k ≥ 2 by either of (i) any nontrivial mixture of Poisson processes; and (ii) a stationary renewal process whose lifetime distribution has finite first moment but infinite second moment. [Hint: Compare Exercises 4.1.1–2 and 4.4.5(c).]

248


12.6.6 Let the stationary random measure ξ on Rd have finite kth moment measure Mk . Show that for bounded Borel functions f of bounded support in (Rd )(k−1) , as n → ∞, E

1 (An )

f (x1 , . . . , xk ) ξ(dx1 ) . . . ξ(dxk ) An

−

X (k−1)

f (x, x + y1 , . . . , x + yk−1 ) ψ˘k (dy1 × · · · × dyk−1 ) → 0.

12.6.7 Suppose the stationary random measure ξ(·) has the mean square continuous nonnegative random function η(·) as density; that is, ξ(A) = A η(u) du a.s. (a) Prove that the covariance measure C2 of ξ(·) has no atom at the origin. (b) When η(·) is stationary and cov(η(x), η(y)) = σ 2 ρ(|x − y|), show that var ξ(0, x] = 2σ 2

x 0

(x − y)ρ(y) dy

(x > 0).

(c) Interpret the results of (a) and (b) in the (degenerate) case that ξ(A) = Y (A) for a r.v. Y with E(Y 2 ) < ∞. 12.6.8 L1 version of Theorem 12.6.VI. Let the strictly stationary random measure ξ of Theorem 12.6.VI be ergodic and have finite kth moment measure. Show that under the conditions of the theorem, as n → ∞, E

1 (An )

˘ k (B1 × · · · × Bk−1 ) → 0. ξ(x + B1 ) . . . ξ(x + Bk−1 ) ξ(dx) − M An

12.6.9 Higher-order moments for marked random measures. Show that for any kth-order stationary marked random measure or MPP on Rd with marks ˘ k such that for any function f ∈ in K there exists a reduced measure M (k) (k) BM(X × K ), and writing xk , κk for (x1 , . . . , xk ), (κ1 , . . . , κk ) and so on,

X (k) ×K(k)

f (xk , κk )Mk (d(xk × κk ))

=

dx X

X (k−1) ×K(k)

˘ k (d(yk−1 × κk )). f (x, x + yk−1 , κk )M

In particular, use this reduced measure to imitate (12.6.5) for the marked case. 12.6.10 Higher-order ergodic theorem for the marked case. (a) Extension of Lemma 12.6.VI. As in Lemma 12.2.III, let ξ be a random measure on the product space X ×K, and let I be the associated σ-algebra of events invariant under shifts Sx , x ∈ X = Rd . Establish the existence ˘ k (xk−1 , κk ) on X (k−1) × K(k) such of an I-measurable random measure Ψ that for bounded Borel functions f on X (k−1) ×K(k) with bounded support

E

X (k) ×K(k)

k

f (xk , κk )

=

dx X

ξ(dxi × dκi ) I

1

X (k−1) ×K(k)

˘ k (d(yk−1 × κk )). f (x, x + yk−1 , κk ) Ψ

(b) Establish a corresponding version of Proposition 12.6.VII for MPPs, with ˘ k replacing ψ˘k . Ψ

12.7.

Long-range Dependence

249

12.6.11 Kagan’s conjectures. On the basis of empirical evidence from current earthquake catalogues [see also Kagan and Knopoff (1980)], Kagan (1981a, b) conjectured that earthquakes in the crust have a type of self-similar distribution in which the second-, third-, and fourth-order factorial moment densities have the respective forms: m[2] (x, y) ∼ 1/D(x, y) ;

m[3] (x, y, z) ∼ 1/A(x, y, z) ;

m[4] (w, x, y, z) ∼ 1/V (w, x, y, z), where D is the distance, A the area, and V the volume described by the respective arguments of the densities. (a) Investigate conditions under which the above conjectures are compatible with the existence of a stationary process with the prescribed densities. [Hint: Consider integrability conditions at the origin.] (b) Investigate conditions under which they might be approximately true for processes of fractal type, meaning, that they are concentrated in ‘random faults’ on lower-dimensional elements such as lines or surfaces. [Remark: These conjectures are still unresolved; for further discussion see Kagan and Vere-Jones (1995).] 12.6.12 Suppose that the process N0 in Example 12.6(b) is Poisson with rate parameter λ0 . Then the output process N2 has covariance density λ20 c(u) = λ20 cov (J(0), J(u)) = λ20 [β(u) − α2 ]. Show that, when N1 is a Poisson process at rate λ1 and the Yi are exponentially distributed with mean 1/µ, α= π(t) =

µ , λ1 + µ λ1 + µe−(λ1 +µ)t , λ1 + µ

β(u) =

µ(λ1 e−(λ1 +µ)u + µ) , (λ1 + µ)2

γ(u) = λ1 (1 − π(u)) =

λ1 µ(1 − e−(λ1 +µ)t ) . λ1 + µ

12.6.13 Replace the inhibitory mechanism of Example 12.6(b) defined via the i.i.d. sequence {Yi } by Zi = min(Tti , ti+1 −ti ), where Tt is the forward recurrence time r.v. of the process N0 . Show that when N0 and N1 are independent stationary simple point processes with intensity λ0 , the output process has intensity ∞

λ2 = λ0

G(t) dF (t), 0

where G is now the d.f. of a lifetime r.v. Y for the process N0 and F is the d.f. of the backward recurrence time r.v. of the process N1 .

12.7. Long-range Dependence Long-range dependence (LRD) of a stochastic process could in principle relate to any of several characteristics of the process, but it has now generally come to be associated with second moments. For stationary point processes and random measures on R, we define it via the following variance property given in Daley and Vesilo (1997) and already referred to in Exercise 4.5.13.

250


Definition 12.7.I. A second-order stationary point process or random measure ξ on R is long-range dependent if lim sup t→∞

var ξ(0, t] = ∞. t

(12.7.1)

When the lim sup here is finite, the random measure is short-range dependent. Definition 12.7.I is less easy to apply in higher dimensions, although essentially the same ideas can be used. One approach, outlined in Exercise 12.7.3, is to consider the growth of var ξ(An ) relative to (An ) as An increases through a convex averaging sequence of sets. In the treatment and examples below, we mostly restrict attention to point processes in R. Recall that the second moment measure of a stationary random measure on an interval (0, t) in R cannot grow faster than O(t2 ) (see Exercise 12.7.1), so that (12.7.2) lim sup t−2 var ξ(0, t] < ∞. t→∞

This property implies that if in the denominator in (12.7.1) we replace t by tα , the ratio can be infinite in the limit only if α < 2. It is therefore convenient to delineate this range by an index as in the next definition [the name recalls early work on long-range dependence in flow records of the river Nile by the British engineer Hurst; see Beran (1994)]. Definition 12.7.II. A long-range dependent stationary random measure ξ has Hurst index var ξ(0, t] =∞ . (12.7.3) H = sup h: lim sup t2h t→∞ Then, the Hurst index must lie in the interval [0, 1], although for longrange dependence it is more narrowly confined to 12 ≤ H ≤ 1. Furthermore, to have H < 1, we can immediately rule out the nonergodic case, for unless the invariant σ-field I is trivial, it follows from Exercise 12.2.9 that (x → ∞), var ξ(0, x] ∼ x2 Γ({0}) where Γ({0}) = var Y and Y = E ξ(U) | I , so H = 1 when var Y > 0. For a stationary Poisson process at rate λ the ratio at (12.7.1) equals λ for all t so it cannot be long-range dependent. The next few examples indicate further possibilities of short- and long-range dependence. Example 12.7(a). For a stationary renewal process N (·) with renewal function U (·) as in Section 4.1, it follows from (3.5.7) that t 2[U (u) − λu] − 1 du (12.7.4) var N (0, t] = λ 0

and [cf. Exercise 4.4.5(d)] that var N (0, t] ≤ (const.)t for some finite constant if and only if supt>0 [U (t) − λt] < ∞, which is the case if and only if the lifetime

12.7.


251

distribution underlying U (·) has its second moment finite. In other words, the independence between intervals (of a renewal process) is not sufficient to eliminate the possibility of long-range dependence. The Hurst index for such N is identified in Exercise 12.7.4. Example 12.7(b) Cluster process. Let V c (x) = var Nc (0, x] denote the variance function [see (8.1.13b)] of the stationary cluster centre process Nc (·) of a cluster process N (·) with independent identically distributed component processes Nm (· | ·) (see Definition 6.3.I). Equation (53) of Daley (1972) shows that when N is stationary and Nc is orderly at rate λc , and the second moment of the component process Nm is finite, the variance function of N is given by ∞ E [Nm ((−u, x − u] | 0)]2 du var N (0, x] = λc −∞ ∞ ∞ ) m1 (dy) m1 (dz) − λc (x − |y − z|)+ + −∞ −∞ * + V c (y − z + x) − 2V c (y − z) + V c (y − z − x) , where m1 (y) = ENm ((−∞, y] | 0) and V c (x) = V c (−x) for x < 0. The three variance terms here equal cov(Nc (0, x], Nc (y − z, y − z + x]); this is dominated by V c (x), which implies that when limx→∞ V c (x)/xα exists finite and equals λ2,α say, the dominated convergence theorem can be applied when α > 1 to conclude that limx→∞ x−α var N (0, x] = m21 λ2,α , where m1 = m1 (∞). In this example, long-range dependent behaviour of the cluster centre process [as shown by V c (x) ∼ const. xα ] is carried over into the cluster process itself, magnified by the square of the mean cluster size. When V c (x)/x → λ2 say (so Nc is not long-range dependent) the same argument [e.g., Daley (1972)] gives instead V (x) ∼ m21 λ2 x + (var Nm (R | 0))λc x

(x → ∞);

that is, variability in the component processes is no longer swamped. Consistent with this fact, having heavy-tailed distributions for the distances from the cluster centre to the cluster members is not sufficient to cause long-range dependence in the process as a whole. For example, the variance in a Neyman– Scott process with Poisson centre process [see, e.g., (6.3.19) and (12.7.5) below] is given by x |u| ˘ var N (0, x] = C[2] (du) = λc [m21 + m[2] F ∗ F − (x)], 1− x x −x where m[2] is the second factorial moment of the cluster size distribution, and F ∗ F − (·) is the distribution function of the distance X1 − X2 between two members of a cluster. When m[2] is finite, this converges to a finite limit irrespective of the character of the distribution of F .

252


Example 12.7(c) Superpositions. Suppose the stationary random measure ξ is expressible as the sum of the two stationary random measures ξ1 and ξ2 , and that all processes have finite second moments; we write ξ(t) = ξ(0, t] and so on. Then for t > 0 and from the Cauchy–Schwarz inequality, 2 var ξ(t) = var ξ1 (t) + var ξ2 (t) + 2 cov ξ1 (t), ξ2 (t) var ξ(t) = 2 var ξ1 (t) ± var ξ2 (t) . (12.7.5) ≤ (≥) Let H, H1 , H2 denote the Hurst indexes of ξ, ξ1 , ξ2 . Using (12.7.5), deduce that either H < H1 = H2 or H = max(H1 , H2 ), and the latter holds when ξ1 and ξ2 are independent. Daley and Vesilo (2000) give details and applications to some queueing examples. Because the variance properties are controlled by the reduced second moment measure C˘2 , we should expect the distinction between long- and shortrange dependence to be expressible in terms of this measure. The next lemma gives a partial resolution of this question; Example 12.7(e) indicates that the complement of these sufficient conditions does not yield a set of necessary conditions. Lemma 12.7.III. Let ξ be a second-order stationary random measure in R, and write C˘2 = C˘2+ − C˘2− for the Jordan–Hahn decomposition of its reduced covariance measure into its positive and negative parts. (a) ξ is short-range dependent if C˘2 is totally finite. When C˘2− is totally finite, ξ is long-range dependent if and only if the positive part C˘2+ is not totally finite. (b) ξ is short-range dependent if its Bartlett spectrum Γ(·) has a bounded density in a neighbourhood of the origin. Proof. The results are proved from the relations [cf. equations (8.1.13) and (8.2.3)] t ∞ |u| var ξ(0, t] = 1− (1 − |u|/t) C˘2 (du) = C˘2 (du) t t + −t −∞ (12.7.6) ∞ sin 12 θ 2 = Γ(t dθ), 1 −∞ 2θ expressing the middle integral as a difference of two integrals (involving C˘2+ and C˘2− , respectively) to which the monotone convergence theorem can be applied. Similarly, if Γ(·) has a density γ(θ) which is bounded in a neighbourhood of θ = 0, it follows from dominated convergence that the final integral remains bounded as t → ∞. Example 12.7(d) LRD Cox process. The Cox process N (·) directed by the and their variance funcstationary random measure ξ(·) on BR is stationary tions are related by var N (0, t] = E ξ(0, 1] t + var ξ(0, t] (Proposition 6.2.II). Thus, N (·) has exactly the same long-range dependence behaviour as ξ(·).

12.7.


253

For example, when ξ(·) accumulates mass at unit rate during just one of the phases of an alternating renewal process with generic lifetimes Xj (j = 1, 2) say, each with finite first moments, whenever one of the Xj has infinite second moment the random measure ξ (and hence N also) is long-range dependent. Details are left to Exercise 12.7.5. Example 12.7(e) Deterministic point process. Example 8.2(e) shows that the reduced covariance measure C˘2 of a stationary deterministic point process N with span a has positive atoms C˘2+ ({ka}) = 1/a for k = ±1, ±2, . . . , C˘2+ (A) = 0 whenever A ∩ {ka} = ∅ for all such k, and C˘2− (du) = (du)/a2 . This process has a periodic variance function, with 0 ≤ var N (0, t] ≤ 14 , so it is not long-range dependent. On the other hand the positive and negative parts of C˘2 are mutually singular [C˘2+ is purely atomic and C˘2− is a multiple of Lebesgue measure], and both C2+ (0, u] and C2− (0, u] increase indefinitely with the length u of the interval, whereas C˘2 (0, ka] = 0 for k = 1, 2, . . . . This example shows that we cannot exclude the possibility of the process being short-range dependent even when C˘2 fails to be totally finite. Long-range dependence is frequently understood in terms of power-law decay of the correlation function. In the point process context, this means looking at the covariance density function rather than the variance function of which it is the second derivative [Exercises 8.1.3(g), 12.7.2]. For example, Ogata and Katsura (1991) discuss situations in which the covariance density shows a power-law decay but describe it in terms of ‘fractal behaviour’. We note here the difficulty of maintaining a coherent and consistent naming system in situations where interest from the media and the general public preponderates. Fractals, with all the striking images associated with them in Mandelbrot’s writings form a case in point. Anything exhibiting some form of scaling behaviour or self-similarity, particularly when it is linked to powerlaw decay, is almost automatically labelled a ‘fractal’ in a nonmathematical context. By contrast, in this book we have tried to establish and maintain distinctions between the concepts of long-range dependence, representing (as in Exercise 4.5.13 and this section) a variance property related to powerlaw decay in a covariance or correlation function; heavy-tailed behaviour in a probability distribution; scale-invariance and self-similarity (discussed in the next section); and fractal behaviour which we take up in Section 13.6 and relate to the behaviour of moment densities at very small distances or time intervals. In a similar vein, the Hurst index is indeed an index and not dependent on any parametric setting.

Exercises and Complements to Section 12.7 12.7.1 When the set A ⊂ Rd is a hyper-rectangle of side-lengths n1 , . . . , nd and ξ is a second-order stationary random measure, express ξ(A) as the sum of

254

12. Stationary Point Processes and Random Measures the measures on νA = di=1 ni unit cubes, and use the Cauchy–Schwartz inequality to conclude that the second moment measure M2 of ξ satisfies 2 )M2 (Ud ). Compare with Exercise 8.1.3(b). M2 (A) ≤ (νA

12.7.2 Show that a stationary point process on R with reduced covariance density c˘(x) ∼ a/xγ (x → ∞) for a > 0 and 0 < γ < 1 has Hurst index 1 − 12 γ. 12.7.3 Call a stationary random measure ξ: B(Rd ) → R+ long-range dependent whenever lim supn→∞ ( var[ξ(An )]/(An )) = ∞ for some convex averaging sequence {An ; n = 1, 2, . . .}. (a) Show that this definition is independent of the convex averaging sequence. (b) Show, analogously to the first part of (12.7.6), that var ξ(An ) = (An )

Rd

(An ∩ Tz An ) ˘ C2 (dz). (An )

Show that the ratio in the integrand is always bounded by 1, and is close to 1 for small z. Hence provide an extension of Lemma 12.7.III to Rd . 12.7.4 Hurst index of a LRD renewal process. Let N be a stationary renewal process as in Example 12.7(a) whose generic lifetime r.v. X with d.f. F and tail F (x) = 1 − F (x) has moment index κ defined by κ ≡ inf{k: E(X k ) = ∞} = lim inf [− log F (x)]/ log x x→∞

[see, e.g., Daley and Goldie (2005) for the equality] for which 1 < κ < 2. Use ∞ (12.7.4) and the asymptotic behaviour U (t) − λt ∼ λ2 0 min(u, t) F (u) du 1 [Sgibnev (1981)] to deduce that its Hurst index H = 2 (3 − κ). [Hint: Daley (1999) gives a proof by contradiction; the proof via (12.7.4) and Exercise 4.4.5(c) as just indicated is direct.] 12.7.5 LRD in ON/OFF processes. Let I(t) be the stationary {0, 1}-valued process generated by the ON phases of a stationary alternating renewal process Nalt with lifetime d.f.s F1 for such ON-phases and F0 for the OFF phases (so I(t)= 1 when the alternating renewal process is in an ON phase, = 0 otherwise). Let T1 (t) denote the accumulated duration during (0, t) for which t I(u) = 1 for 0 < u < t, so that T1 (t) = 0 I(u) du. Let T0 (t) = t − T1 (t) denote the corresponding duration for which I(·) is in the OFF phase. Observe that var Tj < ∞ for j = 0, 1 because these Tj are bounded r.v.s, and in fact

cov (T1 (t), T0 (t)) = cov (T1 (t), t − T1 (t)) = − var T1 (t) = − var T0 (t). Consequently, if the Cox process NCox directed by the ON phases of Nalt is LRD (i.e., lim supt→∞ t−1 [var T1 (t)] = ∞), then so too is the Cox process directed by the OFF phases of Nalt . Show that NCox , Nalt , and the stationary random measure ξ with I(·) as its density, all have the same Hurst index. [Hint: Daley (2007) studies this example further, showing that for t → ∞, ( var NCox (0, t])/( var Nalt (0, t]) has a limit if one of the lifetime distributions Fj (j = 0, 1) has finite second moment but if both have infinite second moment the ratio can oscillate indefinitely. Daley, Rolski and Vesilo (2007) extend this work to a Cox process driven by a LRD semi-Markov process.]

12.8.

Scale-invariance and Self-similarity

255

12.8. Scale-invariance and Self-similarity In this last section we look at processes where invariance under multiplicative actions (scale changes) rather than additive actions (translations) plays the key role. Such processes include the important class of self-similar processes (also called auto-modelling in the older Russian literature). In the context of the present book, the most relevant concept is that of a self-similar random measure, where nonnegativity plays a dominant role, and the theory is rather different from that of the fractional Brownian motions and related two-sided processes which have become familiar in finance and other application areas. The self-similar random measures we consider are purely atomic, and can be described by the marked point process of the locations and sizes of the atoms. Moreover, self-similarity of the random measure can be restated in terms of a corresponding invariance property of this marked point process; we call this property biscale invariance. We first consider the simpler case of point processes invariant under scale changes about a fixed origin. A process on X = Rd is called scale-invariant if it is invariant under the group of scale changes {Tα : 0 < α < ∞},

where for x ∈ Rd ,

Tα x = αx.

(12.8.1)

This group splits Rd into equivalence classes, one of which is the origin, and the others can be identified with rays originating from the origin. Now Rd \{0} can be written as the product Sd × R+ 0 , where Sd , the group of d-dimensional rotations, can in turn be identified with the surface of the d-dimensional unit sphere, and R+ 0 denotes the open half-line (0, ∞) = R+ \ {0}. Note that S1 is just the two-point group T2 = {−1, 1} under multiplication. R+ 0 is a group under multiplication, and it has the unique invariant measure h(dx) = dx/x. It is now obvious, but also follows formally from Lemma A2.7.II, that any measure on R1 that is invariant under scale changes, can be represented as the sum of a point mass at the origin, and the direct product of a two-point mass on T2 and the measure h(·) on R+ 0 . Similarly, a scaleinvariant measure on Rd can be represented as the sum of a point mass at the origin and a measure κ(dθ) dr/r on Rd \ {0} = Sd × R+ 0 , where κ(·) is an arbitrary totally finite measure on Sd . The position of the origin is unimportant here, but it is clear that the above structure is incompatible with translations in Rd , so that a measure on Rd cannot be invariant under both translations and scale changes, a result with important consequences for both scale-invariant and self-similar random measures. Example 12.8(a) Scale-invariant Poisson processes on Rd . As in Example 12.1(a) we deduce that if a Poisson process on Rd is invariant under scale changes, its parameter measure must have the same property, and must therefore have the structure described above, namely, the sum of a point mass at the origin and a measure κ(dθ) dr/r on Sd × R+ 0.

256


Part (a) of the proposition below is an immediate consequence of this discussion. A similar argument applied to the first moment measure, whenever it exists, yields part (b). Proposition 12.8.I. (a) A Poisson process on Rd cannot be simultaneously invariant under both scale changes and translations. (b) No stationary random measure on Rd with finite expectation measure can be scale-invariant. We turn next to a discussion of self-similarity for random measures. Here, a change in scale is balanced by a compensating change in mass. Definition 12.8.II. Let D be a finite positive constant. A random measure is self-similar with similarity index D, (self-similar or D-self-similar for short), (D) if its distribution is invariant under the group of transformations {Rα : α ∈ # + R0 } defined on boundedly finite measures MX by (D) µ(A) = α−D µ(αA) Rα

(A ∈ BX ).

(12.8.2)

Zähle (1988) uses the shorter terminology above, and includes discussion of nonprobability measures on M# X . Even more extensive work is covered in Z¨ ahle (1990a, b, 1991). Note that self-similarity, like scale-invariance, refers in the first instance to invariance relative to a fixed origin; it is only under stationarity, or some explicit rule describing how the invariance properties alter as we shift the origin, that the concept extends beyond this case. (D) The transformations Rα do not result directly from transformations of the phase space X into itself, but do still induce a group, the renormalization group, of bounded continuous transformations of M# (Rd ) into itself (see Exercise 12.8.1). We start with two negative results. Because of the change in mass, the renormalization group does not map N # (Rd ) into itself. This justifies the following. Proposition 12.8.III. A point process cannot be self-similar. (D)

The class of (deterministic) measures invariant under Rα is not a rich family: on R+ it is confined to measures with power-law densities [hyperbolic densities in the usage of Mandelbrot (1982, p. 204)] fD (x) = Cx1−D

(C > 0, x > 0).

Only in the trivial case D = 1 is the measure invariant under both the simi(D) larity transformations Rα and translations; it reduces then to a multiple of Lebesgue measure. The situation for general random measures is more rewarding. To give a preview of the issues which arise, we examine first, without assuming stationarity, the consequences of self-similarity on the representation for completely random measures given in Theorem 10.1.III.

12.8.


257

Suppose that ξ is a completely random measure defined on X = Rd , so that, as in (10.1.4), it can be represented in terms of a drift, a set of fixed atoms, and a Poisson process N in Rd × R+ 0 . Operating on this representation (D) by Rα , and using differential notation for brevity, we find ∞ (D) Rα ξ(dx) = α−D ν(α dx) + y α−D N (α dx × dy) + Uk α−D δxk /α (dx) 0

= α−D ν(α dx) +

0

k

∞

y N (α dx × αD dy) +

Uk α−D δxk /α (dx).

k

This last expression again corresponds to a completely random measure: the (D) measure ν has been transformed by Rα as at (12.8.2), the Poisson process (D) N has been subjected to the biscale transformation Sα on X × R+ 0 given by Sα(D) N (A × K) = N (αA × αD K),

(12.8.3)

and the fixed atoms have been transformed both in mass and in location. If the distribution of the completely random measure is to remain invari(D) ant under all transformations Rα (α > 0), then it is clear that there can be no fixed atoms, that the (deterministic) measure ν must be invariant under (D) the transformations Rα , and that the parameter measure µ of the bivari(D) ate Poisson process N must be invariant under the transformations Sα . Thus, we have reduced the problem of characterizing the class of self-similar completely random measures to the problem of characterizing the classes of measures invariant under these two groups of transformations. For simplicity we consider only the case X = R1 ; the details for X = Rd are similar (see Exercise 12.8.2 for the case d = 2). As in Example 12.8(a), it is necessary to consider separately the action − + of the transformations on R+ 0 , {0} and R0 = {x: −x ∈ R0 }. Because the processes have no fixed atoms, ν{0} = 0 and µ, the parameter measure of N , (D) on measures has µ({0} × R+ 0 ) = 0. Thus, we may consider the effect of Rα (D) ν acting on BR+ , and of Sα on measures µ on BR+ ×R+ , with similar results following for the components on R− and R− × R+ . (D) From (12.8.2), invariance of ν under Rα implies that ν is absolutely + continuous on R0 ; its density with respect to Lebesgue measure is given by dν (x) = c1 xD−1 (x ∈ R+ 0 ). d Similarly, on R− 0,

dν (x) = c2 |x|D−1 (x ∈ R− 0 ). d Next, consider the representation of the parameter measure µ on the quad+ rant R+ 0 × R0 . Invariance of the distribution of N under (12.8.3) implies that

258

12. Stationary Point Processes and Random Measures (D)

µ itself is invariant under the biscale shifts Sα , and hence is invariant under shifts along the curves xD/y = constant. This suggests writing v = D log x − log y,

u = log x,

so that in the (u, v)-plane the transformation (12.8.3) becomes (u, v) → (u + log α, v).

(12.8.4)

We now deduce from Lemma A2.7.II that µ ˜, the image of µ under this mapping, reduces to a product of Lebesgue measure along the u-axis and an arbitrary σ-finite measure ρ˜1 along the v-axis. Thus, integration with respect to µ in this quadrant can be represented in the form ∞ ∞ f (x, y) µ(dx × dy) = f (eu , eDu−v ) du ρ˜1 (dv). (12.8.5) + R+ 0 ×R0

−∞

−∞

+ Similar considerations may be applied on R− 0 × R0 , with a possibly different measure ρ˜2 replacing ρ˜1 . The finiteness constraints (10.1.5), when expressed in the form ∞ (1 − e−y ) µ(A × dy) < ∞ (bounded A ∈ B(R)), 0

lead to the requirements that for i = 1, 2, 0 (1 + |v|) ρ˜i (dv) < ∞ and −∞

∞

e−v ρ˜i (dv) < ∞.

(12.8.6a)

0

If, in particular, ρ˜1 is absolutely continuous with respect to Lebesgue mea+ sure, then it is more convenient to write, for (x, y) ∈ R+ 0 × R0 and with ρ˜1 (dv) = ρ1 (v) dv, µ(dx × dy) =

η1 (xD/y) ρ1 (log(xD/y)) dx dy ≡ dx dy xy xy

for some nonnegative, locally integrable function η1 on R+ 0 . If also a similar + × R , then an analogous representation representation holds for (x, y) ∈ R− 0 0 and |x| in place of x. holds for some similar function η2 on R+ 0 When such absolute continuity conditions hold, the Laplace functional L[f ] for the random measure ξ can be written (f ∈ BM+ (R)), L[f ] = exp L1 [f ] + L2 [f ] where L1 [f ] equals ∞ D−1 x f (x) dx + −c1 0

0

∞

0

∞

(eyf (x) − 1)

η1 (xD/y) dx dy, xy

(12.8.6b)

and a similar expression holds for L2 with c1 replaced by c2 , η1 (xD/y) by η2 (|x|D/y), and integration of x is over (−∞, 0) in place of (0, ∞). Conditions (12.8.6a) transform to 1 ∞ 1 + | log z| ηi (z) ηi (z) dz < ∞ and dz < ∞. (12.8.7) z z2 0 1 We thus have a complete answer to the representation problem in the onedimensional case.

12.8.


259

Proposition 12.8.IV. A completely random measure on R is self-similar if and only if, in terms of the representation in Theorem 10.1.III, there are no fixed atoms, and the measures ν, µ can be written, as in (12.8.5), in terms of positive constants c1 , c2 and measures ρ˜1 , ρ˜2 satisfying (12.8.6a). In particular, when these measures have densities, the Laplace functional of the random measure can be written in the form (12.8.6b) for nonnegative, locally integrable functions η1 , η2 satisfying (12.8.7). For applications we generally require the random measure to be stationary as well as self-similar, in which case the representation must also be invariant under shifts along the x-axis. Then, the first term must vanish unless D = 1, when it reduces to a constant multiple of Lebesgue measure along the whole real axis. For the measures in the second term, the additional condition is easily seen to be satisfied if and only if η1 (v) = ρv 1/D = η2 (v), corresponding to (12.8.8) µ(dx × dy) = ρy −(1+1/D) dx dy. Then the constraints at (12.8.7) require D < ∞ and D > 1 respectively. Hence, it follows that the class of completely random measures that are both stationary and self-similar reduces, for D = 1, to the trivial example of a constant multiple of Lebesgue measure, and for 1 < D < ∞, to the stable processes with index α = 1/D and Laplace functional of the form [cf. Example XIII.7(c) of Feller (1966)]

∞

∞

1 − e−yf (x) dy y 1+1/D −∞ 0 ∞ [f (x)]1/D dx. = ρDΓ(1 − D−1 )

− log L[f ] = ρ

dx

(12.8.9)

−∞

Corollary 12.8.V. A completely random measure on R is both stationary and self-similar if and only if there is no drift or atomic component, and the Poisson process in representation (10.1.4) has density (12.8.8), with 1 0, 1 < D < ∞).

(12.8.11)

Proof. Provided it is understood that K ∈ BR+ is bounded away from both 0 0 and ∞, it follows from Proposition 9.1.VIII that κ N (dt × dκ) ζ(A × K) = A×K

defines a boundedly finite random measure ζ on R × K. Then a.s. convergence of the integral (12.8.10) to a finite value is enough to ensure that ξ, the ground measure for ζ (set K = K), is also a boundedly finite random measure. If the expectation measure M for N exists, and A is bounded, then con dition (10.1.5a) implies that the integral 0 κ N (A × dκ) converges a.s., and condition (10.1.5b) implies that N (A × ( , ∞)) is a.s. finite, the two together being sufficient to imply the convergence a.s. of (12.8.10).

12.8.


261

Assertion (b) is an easy consequence of the definitions of stationarity, and (c) follows from the following equations, where we first consider the distribution of µ(A) for some bounded set A ∈ BRd :

∞

∞

(D) ξ(A) = Rα 0

=

0

=

∞

κ α−D N (αA × dy) κ N (αA × αD dκ) κ N (A × dκ) = ξ(A),

0

in which the last line follows from the assumed biscale invariance of N . (D) Thus the one-dimensional distributions of ξ are invariant under Rα . For all r>1, similar arguments can be applied to the r-dimensional distributions of ξ(A1 ), . . . , ξ(Ar ) for bounded Borel sets A1 , . . . , Ar . Such fidi distributions (D) are sufficient to determine the distributions of ξ and Rα ξ completely, so it follows that the two random measures must be equal in distribution. Finally, (d) follows on taking expections through the equations expressing invariance of N under shifts in its first component and under biscale invariance, and repeating the argument leading to (12.8.8). The representation implies that, as in the Poisson case of Proposition 12.8.I, the power-law form (12.8.11) is the only possible form for the stationary mark distribution, even though it is unbounded and cannot be normalized to form a probabilitiy distribution. Of course, over any lower threshold κ0 > 0, it can be normalized to form a Pareto distribution with distribution function F (κ | κ0 ) = 1 − (κ/κ0 )−(1/D)

(κ ≥ κ0 ).

Our main interest now is in exhibiting random measures of the above type that are both self-similar and stationary. The stable processes form one such example: constructing additional examples is not a trivial exercise. We attempt this only for processes in one dimension (X = R), where we can specify the model via its conditional intensity function (see Section 7.3 and the more extended discussion in Chapter 14). Because we are concerned with processes with an infinite past, the appropriate version of the conditional intensity is the complete conditional intensity, λ∗ (t, κ), representing the current risk (of a point in [t, t + dt) with mark in [κ, κ + dκ)) given the whole past back to −∞. The next result gives necessary conditions which must be satisfied by the complete conditional intensity if the underlying point process is to be stationary and D-biscale invariant. In the present context it is desirable to reflect the dependence on the past explicitly in the notation, so anticipating what we use in Section 14.7, we therefore write λ∗ (t, κ) = ψt (Nt− , κ),

262


where ψt is a functional of the point process realization N− t− on (−∞, t) and (D) the mark κ. We also use Sα Ht to denote the past at time t of a transformed process where, taking t as the time origin, the times of past events in the original process are inflated by a factor α and their marks by a factor αD . Similarly, we use Tτ Ht to denote the past at time t of a version of the original process shifted through τ . Lemma 12.8.VIII. Let N be an extended MPP with state space R × R+ 0, and complete conditional intensity function λ∗ (t, κ) dt dκ = E[N (dt × dκ) | Ht− ] = ψt (Nt− , κ) dt dκ. (a) If N is stationary then for all real t, ψt (Nt− , κ) = ψ0 (St Nt− , κ) ≡ ψ(St Nt− , κ) is independent of t so that λ∗ (t, κ) = ψ(St Nt− , κ) for all t > 0. (b) If N is also D-biscale invariant, then for all real α > 0, ψ(Sα(D) N0− , αD κ) = α−(1+D) ψ(N0− , κ), and for all t, α > 0, " (D) Ht = ψ(St Nt− , κ) dt dκ = λ∗ (t, κ) dt dκ. E N d(αt) × d(αD κ) " Rα Proof. Conditions (a) and (b) are to be understood as equalities of functionals of the infinite past, suitably adjusted where appropriate. Thus, condition (a) means that if there are previous occurrences at {ti : ti < t}, and the times are shifted so that these become {ti + τ : ti < t}, then the value of the conditional intensity for the shifted process at time t + τ coincides with the value of the conditional intensity for the original process at time t. Now, under the assumptions, the conditional intensities can be expressed in terms of the fidi distributions and vice versa, so the statement is equivalent to equality of the fidi distributions under shifts and hence is a consequence of stationarity. It can be satisfied only if the conditional intensity depends on the past occurrence times through the differences t − ti and not on the absolute values ti . It is a necessary condition (but not sufficient) for the conditional intensity itself to be a stationary process in time. Similarly, to justify condition (b), consider a simultaneous inflation of the time scale (from origin t = 0 back into the past) by a factor α, and of the mark scale by a factor αD . If the underlying process is D-biscale invariant, the conditional intensity at t = 0 for the inflated process must have the same value as the conditional intensity for the original process at t = 0, yielding the condition (b) for t = 0. It implies that, as a function of past events, the 1/D conditional intensity at time t = 0 must be a function of the ratios ti /κi . Because by assumption the underlying process is also stationary, a similar

12.8.


263

condition must hold for all t. Note that although the left-hand side of the last equation in condition (b) refers to a conditional intensity at time αt it cannot be equated with λ∗ (αt) [as wrongly asserted in Vere-Jones (2005)] because the conditioning histories are different. Conditions (a) and (b) are both satisfied when the conditional intensity can be expressed in the form κ (t − ti )D , (12.8.12) , λ∗ (t, κ) = κ−(1+1/D) h κi κi where h is a function of the infinite set of pairs of arguments (κ/κi , (t−ti )D /κi ) involving the times and marks ti , κi of past events (i.e., with ti < t). To understand the form of the arguments of h, note that, in general, the complete conditional intensity should be a function of (t, κ) and the infinite set of pairs (ti , κi ). Setting τ = −t in condition (a) shows that the arguments of h can be reduced to κ and the pairs (t − ti , κi ). Then setting α = 1/κ and using (b) the arguments reduce to the pairs ((t − ti )/κ1/D , κi /κ), which is equivalent to the form in (12.8.12). Note that the initial term κ−(1+1/D) arises from the inflation of the infinitesimal elements dt dκ. Unfortunately, the conditions of the lemma provide no guarantee that in any particular case a process with the proposed form of conditional intensity function exists, or, if it does exist, that it is uniquely specified and inherits the invariance properties of the conditional intensities. To illustrate the latter point, consider a Hawkes process, with conditional intensity as set out in Example 7.2(b), but with criticality constant ν ≥ 1. The proposed conditional intensity satisfies condition (a), but the only corresponding point process is explosive and does not admit any stationary version. In developing a potential model, therefore, it is necessary to check two points: that the proposed conditional intensity satisfies the conditions of the lemma, and that a process with this conditional intensity exists and is both stationary and self-similar. The stable processes correspond to the choice h ≡ const. The next example demonstrates that the proposed class of processes is not limited to the stable processes. Example 12.8(b). Self-similar ETAS model. Recall from Examples 6.4(d) and 7.3(b) that the standard ETAS model has conditional intensity of the form, for M > M0 , p > 0, $ ' p cp ∗ −β(Mi −M0 ) α(M −M0 ) µc + A , e λ (t, M ) = βe (c + t − ti )1+p i:t 0),

12.8.


265

and the offspring from this and all later generations independently follow the same Poisson process relative to their own parent. Using this formulation, we can make use of the general criterion for the existence of a Poisson cluster process at Proposition 6.3.III, namely, the convergence, for each bounded Borel set B ∈ X ≡ R × R+ 0 (the set B should also be bounded away from 0 in K = R+ 0 ), of the integral X

Pr{N (B | x) > 0} µc (dx),

(12.8.18)

where N (· | x) is the cluster member process from a cluster centre at x and µc (·) the expectation measure for the process Nc (·) of cluster centres. In addition to being nonnegative, the essential characteristic of the kernel θ is that it should admit the function ψ(t , κ ) = (κ )−(1+1/D) as the eigenfunction corresponding to a positive (and hence maximum) eigenvalue ρ < 1. To clarify its behaviour in this regard, it is convenient to rewrite θ in the more general form θ(t, κ | t , κ ) = ρ f (t − t , κ ) P (κ, κ )

κ −(1+1/D) κ

,

(12.8.19)

where f is normalized to be a probability density function in u = t − t and P is normalized to be a Markov transition kernel in κ . Straightforward computations show that this is achieved in the present instance by setting −(1+p) t − t p 1+ ) , f (t − t , κ ) = C(κ )1/D C(κ )1/D κ κ *δ/2 δ) P (κ, κ ) = min , , κ κ κ ρ = ηC/δp.

For a branching process with Poisson located offspring, the first generation of offspring from an ancestor at (tc , κc ) follows a Poisson process with intensity θ(t, κ | tc , κc ), the second generation follows a Poisson process with intensity θ(2) (t, κ | tc , κc ) =

X

θ(t, κ | t , κ ) θ(t , κ | tc , κc ) dt dκ ,

and in general the kth generation follows a Poisson process with intensity given by the kth iterate of θ, say θ(k) (· | ·). Then, treating tc and κc as the coordinates of a cluster centre (independent arrival) and writing B for a Borel

266


subset of R × R+ 0 that is both bounded and bounded away from 0 in its second argument, we can estimate the integral by X

Pr{N (B | x) > 0} µc (dx) ≤ ≤

X

k

X

k

=µ

Pr{Nk (B | x) > 0} µc (dx) E[Nk (B | x)] µc (dx)

k

R

R+ 0

θ(k) (B | tc , κc ) ψ(tc , κc ) dtc dκc ,

(k) where (k) Nk (· | x) is the process of kth generation offspring, θ (B | ·) = θ (t, κ | ·) dt dκ, and the intensity µc (dx) of cluster centres has been reB placed by the specific form µψ(tc , κc ) dtc dκc . But because ψ(t, κ) is an eigenfunction, this last sum reduces to

M (B) = µ

∞ k=0

κ−(1+1/D) dt dκ .

ρk B

If ρ < 1 and B is bounded away from 0 on the κ axis, this sum is certainly finite, and then represents the expected number of points falling into B, namely, µ κ−(1+1/D) dt dκ. (12.8.20) M (B) = 1−ρ B The argument shows that, although the number of cluster members is infinite in total, most are very small, and only a finite number fall into a bounded set B bounded away from 0 on the energy axis. A similar statement holds for the overall process. Finally, given that such a cluster process exists, and has the above mean rate, it follows that the series defining λ∗ (t, κ) converges almost surely, and then represents the total risk of an event in dt × dκ, given the locations of all points with ti < t, that is, given the complete history of the process up to time t. The model can be modified and extended in various ways. In particular, the power-law form for the density function f of time-delays is not an inevitable feature of self-similarity. It can just as well be replaced by an exponential (short-tailed) form without affecting either stationarity or self-similarity. An outline is given in Exercise 12.8.4. Thus there is no necessary connection between self-similarity and long-range dependence, as there is in the case of the fractional Brownian motions. On the other hand, self-similarity does imply power-law growth in some sense, as is apparent from the very definition. However, because the moments are infinite, this sense cannot be expressed in terms of the rate of growth of the moment functions.

12.8.


267

A related and important approach to self-similarity for random measures is developed in the papers by Z¨ ahle already quoted. Z¨ ahle suggests basing the property, not on absolute locations relative to the state space X , but on locations relative to a given point of the realization, that is, on the Palm distributions of the process. This allows the treatment of some examples, such as Lévy dust [Example 9.1(g)] which lie outside the more restricted development in the text above. Z¨ ahle gives a general algebraic treatment, and develops many important properties of self-similar random measures, such as the dimensionality of the set of atoms, but he does not examine the probabilistic structure of the atoms from the point of view considered in this section.

Exercises and Complements to Section 12.8 (D)

12.8.1 Show that the renormalization group acts boundedly and Ê Rα (D) Ê continuously −D . [Hint: For suitable f , f (x)R µ(dx) = α f (y/α) µ(dy). on M# α X Now imitate the proof of continuity of the shifts Sx in Lemma 12.1.I.] 12.8.2 Develop a representation for self-similar completely random measures in R2 analogous to that set out in Proposition 12.8.IV for such measures in R. Then consider the simplifications which occur on assuming that the random measure is in addition (a) homogeneous (i.e., stationary with respect to shifts in R2 ), (b) isotropic, or (c) both. [Hint: Consider the effects on the intensity of the Poisson process in the representation (10.1.4).] 12.8.3 Let N be a self-similar extended MPP on Rd × R+ ; transform the mark-scale by setting q = log κ. Show that D-self-similarity is equivalent to requiring the transformed process N ∗ on Rd × R to have fidi distributions that are invariant under the transformations Eα(D) (A × Q) = αA × (SD log α Q). Restate Proposition 12.8.VII and Lemma 12.8.VIII in terms of the transformed process N ∗ . 12.8.4 Investigate in detail the properties of a version of the self-similar ETAS model where the normalized density for the time delay in (12.8.19) has the exponential form f (t − ti ) = cκ ecκ (t−ti ) . In particular check that the form (12.8.12) can be sustained, and that the existence of a stationary process can be established as in Example 12.8(b). 12.8.5 Long-range dependence of self-similar ETAS model. (a) In the model of Example 12.8(b), show that the first- and higher-order moment measures of self-similar random measures do not exist, so that long-range dependence in the sense of Section 12.7 cannot be defined. (b) Investigate conditions for long-range dependence of the process NK restricted to any mark set K bounded away from 0 and ∞.

CHAPTER 13

Palm Theory

13.1 13.2 13.3 13.4

Campbell Measures and Palm Distributions Palm Theory for Stationary Random Measures Interval- and Point-stationarity Marked Point Processes, Ergodic Theorems, and Convergence to Equilibrium 13.5 Cluster Iterates 13.6 Fractal Dimensions

269 284 299 317 334 340

In Section 3.4 we gave a brief introduction to Palm–Khinchin equations and noted that, for a stationary point process on the line, they provide a link between counting and interval properties. In this chapter we study this link both in more detail and in a more general setting. It is a topic that continues to find new applications, both within point process theory itself, and in the applications of that theory to ergodic theory, queueing theory, stochastic geometry and many other fields. Its continuing relevance is linked to the shift of viewpoint that it entails: from an absolute frame of reference outside the process under study, to a frame of reference inside the process (meaning, for a point process, relative to a point of the process). Such a change of viewpoint is usually insightful, and sometimes essential, in seeking an understanding of point process properties. Early contributions by Palm (1943) and Khinchin (1955) have already been noted in Chapter 3. Subsequently, the general theme was taken up by Kaplan (1955), who was influenced by Doob’s (1948) work on renewal processes, and Slivnyak (1962, 1966). This work examined point processes in R with the property of interval-stationarity; successful extension of this idea to point processes in Rd has been much more recent. A critical development in the study of stationary random measures and point processes was the formulation by Kummer and Matthes (1970) of what they called Campbell measure, in essence a refinement of the Radon–Nikodym approach that Ryll-Nardzewski (1961) and Papangelou (1970, 1974a) used earlier. The later evolution of their work can be traced through the three editions—in German, English, and Russian—of Matthes, Kerstan, and Mecke, 268

13.1.

Campbell Measures and Palm Distributions

269

referred to as MKM (1974, 1978, 1982). The relation with ergodic theory (the theory of flows and of flows under a function) was studied by Neveu (1968, 1976), Papangelou (1970), and Delasnerie (1977). Baccelli and Brémaud (1994) exploit the links between material in this chapter and the martingale approach outlined in Chapter 14, while Sigman (1995) and Thorisson (2000) use shift-coupling arguments in an alternative approach to the main limit theorems and their applications in queueing theory and elsewhere. More recently, Thorisson (2000), Tim´ ar (2004), Heveling and Last (2005), and others, have shown how to extend the concept of interval-stationarity for a point process in R to a more general concept of point-stationarity in Rd for d ≥ 2, although it dates back at least to Mecke (1975). The theory has many applications, notably in queueing theory, where work was initiated by K¨ onig and Matthes (1963); see also K¨ onig, Matthes, and Nawrotski (1967) and Franken (1975), Franken et al. (1981), and Brandt, Franken, and Lisek (1990), among many others. Related applications in stochastic geometry are presented in Stoyan and Mecke (1983) and, more profusely, in Stoyan, Kendall, and Mecke (1987, 1995) [SKM (1987, 1995) below]. In our discussion, which has been strongly influenced by MKM (1978), the major emphasis concerns the stationary case. The main results are derived by a factorization of the Campbell measure, which parallels the factorization of the moment measures given in Section 12.6. Indeed, the reduced moment measures reappear in this chapter as multiples of the moment measures of the Palm distribution. The definition of Campbell measure and a brief account of the Radon– Nikodym approach is given in Section 13.1. The main results for stationary random measures are set out in Section 13.2, and Section 13.3 develops the basic relationships between stationarity of the measure, and stationarity relative to points of the process. This includes the interpretation of the Palm distribution as the distribution ‘conditional on a point at the origin,’ the equivalence between stationarity of the point process and stationarity of the intervals for a one-dimensional point process, and its recent extensions to point-stationarity in higher dimensions. Ergodicity and convergence to equilibrium from the Palm distribution are discussed in Section 13.4, which also outlines extensions to MPPs. Section 13.5 gives the discussion of cluster iterates deferred from Chapter 11, and Section 13.6 looks at an interpretation of fractal dimensions in terms of moments of the Palm distribution.

13.1. Campbell Measures and Palm Distributions For any random measure ξ, including possibly a point process, on the c.s.m.s. X , we introduce a measure CP (· × ·) on the product space W ≡ X × M# X by # setting, for A ∈ BX , U ∈ B(M# ), and P the distribution of ξ on M , X X

270

13. Palm Theory

CP (A × U ) = E[ξ(A)IU (ξ)] =

ξ(dx) P(dξ). U

(13.1.1a)

A

It represents a refinement of the first moment measure M (A) = CP (A×M# X ), which results when U is expanded to cover the full space M# . X Write BW for the product Borel σ-field BX ⊗ B(M# X ); that is, BW is generated by all A × U ∈ W with A ∈ BX and U ∈ B(M# X ). The set function CP (·) is clearly countably additive on such product sets but is totally finite if and only if the first moment measure exists and is totally finite. To see that CP (·) is always at least σ-finite, let {Am } (m = 1, 2, . . .) be a sequence of bounded Borel sets covering X , and define Umn = {ξ: ξ(Am ) ≤ n}

(n = 1, 2, . . .).

Then the inequalities CP (Am × Umn ) =

ξ(dx) P(dξ) ≤ nP(Umn ) ≤ n

Am

Umn

imply that CP is certainly finite on each set Am × Umn . These sets cover W, because for any given (x, ξ) ∈ W we can select Am x and then, because any ξ ∈ M# X is a.s. boundedly finite, given Am we can find n such that ξ(Am ) ≤ n, so (x, ξ) ∈ Am × Umn . It then follows that the set function CP extends uniquely to a σ-finite measure on BW . We continue to use CP for this extension. It is also convenient to introduce here the modified Campbell measure1 ! CP (· × ·) which plays an important role in the analysis of spatial point processes in Chapter 15. It is defined much as in (13.1.1a) but specifically for simple point processes N , and with the special feature of excluding the point at the origin which is characteristic of the ordinary Palm distributions. For A ∈ BX and U ∈ B(NX#∗ ) and P the distribution of N on NX#∗ , we write IU N \ x N (dx) , CP! (A × U ) = E

(13.1.1b)

A

where N \ x denotes the realization of N modified by the removal of any point that there may be at location x (sometimes N \x is written loosely as N −δx ). Definition 13.1.I. (a) The Campbell measure CP associated with the random measure ξ on the c.s.m.s. X , having distribution P on M# X , is the unique extension of the set function defined at (13.1.1a) to a σ-finite measure on BW . 1 Kallenberg (1983a, §12.3) uses the term ‘reduced Campbell measure,’ which we avoid here to eliminate any confusion with the term ‘reduced moment measure’ used, for example, onwards from Proposition 12.6.III.

13.1.


271

(b) The modified Campbell measure CP! associated with the simple point process N on the c.s.m.s. X with distribution P on NX#∗ , is the unique extension of the set function defined at (13.1.1b) to a σ-finite measure on BW . For the remainder of this chapter we deal only with the ordinary Campbell measure and associated Palm measures, leaving until Chapter 15 any results we need from the analogous development of modified Campbell and Palm measures (but see Exercise 13.2.7). By following the usual route from indicator functions to simple functions and limits of simple functions, the quantity defined initially at (13.1.1a) extends to the following integral form. Lemma 13.1.II. For BW -measurable functions g(x, ξ) that are either nonnegative or CP -integrable, g(x, ξ) CP (dx × dξ) = E g(x, ξ) ξ(dx) W X (13.1.2) = g(x, ξ) ξ(dx) P(dξ). M# X

X

We have already noted the connection between Campbell measure and the first moment measure which results on setting U = M# X in (13.1.1a); it yields CP (A × M# X ) = E[ξ(A)] = M (A) whenever the first moment measure M (·) exists. The link with Campbell’s theorem noted around (9.5.2) follows most easily from (13.1.2). When M (·) exists, and g is a function of x only, (13.1.2) reduces to g(x) ξ(dx) = g(x) ξ(dx) P(dξ) = g(x) M (dx), E X

X

M# X

X

that is, precisely (9.5.2), of which Campbell’s (1909) original result is the special case for a stationary Poisson process. No doubt it was this link with (9.5.2) that Kummer and Matthes (1970) had in mind in coining the term Campbell measure for the measure CP . Several further comments should be made concerning this definition. As , P) can in Chapter 9, the role of the canonical probability space (M# X , BM# X be replaced by a more general probability space (Ω, E, P) without altering the basic character of the definition, provided only that the probability space is rich enough to support the random measure. In this case the product measurable space W above is replaced by the product measurable space (W ∗ , BW ∗ ) ≡ (Ω × X , E ⊗ BX ), and the defining property (13.1.1a) of the Campbell measure becomes ξ(dx, ω) P(dω) (U ∈ E). CP (A × U ) = E[ξ(A)IU ] = U

A

272

13. Palm Theory

In this chapter, as in the last, we develop the basic theory for the canonical probability space. In Chapter 14, however, the more general definition as above is needed when we consider conditioning on random variables external to the point process itself. A second point to note is that refinements of higher-order moment measures can be defined in a similar way to the Campbell measure itself [see, e.g., Kallenberg (1975, p. 69; 1983a, p. 103)]. For example, a second-order (2) Campbell measure CP can be defined on X (2) × M# X by setting (2)

CP (A × B × U ) = E[ξ(A)ξ(B)IU (ξ)]

(13.1.3)

for A, B ∈ BX , U ∈ B(M# X ). Clearly, the second moment measure M2 (A×B), when it exists, appears as the marginal distribution on integrating out P (see Exercise 13.1.1). Finally, observe that the construction is not restricted to measures P on M# X which have total mass one, but can be carried through for any measure Q on M# X for which (i) Q is σ-finite, and (ii) there exists a suitable family {Am } covering X such that, for all (m, n), Q(Umn ) = Q({ξ: ξ(Am ) ≤ n}) < ∞ [see below (13.1.1a)]. Such a construction, of CQ say, starting from CQ (A × U ) = U A ξ(dx) Q(dξ) with A and U as in (13.1.1a), is important (and always possible) for the associated with an infinitely divisible random measure (see KLM measures Q Exercise 13.1.2). We briefly digress to examine its definition in this more general case, for while it is clear from the construction that Q determines CQ uniquely, the converse is true only if some additional information is given. The situation is summarized in Lemma 13.1.III. A characterization of measures that can appear as Campbell measures, based on Wegmann (1977), is outlined in Exercise 13.1.3. Lemma 13.1.III. When the measure Q on M# X satisfies (i) and (ii) above, the corresponding Campbell measure CQ determines Q uniquely on M# X \{∅}; # it determines Q uniquely on MX if, in particular, either Q is a probability measure, or Q({∅}) = 0. Proof. Using the assumptions, choose a bounded set A within a finite union of the Am . Then ξ(A) is Q-a.e. finite, and setting Vx = {ξ: ξ(A) ≤ x} for arbitrary x > 0, we can define FA (x) = Q(Vx ) < ∞. Clearly FA (·) is the d.f. of a probability distribution when Q is a probability measure, but will not be so in general. Then x y dFA (y) = GA (x) say; CQ (A × Vx ) = 0

13.1.


273

that is, the Campbell measure determines GA (·), a weighted version of FA (·). Furthermore, apart from the value of FA (0+), FA (·) can be recovered from CQ via the relation dFA (x) = x−1 dGA (x). By varying the choice of A we see that CQ determines all the one-dimensional ‘distributions’ of Q. Analogous arguments apply to the multivariate fidi distributions [recall that the basic sample functions ξ(·) are Q-random measures] and show that the Campbell measure determines all the fidi distributions of Q, hence Q itself, up to the value of Q({∅}). This last is clearly determined if Q(M# X ) = 1, or if its value is explicitly prescribed. The key to the introduction of Palm distributions in general is the relation between the Campbell measure CP and the first moment measure M (·). Whenever M (·) exists as a boundedly finite measure, for each fixed U ∈ B(M# X ), CP (· × U ) is then absolutely continuous with respect to M (·). We can thus introduce the Radon–Nikodym derivative as a BX -measurable function Px (U ) satisfying, for each A ∈ BX , Px (U ) M (dx) = CP (A × U ), (13.1.4) A

and Px (U ) is defined uniquely up to values on sets of M -measure zero. Moreover, for each fixed, bounded Borel set A, CP (A × U )/M (A) = CP (A × U )/CP (A × M# X) is a probability measure on M# X . Just as in the discussion of regular conditional probabilities (see Proposition A1.5.III), it follows that the family {Px (U )} can be chosen so that (A) for each fixed U ∈ B(M# X ), Px (U ) is a measurable function of x that is M -integrable on bounded subsets of X ; and (B) for each fixed x ∈ X , Px (U ) is a probability measure on U ∈ B(M# X ). We call each such measure Px (·) a local Palm distribution for ξ, and the family of such measures satisfying (A) and (B) the Palm kernel associated with ξ. Then the discussion above implies the following result, in which (13.1.5) follows from (13.1.4) by the usual extension arguments. Proposition 13.1.IV. Let ξ be a random measure whose first moment measure M exists. Then ξ admits a Palm kernel, that is, a regular family of local Palm distributions {Px (·): x ∈ X } which are defined uniquely up to values on M -null sets, and for all BW -measurable functions g they are either nonnegative or CP -integrable, and satisfy E g(x, ξ) ξ(dx) = g(x, ξ) CP (dx × dξ) = Ex [g(x, ξ)] M (dx), X

X ×M# X

X

(13.1.5)

274

13. Palm Theory

where Ex [g(x, ξ)] =

M# X

g(x, ξ) Px (dξ)

(x ∈ X ).

Note that this proposition holds equally for random measures and point processes, nor does it require ξ to be stationary. When ξ is stationary, the local Palm distributions become translated versions of a single basic distribution, so that (-a.e. x). (13.1.6) Px (Sx U ) = P0 (U ) A more general version of (13.1.6) is given in Section 13.2 where a factorization argument is used and there is no requirement for the existence of first moments. An outline proof in the present setting, using arguments similar to those of Exercise 12.1.9, is sketched in Exercise 13.1.4(a). We turn to illustrate the nature of local Palm distributions, first for a random measure with density. Example 13.1(a) Palm distributions for a random measure with density. Suppose that the random measure ξ on R has trajectories with a.s. continuous locally bounded derivatives dξ(x, ω)/dx = X(x, ω) and that the first moment measure M (·) has a continuous locally bounded density dM (x)/dx = m(x) = E[X(x)]. Here we use ω ∈ Ω for the probability space, rather than ξ ∈ M# X, merely to avoid ambiguity of notation. In (13.1.5) let g(·) run through a sequence of functions of the form gn (x, ω) = hn (x)IU (ω), where U ∈ B(Ω) is fixed and {hn (x)} is a sequence of functions converging to δx0 for some x0 ∈ R, with m and X continuous at x0 . From (13.1.5) we obtain P(dω) hn (x)X(x, ω) dx = m(x)hn (x) Px (U ) dx. U

X

X

Using the a.s. continuity and local boundedness, the left-hand side converges as n → ∞ to X(x0 , ω) P(dω), U

and the right-hand side converges to m(x0 ) Px0 (U ), where these functions are continuous in x at x0 by assumption, so 1 X(x0 , ω) P(dω). (13.1.7) Px0 (U ) = m(x0 ) U Thus, the measure Px0 (·) appears as a reweighted version of the probability measure P, the weight for the particular realization ξ(ω) being taken as proportional to the value of the density X(x0 , ω) at the chosen point x0 . Alternatively, if Y is any random variable defined on the process, and Ex0 (·) denotes expectations with respect to the Palm distribution at x0 , (13.1.7) is equivalent to E[X(x0 )Y ] . Ex0 (Y ) = E[X(x0 )]

13.1.


275

Some condition such as the assumed continuity in x at x0 of Px (U ) is essential, as can be shown by counterexamples where the right- and leftlimits at x0 of Px (U ) exist but are different [see Leadbetter (1972) for the point process context]. Greater interest attaches to random measures that are a.s. purely atomic. In this case the Campbell measure inherits a singular structure from ξ, in that its support is restricted to the subset of X × M# X defined by # U = {(x, ξ) ∈ X × M# X : ξ ∈ MX , ξ({x}) > 0}.

(13.1.8)

For point processes the support is further restricted to the subset of X × NX# defined by V = {(x, N ) ∈ X × NX# : N ∈ NX# , N ({x}) ≥ 1}.

(13.1.9)

The relevant properties of U and V are summarized in the next proposition. Proposition 13.1.V. Let U and V be defined by (13.1.8–9). # (i) U is a Borel subset of X × M# X and V is a Borel subset of X × NX . (ii) A random measure ξ with distribution P on M# X is purely atomic if and only its Campbell measure CP satisfies CP (U c ) = 0. (iii) ξ is a point process if and only if CP (V c ) = 0. Proof. Consider for each n > 0 a partition of X into a countable family Tn of Borel subsets {Anm : m = 1, 2, . . .} of diameter ≤ n−1 , and set Vnmk = {(x, ξ): x ∈ Anm , ξ(Anm ) ≥ 1/k}. Clearly, Vnmk ∈ BX ⊗ B(M# X ). We assert that U=

∞ ∞ ∞ (

Vnmk ,

(13.1.10)

k=1 n=1 m=1

implying inter alia that U is a measurable subset of X × M# X. To justify (13.1.10), consider any (x, ξ) ∈ U. Because ξ({x}) > 0 there exists k such that ξ({x}) ≥ 1/k . For each n, x belongs to just one element of Tn , Anm say, for which ξ(Anm ) ≥ ξ({x}) ≥ 1/k . Hence (x, ξ) ∈ Vnm k . Thus every element in U is an element in theright-hand side of (13.1.10). Conversely, for fixed k, suppose (x, ξ) ∈ n m Vnmk , and for each n let An (x) denote the unique Anm containing x. Then ξ({x}) = lim ξ An (x) ≥ 1/k, n→∞

implying that (x, ξ) ∈ U, so (13.1.10) holds as asserted. Part (i) is shown. To check (ii), let Tn = {Anm : m = 1, . . . , rn } now be a dissecting system for an arbitrary bounded A ∈ BX , and let UA denote the analogue of U at

276

13. Palm Theory

(13.1.10) with these redefined {Anm }. Use the representation of ξ in terms of the MPP Nξ to write, for such bounded A, κ N (dx × dκ) < ∞, (13.1.11) M (A) = EP [ξ(A)] = EP ∗ A×(0,∞)

where expectations are written with respect to the measures P for ξ and P ∗ for N under the one-to-one measurable mapping ξ ↔ Nξ [cf. Proposition 9.1.V(v)]. Set 1 ∗ . κ N (dx × dκ) ≥ Vnmk = (x, N ): k Anm ×(1/k, ∞) ∞ ∗ c We assert that for each n and k, CP ∗ vanishes. Indeed, m=1 (Vnmk ) CP ∗

( ∞

∗ (Vnmk )c

= EP ∗

m=1

and the right-hand side equals ∞ EP ∗ m=1

∞

∗ [1 − IVnmk (x, N )] κ N (dx × dκ) ,

X ×(0,∞) m=1

Anm ×(0,∞)

∗ [1 − IVnmk (x, N )] κ N (dx × dκ) ,

because for x ∈ Anm , only the term involving Anm can contribute a term different from unity to the infinite product, so that the integral of the product re∗ ∗ , IVnmk (x, N )= 1, duces to the sum of integrals as shown. But if (x, N ) ∈ Vnmk so the integrand in that term, and hence its integral, vanishes. If, alterna∗ tively, (x, N ) ∈ / Vnmk , then there are no atoms in Anm of mass > 1/k [i.e., N Anm × (1/k, ∞) = 0], so again the integral vanishes. Thus all terms in the sum vanish, and our assertion is justified. Let

∗ = (x, N ): x ∈ A, N {x} × (1/k, ∞) = 1 . UA,k Then, as for (13.1.6), we can show that ∗ = UA,k

rn (

∗ Vnmk ,

n m=1

and it follows from the assertion just proved that ( rn ∗ c c ∗ CP ∗ (UA,k ) = CP ∗ (Vnmk ) = 0. n m=1 ∗ DefineUA analogously to UA but in terms of the counting measures, that ∞ ∞ rn ∗ ∗ , and consider the difference is, UA = k=1 n=1 m=1 Vnmk ∗ ∗ CP ∗ (UA ) − CP ∗ (UA,k ) = EP ∗ κ IUA∗ (x, N ) N (dx × dκ) . A×(0,1/k]

13.1.


277

Letting k → ∞, the difference to zero by (13.1.7) ∗ cconverges and dominated ) = 0, equivalently, CP (UA )c = 0. convergence, so that CP ∗ (UA Because the space X is separable, we it by a countable family of can cover bounded sets Ai on each of which CP (UAi )c = 0. Also, because U c = {(x, ξ) : ξ({x}) = 0} =

c (x, ξ): x ∈ Ai , ξ({x}) = 0 = UA i ,

i

i

assertion (ii) in the Proposition now follows. Assertion (iii) follows by a similar (and simpler (!)) argument applied to the space of counting measures. The proposition above shows that for random measures that are a.s. purely atomic, each local Palm measure Px inherits the singular structure of the Campbell measure, in the form of an atom at the point x selected as a local origin. In the point process case, this leads directly to the interpretation of the Palm distribution Px as a distribution conditional on the occurrence of a point at x. For a direct approach to this definition, under some assumptions, see Exercise 13.1.9. The stationary case is discussed in detail in Section 13.3. More generally, for a purely atomic random measure, it follows from the proposition that the only contributions to the local Palm distribution Px come from realizations of ξ which have an atom at x. The relationships are most easily explored in the case of a random measure with only finitely many atoms, or equivalently, a finite point process with positive marks, as in the next example. The special case of a nonsimple point process is illustrated in Exercise 13.1.6; the general relationship is sketched in Exercise 13.1.7. Example 13.1(b) Purely atomic random measure with finitely many atoms. On a state space X we suppose given a random measure ξ that is purely atomic and which a.s. has only finitely many atoms, so ξ(X ) < ∞ and we also assume that M (X ) = E[ξ(X )] < ∞.Denote P{ξ has n atoms} = pn for some npn = µ denoting the mean number of probability distribution {pn } with atoms. Let {xi : i = 1 . . . , n} denote the locations of the atoms, and {κi } their masses, supposing that the locations are i.i.d. with distribution F (·), and that the masses are independently distributed with distributions Π(· | x) conditional on the locations. We can thus describe a realization ξ by means of the subset {y1 , . . . , yn } for some finite integer n and the pairs yi = (xi , κi ) that are i.i.d. on X × R+ 0 with distribution Ψ(dy) = Ψ(d(x, κ)) = Π(dκ | x) F (dx). With this notation and recalling (9.1.4) and Proposition 9.1.III(v), the process can alsobe identified as an MPP N on X with positive marks, and for A ∈ BX , ξ(A) = A ξ(dx) = xi ∈A κi . To identify the Palm kernel for ξ, consider first the left-hand side of the defining equation (13.1.5), taking the function g(x, ξ) there to be of product form α(x)h(ξ), where h on Y ∪ is defined piecewise as hn (y1 , . . . , yn ) on Y (n) where it is symmetric in the indices (1, . . . , n) for positive n; h can be arbitrary

278

13. Palm Theory

for n = 0 because the integral vanishes when ξ(X ) = 0. Then n

∞ E g(x, ξ) ξ(dx) = pn X

Y (n)

n=1

n

κj α(xj )hn (y1 , . . . , yn )

j=1

Ψ(dyi ).

i=1

Because hn and the joint distribution are symmetric, this can be rewritten as µ Y

κα(x) Π(dκ | x) F (dx)

n−1

Y (n−1)

n−1

κj hn−1 (y1 , . . . , yn−1 )

j=1

Ψ(dyi ),

i=1

∞ where µ = n=1 npn denotes the mean number of atoms. Introducing the mean atomic mass m(x) = R+ κ Π(dκ | x), the first moment measure is then 0 given by M (dx) = µ m(x) F (dx). Inspecting the right-hand side of (13.1.5), define for x ∈ X , κ ∈ R+ 0 , n ∈ Z+ , ! ! p∗n = npn µ and Π∗ (dκ | x) = κ Π(dκ | x) m(x), and identify Ex [g(x, ξ)] as α(x)

∞ n=1

p∗n

R+ 0

∗

Π (dκ | x)

Y (n−1)

n−1

κj hn−1 (y1 , . . . , yn−1 )

j=1

n−1

Ψ(dyi ).

i=1

We can now see that (13.1.5) is indeed satisfied if we take for Px (·) on Y (n) for n = 1, 2, . . . the symmetrized version of the measure Pn∗ (x; dy1 × · · · × dyn ) = p∗n δx (dx1 ) Π∗ (dκ1 | x1 )

n

Ψ(dyi ).

(13.1.12)

i=2

In comparison with the corresponding component of the original measure, the above component is ‘tilted’ in two respects: the distributions of the realizations are weighted by the number of points they contain (a realization with n points has n different possibilities of locating a point at a given origin), and the distribution of the mass of the atom at x is weighted by the mass of original atom at x. Exercise 13.1.7 extends this example to a more general setting. Palm distributions for an MPP are best introduced by treating the MPP as a point process on X × K. Each local Palm distribution P(x,κ) then represents the behaviour of the process given the occurrence of a point at x with mark κ. The relation of the local Palm distributions to the Palm distribution of the ground process which we assume has finite first moment measure, can be deduced from the following representation, where the function g(·, ·) of (13.1.5) is taken to have the special form g(x, N ) for x ∈ X and N ∈ NX#×K :

13.1.


E

g(x, N ) N d(x, κ) =

X ×K

X ×K

=

X ×K

X

M g (dx)

= X

# NX ×K

M g (dx) Π(dκ | x)

M g (dx)

=

M d(x, κ)

279

# NX ×K

g(x, N ) P(x,κ) (dN )

g(x, N ) P(x,κ) (dN )

# NX ×K

# NX ×K

g(x, N ) K

Π(dκ | x) P(x,κ) (dN )

g(x, N ) P x (dN ).

(13.1.13)

In this chain of relations, M g (·) = M (· × K) is the first-moment measure for the ground process, and we have used the disintegration M (d(x, κ)) = M g (dx) Π(dκ | x). The measure Px (·) defined at the last step can be interpreted as an ‘average’ local Palm distribution representing the behaviour of the process given the occurrence of a point at x with unspecified mark. The ground measure of the Palm distribution can now be obtained as the projection of Px (·), which lives on the Borel sets of NX#×K , onto the Borel sets of g NX# . Indeed, denoting this projection by Px and taking g(x, N ) in (13.1.13) to be a function g ∗ (x, Ng ) of x and Ng only, (13.1.13) reduces to g ∗ g g (x, Ng ) Ng (dx) = M (dx) g ∗ (x, Ng ) Px . E X

# NX

X

Exercise 13.1.10 gives some further details. To illustrate this situation, suppose in Example 13.1(b) above, we treat the process not as a random measure ξ but as an MPP. Then the representation (13.1.12) for the local Palm distribution of ξ should be replaced by Pn∗ (y; dy1 × · · · × dyn ) = p∗n δy (y1 )

n

Ψ(dyi )

(13.1.14a)

i=2

for the local Palm distribution for the MPP, and for its ground process, g

P n (x; dx1 × · · · × dxn ) = p∗n δx (x1 )

n

F (dxi ).

(13.1.14b)

i=2

We conclude this section with a characterization of Palm distributions via Laplace functionals, and give some results that can be deduced via this characterization. Much as at (9.4.18), write L[f ], where f ∈ BM+ (X ), for the Laplace functional of a random measure with distribution P, and {Lx [f ]} for the family of Laplace functionals derived from the associated Palm kernel at (13.1.5), so that for x ∈ X and f ∈ BM+ (X ), exp − X f (y) ξ(dy) Px (dξ). (13.1.15) Lx [f ] = M# X

280

13. Palm Theory

Proposition 13.1.VI. Let ξ be a random measure with finite first moment measure M and L[f ], Lx [f ] the Laplace functionals associated with the original random measure and its Palm kernel, respectively. Then the functionals L[f ] and Lx [f ] satisfy the relation, for f, g ∈ BM+ (X ), L[f ] − L[f + εg] = g(x)Lx [f ] M (dx). (13.1.16) lim ε↓0 ε X Conversely, if a family {Lx [f ]} satisfies (13.1.16) for all f, g ∈ BM+ (X ) and some random measure ξ with Laplace functional L[·] and first moment measure M (·), then the functionals {Lx [f ]} coincide M -a.e. with the Laplace functionals of the Palm kernel associated with ξ. Proof. Because the first moment measure exists, a finite Taylor expansion (see Exercise 9.5.8) for ε > 0 and f, g ∈ BM+ (X ) yields L[f + εg] = L[f ] − ε E g(x) exp − X f (y) ξ(dy) ξ(dx) + o(ε). X

(13.1.17)

From (13.1.6), g(x) exp − X f (y) ξ(dy) ξ(dx) E X = g(x) M (dx)

X

= X

M# X

exp − X f (y)ξ(dy) Px (dξ)

g(x)Lx [f ] M (dx);

substitution into (13.1.17) followed by rearrangement leads to (13.1.16). To prove the converse, suppose that (13.1.16) holds for the family of func˜ x [f ]}; because (13.1.16) holds for all g ∈ BM+ (X ), the measures tions {L ˜ x [f ]M (dx) coincide, so Lx and L ˜ x agree for M -a.e. x. Lx [f ]M (dx) and L The relation at (13.1.16) is useful in identifying the form of the Palm kernel in some simple cases. Example 13.1(c) The Palm kernel for a Poisson process. For a Poisson process with parameter measure µ(·), we have from below (9.4.18) that log L[f ] = − (1 − e−f (x) ) µ(dx). X

Then

d dL[f + εg] −f (x)−εg(x) = −L[f + εg] (1 − e ) µ(dx) dε dε X −f (x)−εg(x) g(x)e µ(dx) = −L[f + εg] X → −L[f ] g(x)e−f (x) µ(dx) (ε → 0). X

13.1.


281

This can be put in the form of (13.1.16) via the identification M (·) = µ(·), and (13.1.18) Lx [f ] = e−f (x) L[f ] = Lδx [f ] L[f ], where on the right-hand side, Lδx [·] denotes the Laplace functional of the degenerate random measure with an atom of unit mass at x and no other mass. The interpretation of (13.1.18) is that the local Palm distribution Px (·) coincides with the distribution of the original process except for the addition of an extra point at x itself to each trajectory. By regarding the local Palm distribution as being conditional on a point at x, the independence properties of the Poisson process then imply that, apart from a given point at x, the probability structure of the conditional process is identical to that of the original process. The relation embodied in (13.1.18) can also be written in the form Px = P ∗ δx ,

(13.1.19)

where δx denotes a degenerate random measure as in (13.1.18). Equation (13.1.19) is the focus of a characterization of the Poisson process. Proposition 13.1.VII [Slivnyak (1962); Mecke (1967)]. The distribution of a random measure with finite first moment measure satisfies the functional relation (13.1.19) if and only if the random measure is a Poisson process. Proof. The necessity of (13.1.19) has been shown above. For the converse, suppose that Px satisfies (13.1.19). Then from (13.1.16) we obtain dL[εf ] = −L[εf ] f (x)e−εf (x) M (dx), dε X where M is the first moment measure, assumed to exist. Using log L[0] = log 1 = 0, 1 f (x)e−εf (x) dε M (dx) = (1 − e−f (x) ) M (dx), − log L[f ] = X

X

0

so that L[f ] is the Laplace functional of a Poisson process with parameter measure equal to M (·).

Exercises and Complements to Section 13.1 ! 13.1.1 (a) Check that modified Campbell measure CP (·) has a unique extension ). from (13.1.1b) to sets in BX ⊗ B(M# X (2)

(b) Show that the set function CP (·) defined at (13.1.3) has a unique extension to a σ-finite measure on X × X × M# X . When the second moment measure exists, define a second-order family of local Palm distributions (2) Px,y (U ) satisfying (2)

A×B

(2) Px,y (U ) M2 (dx × dy) = CP (A × B × U ).

282

13. Palm Theory

13.1.2 Use the analogue of (13.1.1) to define the Campbell measure CQ (·) of the KLM measure Q, using (10.2.8) to establish the σ-finiteness property of CQ (·)

on appropriate subsets of X × NX# . Appeal to Lemma 13.1.III to establish that CQ (·) determines Q uniquely.

13.1.3 A measure C(·) on X × M# X is the Campbell measure CP of some random measure ξ with σ-finite first moment measure if and only if the following three conditions hold: # (i) C(A × MX ) < ∞ for bounded A ∈ BX ; (ii) X ×M# g(x, η) C(dx × dη) = 0 whenever X g(x, η) η(dx) = 0 for each X

η ∈ M# X ; and (iii) 1 − φA ≡ A×{η:η(A)>0} [η(A)]−1 C(dx × dη) ≤ 1 for bounded A ∈ BX . When these conditions hold, inf A φA = P{ξ = ∅}. [Hint: For the converse, define a measure P on M# X by

#

MX

f (η) P(dη) =

X

#

MX

k(x, S−x η) f (S−x η) C(dx × dη),

where k(·) satisfies (13.2.9); then verify that C and CP coincide. See Wegmann (1977) for details.] 13.1.4 (a) Use arguments analogous to those of Exercise 12.1.9 to show that (13.1.6) holds when the process N is stationary and has boundedly finite first moment measure. (2) (b) What can be said about the local second-order Palm measure Px,y (·) when the process is stationary? [Hint: Use Exercise 12.1.8 much as in Exercise 12.1.9.] 13.1.5 Let ξ be a random measure supported by Z = {0, ±1, . . .} and with finite first moment measure. Describe its Palm and Campbell measures [see (13.1.4)]. When ξ is a simple point process, reinterpret the Palm measure as a conditional distribution. 13.1.6 Discuss two possible interpretations for the local Palm distributions for a nonsimple point process, the first based on the occurrence of a point at x, and the second based on the occurrence of a point of given multiplicity at x. Show that the first can be represented as an average of the second. + # 13.1.7 Let Ψ: M# X → N (X × R0 ) denote the mapping of a purely atomic boundedly finite random measure ξ on X into an integer-valued extended random measure Nξ on X × R+ 0 as in Proposition 9.1.III(v). + ∗ # (a) Show that a measure P on M# X induces a measure P on N (X × R0 ) and conversely. (b) Let CP and CP ∗ denote the corresponding Campbell measures. Show that

#

X ×MX

g(x, ξ) CP (dx × dξ) = EP

g(x, ξ) ξ(dx) X

= EP ∗

X ×R+ 0

κ g(x, Ψ(N )) N (dx × dκ) .

13.1.


283

(c) Writing h(x, κ, N ) = κ g(x, Ψ(N )) as in part (b), and Y = X × R+ 0 , show that EP ∗

X ×R+ 0

=

h(x, κ, N ) N (dx × dκ)

#

Y×NY

=

Y

h(x, κ, N ) CP ∗ (dx × dκ × dN )

M ∗ (dx × dκ) E(x,κ) [h(x, κ, N )],

where M ∗ is the first moment measure of the extended MPP on Y and

E(x,κ) [h(x, N )] =

#

NY

h(x, N ) P(x,κ) (dN ),

where the Palm kernel P(x,κ) (·) is conditioned both on the location and the mass of a given atom. Simplifications of the relations above depend on particular features in the model as in Example 13.1(b) and (13.1.15). 13.1.8 Show that, although the arguments leading to Proposition 13.1.V can fail when CP (A × M# X ) = M (A) = ∞ for some bounded A ∈ BX , they can be recovered by introducing a nonnegative function h(·) such that

#

h(x, ξ) CP (dx × dξ) < ∞

(bounded A ∈ BX ),

A×MX

and defining a modified Campbell measure H(dx×dξ) = h(x, ξ) CP (dx×dξ). [Hint: Apply the arguments leading to Proposition 13.1.V to H, and use the results for H to derive corresponding results for CP itself. A possible function for h is IB (x)/[1 + ξ(B)] for some fixed bounded B ∈ BX . Similar arguments occur in the discussion following (13.2.4).] 13.1.9 Establish conditions for a limit interpretation of the local Palm distributions Px as conditional distributions, given the occurrence of a point in a neighbourhood of x. [Hint: As in Example 13.1(a), consider the behaviour of functions of the form gn (x, ω) = hn (x)IU (ω), where hn (x) is a δ-sequence. First take U to be the event that a point occurs within a neighbourhood of x, then a compound event incorporating the occurrence of a point near x with additional conditions. Find appropriate continuity conditions for the ratio to converge. The stationary case is discussed in detail in Section 13.3; see also Leadbetter (1972).] 13.1.10 Local Palm measures for an MPP and its ground process. Let N ≡ {(xi , κi )} be an element of NX#×K . Denote by ψg the projection of N onto the corresponding realization Ng ≡ {xi } of its ground process which has measure Pg [for this projection, regard N as the pair (Ng , S) where S is the ordered sequence of marks {κi } from the space K∞ of such sequences]. (a) Verify that every probability measure P on B(NX#×K ) admits the disintegration P(dN ) = Pg (dNg ) ν(dS | Ng ), where ν is a regular probability kernel on the product space NX# × K∞ .

284

13. Palm Theory (b) Use this kernel to express formally the link between the Palm measures P0 and P0g of an MPP and its ground process respectively, namely, #

NX ×K

g(x, κ, N ) P0 (dN ) =

#

NX

P0g (dNg )

−1 ψg (Ng )

g(x, κ, Ng , S) ν(dS | Ng ).

g

(c) Check that the measure Px introduced below (13.1.13) is the marginal distribution, in the above sense, corresponding to the measure Px . 13.1.11 Higher-order versions of local Palm measure. (a) Let the random measure ξ have boundedly finite second-order moment measure M2 . Show that for any bounded measurable function g(x, y, ξ), (2) the second-order kernel Px,y (·) on M# X , which exists by Exercise 13.1.1, satisfies #

MX

P(dξ)

X (2)

g(x, y, ξ) ξ(dx) ξ(dy) = X (2)

M2 (dx × dy)

#

MX

(2) g(x, y, ξ) Px,y (dξ).

(b) Show that when ξ is stationary, there exists a one-parameter kernel (2) P˘x (·) such that (2) (2) Px,y (·) = S−x P˘y−x . (2) (2) (c) Evaluate explicitly the forms of the kernels Px,y (·) and P˘u (·) for the simple i.i.d. model of Example 13.1(b).

13.2. Palm Theory for Stationary Random Measures Throughout this section we suppose that X = Rd and that the random measure ξ is stationary (Definition 12.1.II), or, equivalently, that its distribution P is invariant under the transformation ξ → Su ξ (all u ∈ X ). Recall also the notation S+u P in (12.1.3) for the transformation on distributions induced by Su : B ∈ B(M# (S+u P)(B) = P(Su B) X) . In fact, the only properties of Rd that are critical for the discussion are the existence and uniqueness of the invariant measure, various standard results such is determined by its integrals f (x) µ(dx) for as that any measure µ ∈ M# X X bounded nonnegative measurable functions f , and the factorization Lemma A2.7.II. Thus, the results hold equally when X is the circle group S, or more generally whenever X is an Abelian σ-group as defined in Section A2.7. Explicit treatments from this more general point of view are given by Mecke (1967) and in MKM (1982). The results below are stated for general random measures, but most of the illustrations concern simple point processes. Extensions to stationary MPPs are given at the beginning of Section 13.4. We start by investigating the effect of stationarity on the Campbell measure, again using W to denote the product space X × M# X.

13.2.

Palm Theory for Stationary Random Measures

285

Proposition 13.2.I. Let ξ be a random measure on X = Rd with distribution P and CP the associated Campbell measure on W. Then ξ is stationary if and only if CP is invariant under the group of transformations Θu : W → W defined for each u ∈ X by (13.2.1a) Θu (x, ξ) = (x − u, Su ξ). The interpretation of (13.2.1) is that, if we shift the origin to u, the Campbell measure of the shifted process relative to u is the same as the Campbell measure of the original process relative to 0. Note also that by + u for the mapping on measures on W induced by Θu , the assertion writing Θ of the proposition is expressed more succinctly as + u CP (all u). (13.2.1b) if and only if CP = Θ P = S+u P (all u) Proof. When CP is the Campbell measure associated with P, the Campbell measure associated with S+u P is to be found from [cf. (13.1.2)] g(x, ξ) ξ(dx) P(Su dξ) = g(x, ξ) Su ξ(dx − u) P(Su dξ) X

M# X

X

= =

M# X

X ×M# X X ×M# X

g(x, ξ) CP Θu (dx × dξ) + u C (dx × dξ) g(x, ξ) Θ P

+ u C . Consequently, if P and S+u P coincide, so do C and is thus equal to Θ P P + and Θu CP . + u C coincide, then it follows from Lemma 13.1.III Conversely, if CP and Θ P that the probabilities P and S+u P from which they are derived coincide. An alternative criterion, due to Mecke (1975), can be deduced as a consequence of the above result. Proposition 13.2.II. P as in Proposition 13.2.I is stationary if and only if its associated Campbell measure CP satisfies g(x, y, Sy ξ) CP (dy × dξ) dx X

X ×M# X

= X

X ×M# X

g(y, x, Sy ξ) CP (dy × dξ) dx

(13.2.2)

for every nonnegative BX ⊗ BX ⊗ B(M# X )-measurable function g(·). + u on a measure Ψ on BX ⊗ B(M# ) Proof. The action of the mappings Θ X can be represented by requiring the equations + h(y, ξ) Θu Ψ(dy × dξ) = h(Θ−u (y, ξ)) Ψ(dy × dξ) X ×M# X

X ×M# X

=

X ×M# X

h(y + u, S−u ξ) Ψ(dy × dξ)

(13.2.3)

286

13. Palm Theory

to hold for any measurable nonnegative h. P being stationary implies by Proposition 13.2.I that CP on X × M# X is stationary. Put Ψ = CP in (13.2.3), set h(y, ξ) = g(x, y, Sy ξ) and u = −x, and then integrate with respect to x over X ; this yields g(x, y, Sy ξ) CP (dy × dξ) dx X

X ×M# X

= X

X ×M# X

g(x, y − x, Sy ξ) CP (dy × dξ) dx.

But because dx is the invariant measure on X , integration over the whole of X for any nonnegative measurable function m(·, ·) satisfies m(x, y − x) dx = m(y − x, x) dx. X

X

Applying this and Fubini’s theorem reduces the right-hand side of the previous equation to X

X ×M# X

g(y − x, x, Sy ξ) CP (dy × dξ) dx.

Then reversing the operation with Θu gives the right-hand side of (13.2.2). Conversely, suppose that (13.2.2) holds; let j(x) be a measurable nonnegative function satisfying X j(x) dx = 1. Apply (13.2.3) with Ψ = CP to general nonnegative h, multiply by j, and integrate over x; this yields + u CP (dy × dξ) h(y, ξ) Θ X ×M# X

= X

X ×M# X

X

X ×M# X

X

X ×M# X

X

X ×M# X

j(x) h(y + u, S−(u+y) Sy ξ) CP (dy × dξ) dx

=

j(y) h(x + u, S−(u+x) Sy ξ) CP (dy × dξ) dx

by (13.2.2),

=

j(y) h(x, S−x Sy ξ) CP (dy × dξ) dx

by invariance of x,

= =

X ×M# X

j(x) h(y, ξ) CP (dy × dξ) dx

h(y, ξ) CP (dy × dξ)

by (13.2.2),

on integrating out x.

We now look for a product representation of W to which Lemma A2.7.II can be applied. Consider the transformation D: W → W for which D(x, ψ) = (x, S−x ψ) for (x, ψ) ∈ W. Much as in Exercise 12.1.1, this transformation is continuous and hence measurable. It is also one-to-one and onto, because the inverse mapping D−1 has D−1 (x, ξ) = (x, Sx ξ). Observe that Θu D(x, ψ) = Θu (x, S−x ψ) = (x − u, Su−x ψ) = D(x − u, ψ),

13.2.


287

so that D provides a representation of W under the actions of the group Θu . is invariant under denoting the image of C under D, C Thus, with C P P P shifts in its first argument. It may appear now that Lemma A2.7.II should be applicable and thereby . However, there is a technical diffiyield the required decomposition of C P culty: it is not obvious in general (and may not be true if the first moments takes finite values on products of bounded subsets of W. are infinite) that C P to construct a To overcome this difficulty, we use the σ-finiteness of C P modified measure, with the same invariance properties, to which we can apply implies the existence of a Proposition 13.2.I. Indeed, the σ-finiteness of C P strictly positive function h(x, ψ) such that, provided P({∅}) < 1, P (dx × dψ) < ∞ h(x, ψ) C (13.2.4) 0< X ×M# X

(see, e.g., the constructions in Exercises 13.1.8 and 13.2.3). Define α(ψ) = h(x, ψ) dx = h(x+y, ψ) dy (all x ∈ X ), so that α(ψ) > 0, and let g(·) be X X any nonnegative Lebesgue integrable function on X . By using the invariance we obtain properties of C P P (dx × dψ) = P (dx × dψ) g(x)α(ψ) C g(x)h(x + y, ψ) dy C X ×M# X

X ×M# X

=

X ×M# X

=

X

X

X

(du × dψ) g(u − y)h(u, ψ) dy C P

g(u − y) dy

< ∞.

X ×M# X

P (du × dψ) h(u, ψ) C

The finiteness of the integral on the left-hand side for all such integrable g (dx × dψ) takes finite values on shows that the modified measure α(ψ) C P products of bounded sets, and indeed on sets A × M# X for bounded A. Inasmuch as the presence of the multiplier α(ψ) does not affect invariance of the measure with respect to shifts in x, the factorization Lemma A2.7.II can still be applied and yields P (dx × dψ) = α C˘P (dψ) (dx) α(ψ) C for some uniquely defined finite measure α C˘P on B(M# X ). We now define the Palm measure being sought by setting ! C˘P (dψ) = α C˘P (dψ) α(ψ). This measure is still σ-finite and satisfies the defining relation P (dx × dψ) = C˘P (dψ) (dx). C Its properties are spelled out in more detail in Theorem 13.2.III where the relations (13.2.5–6) are referred to as the refined Campbell theorem.

288

13. Palm Theory

Theorem 13.2.III. (a) Let ξ be a stationary random measure on X = Rd . Then there exists a unique σ-finite measure C˘P on M# X such that for any BX ⊗ B(M# )-measurable nonnegative, or C -integrable, function g(·), P X g(x, ξ) ξ(dx) = g(x, ξ) CP (dx × dξ) E X

X ×M# X

=

dx X

M# X

equivalently, E g(x, Sx ξ) ξ(dx) = dx X

X

M# X

g(x, S−x ψ) C˘P (dψ), (13.2.5)

g(x, ψ) C˘P (dψ).

(13.2.6)

(b) The measure C˘P is totally finite if and only if ξ has finite mean density m, in which case m = C˘P (M# the measure m−1 C˘P (·) coincides with X ) and the regular local Palm distribution Px S−x (·) for -a.e. x. Proof. Equations (13.2.5–6) are applications to the present setting of the factorization theorem, specifically, of (A2.7.5). The σ-finiteness of C˘P follows from the construction preceding the theorem. To prove part (b), set g(x, ξ) = IA (x) for some bounded A ∈ BX . Then (13.2.5) yields C˘ (dψ), M (A) = E[ξ(A)] = (A) M# X

P

so that if M (A) < ∞, C˘P must be totally finite with C˘P (M# X ) = M (A)/(A) # ˘ = m. Conversely, if CP (MX ) < ∞, and equal to m say, then M (A) = m (A) < ∞. Furthermore, setting g(x, ξ) = f (x)h(Sx ξ) in (13.1.5) and (13.2.5) above yields f (x)h(Sx ξ) Px (dξ) dx = f (x) dx h(ψ) C˘P (dψ) m X

M# X

X

M# X

for all measurable Lebesgue-integrable f , so for all C˘P -integrable h it follows that h(Sx ξ) Px (dξ) = h(ψ) C˘P (dψ) (-a.e. x), m M# X

M# X

which in turn implies that mPx (S−x dξ) and C˘P (dψ) must coincide for -a.e. x as measures. Definition 13.2.IV. The Palm measure associated with the stationary random measure ξ on Rd or its distribution P is the measure C˘P (·) on B(M# X) defined by (13.2.5); when the mean density m is finite, the probability measure P0 (·) = m−1 C˘P (·) is the Palm distribution for ξ. Remark. In the point process case, it follows from Proposition 13.1.V that the measure C˘P is supported by the subspace of counting measures N on Rd

13.2.


289

for which N {0} ≥ 1; we denote this space N0# (Rd ) in general, and N0#∗ when the counting measures are also simple with the space Rd omitted when the dimension d is clear from the context. Thus, a generic element N0 ∈ N0#∗ is boundedly finite, simple, and has N {0} = 1. If there is danger of ambiguity, we may use the phrase ‘stationary Palm measure’ (or ‘distribution’) to distinguish P0 (·) from the local versions described onwards from (13.1.4) in the last section. As there, we can gain insight into the character and interpretation of the Palm measure by looking first at a finite point process. Example 13.2(a). Stationary point process on S [see Example 12.1(f)]. A stationary point process on S has probability distributions that are invariant under rotation, and we showed in Example 12.1(f) that this implies that the symmetrized probability measures Πn describing such a process have re˘ n satisfying (12.1.17); they describe the positions, relative to duced forms Π an arbitrarily selected initial point of the realization, of the other n−1 points. It is obvious that the Palm measure for the process should be closely related ˘ n . The diagonal decompositions summarized to the reduced distributions Π by (12.1.17) for a given value of n can be expressed in portmanteau form via a mapping of X ∪ \{∅} → X × X ∪ , similar to the mapping D preceding (13.2.4), where on the component X (n) (θ1 , . . . , θn ) → (θ1 ; θ2 − θ1 , . . . , θn − θ1 ). Any measure P on X ∪ satisfying P({∅}) = 0 is thereby mapped into a product of uniform measure on X and a reduced measure on X ∪ , P˘ say, which consists ˘ n with weightings pn = P{ξ(X ) = n}. However, this is not of the measures Π quite the Palm distribution, as we can see by reference to any of the previous formulae, such as (13.2.6), from which follows the relation 2π 1 ˘ E IΓ (Sθ N ) N (dθ) . CP (Γ) = 2π 0 Then for a set Γ determined by a realization with n points of the form (0, φ1 , . . . , φn−1 ), n " pn " E IΓ (Sθi N ) " n C˘P (Γ) = 2π i=1 n pn IΓ (Sθi N ) Πn (dθ1 × · · · × dθn ) = 2π X (n) i=1 npn ˘ n (dφ1 × · · · × dφn−1 ), dθ IΓ (N0 ) Π = 2π X X (n−1) where now N0 is a generic counting measure with points at 0, φ1 , . . . , φn−1 . The factor n arises because the n terms in the sum give identical integrals

290

13. Palm Theory

on account of the symmetry properties of Πn . For such Γ we therefore have ˘ n (Γ). Thus, just as in the more general situation of Exercise C˘P (Γ) = npn Π 13.1(b), the Palm measure requires a weighting by the factor n. The Palm distribution ∞ is then obtained by normalizing this weighted form, which requires n=1 npn = E[N (X )] < ∞. The intuitive explanation is as before: a realization with n points is n times more likely to locate a point at the origin than a realization with just one point, and must be weighted accordingly in taking the expectation. Equations (13.2.5) and (13.2.6) yield a range of striking formulae as special cases and corollaries. One of the simplest is the following interpretation of the Palm probabilities for point processes. Example 13.2(b) Palm probabilities as rates. An important interpretation of the Palm measure for a point process comes from (13.2.6), which yields for Γ ∈ N0#∗ , C˘P (Γ) = E #{i: xi ∈ Ud and Sxi N ∈ Γ} , (13.2.7) where on the right-hand side it is to be understood that each xi is a point of the realization N . Thus, C˘P (Γ) is the expected number of points of the process in the unit cube (or, because the process is stationary, their expected rate), which, when the origin is transferred to the point in question, are associated with the occurrence of Γ. As Matthes and others have suggested, we can regard this as the rate of occurrence of marked points where a point is marked if and only if Γ occurs when the origin is shifted to the point in question. The Palm distribution then appears as a ratio of rates P0 (Γ) = m(Γ)/m,

(13.2.8)

where m(Γ) is the rate of the marked process and m is the rate of the original process. Next we use (13.2.5–6) to give more explicit formulae expressing the Palm measure C˘P in terms of P and vice versa. Setting g(x, ξ) = j(x)h(ξ) in (13.2.5), where X j(x) dx = 1, we obtain at (13.2.10) below the expression for C˘P in terms of P. For an inverse relation, more subtlety is needed: suppose that k: X × M# X → R+ is a normalizing kernel for the realizations ξ, that is, a nonnegative measurable function satisfying k(x, ξ) ξ(dx) = 1 for each ξ = ∅. (13.2.9) k(x, ∅) = 0 and X

Substituting g(x, ξ) = k(x, ξ)f (ξ) for some nonnegative B(M# X )-measurable f (·) in (13.2.5), equation (13.2.11) is obtained. We summarize as below. d ˘ Proposition 13.2.V. Let P be stationary on B(M# X ) for X = R and CP its Palm measure.

13.2.


291

# (a) For any nonnegative B(MX )-measurable function h(·), and nonnegative BX -measurable j with X j(x) dx = 1,

M# X

h(ψ) C˘P (dψ) =

M# X

= EP

P(dξ)

X

X

j(x)h(Sx ξ) ξ(dx)

j(x)h(Sx ξ) ξ(dx) .

(13.2.10a) (13.2.10b)

(b) For any nonnegative B(M# X )-measurable function f (·) and for k(·) satisfying (13.2.9), EP f (ξ) =

M# \{∅} X

X

k(x, S−x ψ)f (S−x ψ) C˘P (dψ) dx + f (∅)P{ξ = ∅}, (13.2.11)

where ∅ denotes the null measure. As a particular corollary of the proposition set h(ψ) = I{∅} (ψ) in (13.2.10). Then (13.2.12) C˘P ({ψ = ∅}) = 0. From (13.2.11) it can be seen that C˘P determines P up to P{ξ = ∅}. For the rest of this section we assume also that P{ξ = ∅} = 0, in which case it follows from Proposition 12.1.VI that P{ξ(X ) = ∞} = 1. Equation (13.2.10) is often presented in the special case j(x) = IUd (x), when it takes the form ˘ h(ψ) CP (dψ) = EP h(Sx ξ) ξ(dx) . M# X

Ud

In the point process case this can be interpreted as follows: shift the origin successively to each point of the process in Ud and sum the values of h obtained from these shifted versions of ξ, thus obtaining an expected rate of contributions to the sum. When h is specialized to the indicator of some event U of the process, the value of the integral on the right-hand side is just the rate of occurrence of points for which U occurs when the origin is shifted to the point in question. The two equations (13.2.10–11) can be specialized in various obvious ways. If we take h or f to be the indicator function of a set U ∈ B(M# X ), we obtain direct expressions for C˘P (U ) in terms of P and for P(U ) in terms of C˘P . When the mean density m exists, the equations can be put in the more symmetrical forms using P0 from Definition 13.2.IV, −1 h(Sx ξ) ξ(dx) , EP0 h(ψ) = m EP d U k(x, S−x ψ)f (S−x ψ) dx . EP f (ξ) = m EP0

X

(13.2.13) (13.2.14)

292

13. Palm Theory

Specific examples of functions k(·) satisfying (13.2.9) are given in Section 13.3; Exercise 13.2.3 outlines Mecke’s (1967) general construction. We now consider the moment measures of the Palm distribution. There is no loss of generality here in speaking of moments of the Palm distribution, rather than of the Palm measure, because the existence of higher moments implies the existence of the first moment, which ensures that the Palm measure is totally finite and so can be normalized to yield the Palm distribution. Following the notation of the earlier sections, let ψ denote a stationary random measure, and write ˚k (A1 × · · · × Ak ) = EP0 ψ(A1 ) . . . ψ(Ak ) (A1 , . . . , Ak ∈ BX ) M for the kth moment measure for P0 . ˚k of Proposition 13.2.VI. For k = 1, 2, . . . , the kth moment measure M the Palm distribution exists if and only if the (k + 1)th moment measure of the original random measure exists, in which case it is related to the reduced ˘ k+1 by (k + 1)th moment measure M ˘ k+1 (·). ˚k (·) = m−1 M M (13.2.15) Proof. The result is a further application of (13.2.5) and Fubini’s theorem. For nonnegative measurable g and h with h(·) on X (k) and g(·) integrable on X , set ··· h(y1 − x, . . . , yk − x) ξ(dy1 ) . . . ξ(dyk ). g(x, ξ) = g(x) X

X

Then the left-hand side of (13.2.5), using also (12.6.6), becomes g(x)h(y1 − x, . . . , yk − x) Mk+1 (dx × dy1 × · · · × dyk ) X (k+1) ˘ k+1 (du1 × · · · × duk ), = g(x) dx h(u1 , . . . , uk ) M X (k)

X

whereas using Fubini’s theorem with the right-hand side yields g(x) dx h(u1 , . . . , uk ) ψ(du1 ) . . . ψ(duk ) m EP0 X X (k) ˚k (du1 × · · · × duk ). =m g(x) dx h(u1 , . . . , uk ) M X

X (k)

But h ≥ 0 is arbitrary, so (13.2.15) follows, together with finiteness (because m is finite). The results of Section 12.4 and as above can be summed up in the following diagram: P(dξ) ξ(dx) | | reduction ↓ m P0 (dξ)

moments

−−−−−−−→

moments

−−−−−−−→

˚k mM

Mk+1 | | reduction ↓ ˘ k+1 ≡M

13.2.


293

Again the results are conveniently illustrated with respect to the finite case of a point process on a circle, as set out in Exercise 13.2.6. Moment measures can also be defined for higher-order and modified Palm distributions derived from the corresponding Campbell measures introduced around (13.1.3). For example, the first-order moment measure for the second˚1(2) (dz | x, y) can be defined as the Radon–Nikodym order Palm distribution M derivative of the third-order moment measure with respect to the second-order moment measure: ˚1(2) (dz | x, y) = M

M3 (dz × dx × dy) . M2 (dx × dy)

(13.2.16)

When the process is stationary this reduces to a function of u = y − x and v = z − x. Further discussion of these quantities and of the corresponding moment measures for the modified Palm distributions are briefly set out in Exercise 13.2.9. We turn finally to a characterization of Palm measures due to Mecke (1975). Keeping to Mecke’s context, we first observe that the relation between a sta˘ tionary measure P on B(M# X ) and its Palm measure CP can be extended to general σ-finite measures P. The definition of stationarity carries over to this context, and equations (13.2.5–6) continue to define a σ-finite measure C˘P in terms of an initial σ-finite measure P, whether or not the initial measure is totally finite. The only step that needs checking is that the argument establishing the σ-finiteness of C˘P remains valid (see Exercise 13.2.8). In this context, neither P nor CP need be totally finite but both may be. The previous results can be succinctly summarized for this extended context as follows. Proposition 13.2.VII. Let Q be a σ-finite measure on B(M# X ) and CQ its associated Campbell measure. Then the following assertions are equivalent. (i) Q is stationary (invariant under shifts S+u , u ∈ X ). + u , u ∈ X defined at (13.2.1b). (ii) CQ is invariant under the transformations Θ (iii) CQ factorizes as in (13.2.6) into a product of Lebesgue measure and an ˘ associated σ-finite measure R on B(M# X ), and R coincides with CQ . Proof. Only the last assertion, equivalent to asserting the uniqueness of the Palm factorization, requires further comment. Suppose in fact that two distinct measures R and R both satisfy (13.2.6) for the same Q. Then there must be some measurable, nonnegative f with M# f dR = M# f dR . But X X each of R and R is associated with an inversion formula (13.2.11), leading to a contradiction if both R and R are associated with the same Q. The third assertion of the proposition invites the further question: which measures R can appear in a Palm factorization. Although the factorization lemma implies that any R inserted in the right-hand side of (13.2.6) yields + u , in general this measure will not a measure on W that is invariant under Θ

294

13. Palm Theory

be a Campbell measure, that is, it will not be generated by some underlying stationary Q on M# X . The constraint which R must satisfy in order to be a stationary Palm measure is given in the following result of Mecke (1975, Theorem 1.7). d Theorem 13.2.VIII. A measure R on B(M# X ) with X = R , is the Palm measure of some stationary, σ-finite (but not necessarily finite) measure Q on # MX , B(M# X ) if and only if the following three conditions hold: (i) R is σ-finite; (ii) R({∅}) = 0; and (iii) for all measurable nonnegative h(·, ·) on X × M# X, h(−y, Sy ψ) ψ(dy) R(dψ) = h(y, ψ) ψ(dy) R(dψ). M# X

M# X

X

If, in addition, η =

M# X

X

X

(13.2.17)

k(y, S−y ψ) dy R(dψ) ≤ 1 for some k(·) satisfy-

ing (13.2.9), then R is the Palm measure of a probability measure P on M# X satisfying P{ξ = ∅} = 1 − η. Proof. Suppose first that R = C˘Q as in Proposition 13.2.VII for some stationary, σ-finite measure Q. Then (i) follows from Theorem 13.2.III, and (ii) is (13.2.12). To establish (iii), let j(x) be nonnegative, measurable, with j(x) dx = 1, and apply (13.2.10) to the left-hand side of (13.2.17), regarded X as a function of ψ. This yields j(x)h(−y, Sy Sx ξ) ξ(dx) Sx ξ(dy) Q(dξ).

M# X

X

X

Because X f (y) Sx ξ(dy) = X f (y − x) ξ(dy) for any nonnegative measurable function f , the displayed expression can be rewritten as j(x)h(x − y, Sy ξ) ξ(dx) ξ(dy) Q(dξ), M# X

X

X

and using now the reverse of this transformation to the integration over x, gives j(x + y)h(x, Sy ξ) Sy ξ(dx) ξ(dy) Q(dξ). M# X

X

X

Rewriting the integration over ψ and y in terms of the factorization as in the right-hand side of (13.2.6) (with y and ξ in place of x and ψ there), gives j(x + y)h(x, ξ) ξ(dx) R(dξ) dy.

X

M# X

X

Finally, using X j(x + y) dy = 1 and integrating out y, yields the right-hand side of (13.2.17).

13.2.


295

For the reverse argument, suppose that R satisfies conditions (i)–(iii) and that k is a normalizing kernel satisfying (13.2.9). Use R to define a measure Q on B(M# X ) by Q({∅}) = 0 and the first term in (13.2.11), so that

f (ξ) Q(dξ) =

M# X

M# \{∅} X

X

k(y, S−y ψ)f (S−y ψ) R(dψ) dy.

We insert this measure into the left-hand side of the fundamental equation (13.2.6) and obtain

= = =

M# X

M# X

X

X

M# X X

X

M# X X

X

X

g(x, Sx ξ) ξ(dx) Q(dξ)

g(x, Sx−y ψ)k(y, S−y ψ) S−y ψ(dx) dy Q(dψ)

g(u + y, Su ψ)k(y, S−y ψ) ψ(du) dy Q(dψ),

putting x = u + y,

g(v, Su ψ)k(v − u, Su−v ψ) ψ(du) dv Q(dψ), putting y = v − u,

where in the last equation we used the invariance of Lebesgue measure under shifts. In general this expression does not simplify, but when (iii) holds, then with h(y, ψ) = X g(v, Su ψ) k(v − u, Su−v ψ) dv, we can write for the righthand side g(v, ψ)k(v + u, S−v ψ) ψ(du) dv R(dψ) = g(v, ψ) dv R(dψ), because

k(v + u, S−v ψ) ψ(du) =

k(x, S−v ψ) S−v ψ(dx) = 1.

The last equation but one shows that the measure Q defined from R factorizes as in Proposition 13.2.VII(iii), and hence first is stationary, and second, identifies R as the reduced Campbell measure of Q. By extension of Definition 13.2.IV we call such an R a Palm measure also. Finally, the normalizing condition entailing η ≤ 1 comes from Proposition 13.2.V(b). Example 13.2(c) Palm factorization of the KLM measure for a stationary infinitely divisible point process. From Proposition 12.4.I we know that an infinitely divisible point process is stationary if and only if its KLM measure Q is stationary. In that case, a Palm factorization can be applied to the measure CQ (A × U ) =

U

(A) Q(d N ) N

U ∈ B(NX# \{∅}) ,

296

13. Palm Theory

giving X

# NX \{∅}

) N (dx) Q(d N ) = g(x, Sx N

dx X

N0#

0 ) C˘ (dN 0 ), g(x, N Q

0 here denotes a generic element of N # . Let us write for brevity where N 0 0 , C˘ = Q Q

0 is defined on and note that for a point process, as in the probability case, Q 0 may or may not be totally finite. B(N0# ); Q 0 for the various types of stationary, We can now examine the properties of Q infinitely divisible point processes. (1◦ ) Suppose that P is regular and therefore has a representation as the regular version of a stationary Poisson cluster process (Proposition 12.4.II). 0 is closely related to the symmetrized measures Pk−1 used in defining Here Q the regular representation. Regard Pk−1 not as a measure on X (k−1) but as a measure on the set Dk of counting measures in N0# containing just k − 1 points in addition to the point at the origin. Then we have k−1 0 ) Q 0 (dN 0 ) = k h(N h δ0 + δxi Pk−1 (dx1 × · · · × dxk−1 ), Dk

X (k−1)

i=1

(du) integration the factor k arising here, as in Example 13.2(a), from the N in the Campbell measure. The normalized measures ! kpk Pk−1 (·) m, where m = kpk and {pk } describes the distribution of the cluster size, can be interpreted loosely as providing the conditional distribution of the other cluster members, given that the point at the origin is arbitrarily chosen from a randomly selected population of i.i.d. families of k members. 0 is supported by the set of We see that in the regular case, the measure Q # counting measures in N0 with finite support. (2◦ ) If the process is strongly singular [see the definitions of Ss at the ) below (12.4.6)], the KLM measure itself is end of Section 12.4 and of I V (N supported on the set of counting measures with positive ergodic limits [i.e., ) > 0], and it follows, as in the discussion of Theorem 13.3.II below, I V (N 0 is concentrated on the space of sequences that also have positive that Q ergodic limits. This is the situation if, in particular, the process is a Poisson 0 = µC˘ , where µ is the parameter of the randomization, in which case Q P Poisson distribution and P is the point process that is being ‘randomized’. 0 measures concentrated (3◦ ) Finally, the weakly singular processes have Q # on the subset of N0 of counting measures that have infinite support but are asymptotically sparse in the sense that their ergodic limits are zero. In summary we have proved the following statement.

13.2.


297

Proposition 13.2.IX. A stationary infinitely divisible point process is regular, strongly singular, or weakly singular, according to whether the Palm 0 is supported, respectively, by the finite counting measures, the measure Q counting measures with infinite support and positive ergodic limits, or the counting measures with infinite support but zero ergodic limits. Exercise 13.2.10 exhibits the Palm measure P of a stationary infinitely divisible random measure with finite first moment measure as a convolution of P with the reduced Campbell measure of its KLM measure.

Exercises and Complements to Section 13.2 13.2.1 Let the random measure ξ on the integers as in Exercise 13.1.5 be stationary. Describe ξ as a stationary sequence {Xn } ≡ {ξ{n}} of nonnegative r.v.s. What aspects of this sequence are described by P and P0 , respectively? When ξ is a simple point process, give analogues and simple cases of equations (13.2.2–3) and (13.2.6–7). 13.2.2 Let {Pn } be a sequence of probability measures on M# X and P a limit measure. If {Pn } and P are stationary and Pn → P weakly, investigate under what conditions and in what sense there is convergence of the Campbell measures CPn to CP . Consider also conditions under which, for a stationary process, weak convergence of the underlying probability measures implies weak convergence of the associated Palm distributions, and vice versa. [Hint: Consider first the convergence of the first moment measures, which is necessary for any meaningful sense of convergence for the Campbell measures. For the second part, use the representation theorems first to establish convergence of the fidi distributions, and then refer to Section 11.1.] 13.2.3 Let {An } be a covering of X by disjoint bounded Borel subsets of X . Define the functions a(·) on W \ (X × {∅}) and k(·) for ξ = ∅ by

1 2 Iξ(A(x)) a(x, ξ) . k(x, ξ) = a(y, ξ) ξ(dy) a(x, ξ) =

∞ n=1

−n An

if x ∈ An and ξ(An ) = 0, otherwise,

n

X

Verify that the function k(·) satisfies (13.2.9). 13.2.4 Let the stationary random measure ξ have finite first moment measure. Show that the Laplace functional result of Proposition 13.1.VI simplifies on writing Lx [f ] = L0 [S−x f ]. 13.2.5 (a) Let ξ , ξ be stationary random measures with probability measures P , P . For nonnegative p, q with p + q = 1, the random measure ξ, which equals either ξ or ξ with probabilities p, q, respectively, has probability measure P = pP + qP . Find its Campbell and Palm measures. (b) Let ξ , ξ as in (a) be independent random measures, and let ξ = ξ + ξ . Using ∗ to denote convolution, show that the Palm measure P0 of ξ is given by P0 = P0 ∗ P + P ∗ P0 .

298

13. Palm Theory

13.2.6 Find explicit expressions for the reduced moment measures for a stationary ˘ n (·) in the point process on the circle in terms of the distributions {pn }, Π notation of Example 13.2(a). In particular, expanding the second moment measure by (5.4.7) and with p∗n = npn /m, m = n npn ,

∞

˘ [2] (B) dθ = M

(n + 1)(n + 2)pn+2 Πn+2 (dθ × S−θ B × X (n) )

n=0

=

m dθ 2π

∞

˘ n+2 (B × X (n) ) (n + 1)p∗n+2 Π

n=0

= (m dθ/2π) EP0 [N (B)]. † introduced in 13.2.7 Investigate properties of the modified Campbell measure CP (13.1.1b) and the corresponding modifications of local Palm distributions. In particular, find the form of these distributions for the process considered in Example 13.1(b). [Hint: See Kallenberg (1983a, §12.3), (MKM, 1978, §5.4).]

13.2.8 Extend the argument of Exercise 13.1.2 from the special case of KLM mea# sures to show that if a stationary, σ-finite measure R on (M# X , B(MX )) is associated with a Campbell measure CR by (13.1.1), then CR is again σ-finite. Show that in this case also, a function h can be found satisfying (13.2.4), and used to establish the existence of a σ-finite reduced version C˘R satisfying (13.2.5) and (13.2.6) of Theorem 13.2.III. 13.2.9 (a) Define a second-order Campbell measure as in (13.1.3), and use it to (2) define a family of second-order local Palm distributions Px,y (·). What simplification can be expected if the underlying random measure is stationary? (b) Suppose that first-, second-, and third-order moment measures M1 , M2 , M3 exist for the random measure ξ. Define the reduced moment measure ˚1(2) (dz | x, y) as the Radon-Nikodym derivative in M M3 (C, A, B) =

˚1(2) (C | x, y) M2 (dx × dy), M

A×B

and show that it can be interpreted as the first moment measure of the second-order local Palm distribution as in (a). What simplications can be expected if the underlying random measure is stationary? [Hint: Both sides equal E[ξ(A)ξ(B)ξ(C)], assumed finite for bounded A, B, C ∈ BX .] (c) Investigate the forms of these measures for (i) Poisson, (ii) renewal, and (iii) two-point cluster Poisson processes.

13.2.10 Let P be the probability measure of a stationary infinitely divisible random measure with finite first moment measure, and let Q be its KLM measure. ˘ . Prove that P0 = Q0 ∗ P, where Q0 is the reduced Campbell measure C Q

[Hint: Let P and Q have Laplace functionals LP and LQ . Relate log LP [f ]

to LQ [f ] and deduce that for f , g ∈ BM+ (Rd ) [see (13.1.15)], Rd

g(x)LP [S−x f ; 0] mP dx = LP [f ]

Rd

g(x)LQ [S−x f ; 0] mQ dx. ]

13.3.

Interval- and Point-stationarity

299

13.3. Interval- and Point-stationarity When the ideas of Section 13.2 are specialized to point processes, a number of new features arise; we review them in this section. In particular, we consider a number of results for the important special case where X is the real line, and the point process is simple with finite mean rate m. The central result here is the correspondence, foreshadowed already in Chapter 3, between such point processes and stationary sequences of intervals. Finding a counterpart for this result in spaces of higher dimension seemed an impossible task at the time of the first edition of this book, but the recent attack on the problem by Thorisson, Last, Heveling, and others has shown this not to be the case, and is introduced at the end of the section. As noted below Definition 13.2.IV, it follows from Proposition 13.1.V that, for a simple stationary point process, the support for its Palm measure can be taken to be the space N0#∗ of simple counting measures with a point at the origin. From Proposition 12.1.VI it follows, because N {0} = 1, that the Palm measure is in fact supported by elements N ∈ N0#∗ which cannot be empty and must therefore satisfy N (Rd ) = ∞. When the state space is the line, this means that N (−∞, 0] = N (0, ∞) = ∞. For this special case, it is clear also that there is a one-to-one both ways measurable mapping Φ between the space N0#∗ [with the σ-algebra of Borel sets inherited from B(NR#∗ )] and the space T + of doubly infinite sequences of positive numbers. Denoting the points of a generic element N0 ∈ N0#∗ by {. . . , t−1 (N0 ), t0 (N0 ) = 0, t1 (N0 ), . . .} with ti (N0 ) < ti+1 (N0 ) (i = 0, ±1, . . .), the mapping Φ associates the points of N0 with the sequence of intervals τi ≡ ti (N0 ) − ti−1 (N0 ); that is, ΦN0 = {τi } ≡ {ti (N0 ) − ti−1 (N0 ): i = 0, ±1, . . .}. For measurability see Exercise 9.1.14. Every measure on N0#∗ , B(N0#∗ ) , and in particular every Palm measure C˘P (·), then induces a measure on (T + , B(T + )), say (C˘P Φ−1 )(·). These remarks pave the way for the results setting up a correspondence between counting properties and interval properties. The correspondence is essentially a restatement of the equations representing the Palm measure in terms of the Campbell measure and hence of its underlying probability measure, and vice versa. We state the theorem first in its most striking form, for the case of finite mean density m. Theorem 13.3.I [Ryll-Nardzewski (1961), Slivnyak (1962, 1966), Kaplan (1955)]. There is a one-to-one correspondence between the distributions P on B(NR#∗ ) of simple nonnull stationary point processes on the line with finite mean density m, and the distributions Π on B(T + ) of stationary sequences of positive random variables with finite mean m−1 . If P0 is the image in

300

13. Palm Theory

B(N0#∗ ) under Φ−1 of the measure Π on B(T + ), this relation is effected by the equations N (0,1] h(Sti N ) (13.3.1) EP0 h(N0 ) = m−1 EP i=1

for nonnegative B(N0#∗ )-measurable h(·), and EP

t1 (N0 ) g(N ) = mEP0 g(St N0 ) dt ∞ 0 =m EP0 g(St N0 )I{t1 (N0 )>t} (N0 ) dt

(13.3.2)

0

for nonnegative B(NR#∗ )-measurable g(·). Proof. Equations (13.3.1–2) are adaptations to the present context of equations (13.2.10–11), but further comments are required. In (13.3.1), the points ti refer to points of N lying in the unit interval (0, 1]; each Sti satisfies (Sti N )({0}) = N ({ti }) = 1, and so with probability 1 it can be identified with an element of U0 whenever h(Sti N ) is well defined. In the exceptional / U0 , the value of h(Sti N ) can be represented P-null set, where (ti , Sti N ) ∈ arbitrarily (and set equal to zero, say). To derive (13.3.2) from (13.2.11) set there k(x, N ) =

1 if x = t0 (N ), 0 otherwise.

(13.3.3a)

Observe that for a simple nonnull counting measure (see Definition 12.1.VII), R

k(x, N ) N (dx) = N ({t0 (N )}) = 1,

(13.3.3b)

so that k(·) satisfies (13.2.9) when also P corresponds to a simple point process, that is, P(NR#∗ ) = 1. Now substituting in (13.2.11), the term k(x, S−x N0 ) in the integral on the right-hand side of (13.2.11) equals unity provided x = t0 (S−x N0 ), which, because the counting measure S−x N0 for N0 ∈ N0#∗ consists of atoms of unit mass at {x + ti (N0 ): i = 0, ±1, . . .}, is true for x ≤ 0 < x + t1 (N0 ), that is, for −τ1 < x ≤ 0. Changing the variable of integration from x to −x leads to (13.3.2). We now show that if P is stationary on B(NR#∗ ) then Π is stationary on B(T + ) and conversely. Following Franken et al. (1981) we argue as follows. First, using the stationarity of P we can extend (13.3.1) to give EP0

m EP h(N0 ) = T

N (0,T ] i=1

h(Sti N )

(0 < T < ∞).

13.3.


301

Define the shift operator ϑ: T + → T + by {ϑτi } = {τi−1 }. Then its image Θ: N0#∗ → N0#∗ , where ΘN0 = Φ−1 {ϑτi } = Φ−1 (ϑ(ΦN0 )), satisfies EP0

m EP h(ΘN0 ) = T

N (0,T ]

h(Sti−1 N ) ,

i=1

from which we have " " "EP h(N0 ) − EP h(ΦN0 )" ≤ (m/T )EP |h(St N )| + |h(St N )| , 0 0 0 N +1 where N = N (0, T ]. Then for all bounded h, the right-hand side → 0 as T → ∞. Consequently, the expectations on the left-hand side coincide for all bounded measurable h, so that the measures Π and Π ◦ ϑ, equivalent to P0 and P0 ◦ Θ, are therefore equal; that is, Π is invariant under ϑ, and thus its iterates are also invariant. Similarly, when Π is stationary, (13.3.2) can be extended by iteration to tr (N0 ) 1 EP0 g(St N0 ) dt (r = 1, 2, . . .). EP [g(N )] = mr 0 Replacing N by St N , subtracting, and letting r → ∞, we find in an analogous fashion that P is stationary under shifts St . We come finally to the question of uniqueness. Suppose we are given a stationary measure Π on T + and that P is constructed from Π via P0 and (13.3.2). Then P, which is clearly a probability measure, has an associated Palm measure C˘P that satisfies the equation analogous to (13.3.2); namely, t1 (N0 ) EP [g(N )] = C˘P (dN0 ) g(St N0 ) dt. (13.3.4) N0#∗

0

Substituting g(N ) = h(St0 (N ) N ), the inner integral in (13.3.4) becomes

t1 (N0 )

h(St0 (St N0 ) St N0 ) dt.

0

Now for 0 < t < t1 (N0 ), St N0 has points at ti (N0 ) − t, so the point of St N0 lying in (−∞, 0) and nearest to the origin is at −t. In this range of t, therefore, the argument of h reduces to N0 , and (13.3.2) yields for this g EP [g(N )] = mEP0 t1 (N0 )h(N0 ) . Similarly, (13.3.3) yields EP [g(N )] =

N0∗

t1 (N0 )h(N0 ) C˘P (dN0 ).

Both these equations hold for nonnegative B(N0#∗ )-measurable h, so it follows that the measures mt1 (N0 ) Π(dN0 ) and t1 (N0 ) C˘P (dN0 ) coincide, thereby

302

13. Palm Theory

identifying mΠ as the Palm measure for P. Thus, Π is determined uniquely by P. Again, if P is given and Π is determined (through P0 ) by (13.3.1), then we know already that Π is the Palm distribution of P, and hence from (13.3.2) that P is uniquely determined by Π. Either equation on its own is enough to imply a one-to-one correspondence. Theorem 13.3.I is a substantial generalization of the Palm–Khinchin equations of Section 3.4, and it provides the most satisfactory approach to the determination of the point process associated with a given process of intervals. The intuitive content of (13.3.2) can be expressed as follows. To embed a stationary sequence of intervals {τn : n = 0, ±1, . . .} with distribution Π as in Theorem 13.3.I into a stationary point process on R, first select a realization {τn }, and choose a number X uniformly at random on a suitably large interval (0, T ) say, with T any τn [roughly speaking, this is like taking r large in the display before (13.3.4)]. Then define a realization {tn } of the point process on R by relabelling the sequence −X + τ1 + · · · + τr (r = 0, 1, . . .), tr = −X − τ0 − · · · − τr+1 (r = −1, −2, . . .), as tn = tr +n (n = 0, ±1, . . .), where we identify r from tr = inf{tr : tr > 0}. Then (all n). tn − tn−1 = τr +n ∞ From the choice of X, Pr{τr > x} = x u Π(du), that is, the length-biased distribution of the common distribution of the stationary sequence {τn }. The next example utilizes a more direct construction that follows Palm’s original suggestion. It incorporates the idea of a point chosen uniformly at random from within an initial interval selected by the length-biased form of the stationary interval distribution spelled out in detail for the Wold process around (4.5.3a). Equation (13.3.2) is a more formal and general way of expressing the same ideas. Example 13.3(a) Renewal and Wold processes. Suppose first that {. . . , L−1 , L0 , L1 , . . .} is a sequence of i.i.d. positive r.v.s, which is therefore stationary and so describes a distribution Π on T + . Indeed, Π is just the product measure on (R+ )(∞) derived from multiple copies of the measure F (dx) associated with each of the Li . To fit into the framework of the theorem we must have ∞ x F (dx) = m−1 < ∞. F (0+) = 0, 0

In (13.3.2) take g(N ) = IΓ (N ), where Γ = Γ1 ≡ {N : t1 (N ) > x}. Then the term g(St N0 ) on the right-hand side of (13.3.2) equals unity for 0 < t < t1 (N0 ) − x and t1 (N0 ) > x, and zero otherwise, so that EP g(N ) = P(Γ1 ) = P{t1 (N ) > x} = m EΠ [(L1 − x)I{L1 >x} ] ∞ ∞ (y − x) F (dy) = m [1 − F (y)] dy, =m x

x

13.3.


303

which is the first of the Palm–Khinchin equations (3.4.9) and shows in the renewal case that the first interval after a fixed origin (and in view of stationarity the choice of origin is immaterial) has the distribution of the forward recurrence time [see Example 4.1(c)]. Next, take Γ = Γ2 ≡ {N : t1 (N ) > x, t2 (N ) − t1 (N ) > y}. We obtain similarly P(Γ2 ) ≡ P{t1 (N ) > x, t2 (N ) − t1 (N ) > y} = m EΠ [(L1 − x)I{L1 >x,L2 >y} ] ∞ [1 − F (u)] du 1 − F (y) , =m x

on account of the assumed independence of the {Li }. The first equality here is the second of the Palm–Khinchin equations. The other equality shows that for a stationary renewal process, the length of the second interval after the origin is independent of the first. In the case of a Wold process (see Section 4.5), the intervals {Li } form a stationary Markov chain with stationary distribution π(·) say and transition kernel P (x, B) = Pr{Li+1 ∈ B | Li = x}. Again we must assume that ∞ π({0}) = 0 = π((−∞, 0]) and 0 x π(dx) = m−1 < ∞. For Γ1 and Γ2 as above we find that t1 (N ) has the same kind of forward recurrence time distribution with π(·) in place of F (·), and that t1 (N ) and t2 (N ) − t1 (N ) have the joint distribution

∞

F2 (dx × dy) = m dx

π(du) P (u, dy). x

Thus, the marginal distribution of t2 (N ) − t1 (N ) is now given by F2 (R+ × dy) = m

∞

∞

dx 0

π(du) P (u, dy), x

and in general neither this interval nor any of the later intervals has exactly the stationary interval distribution. The analysis of Section 13.2 allows us to construct a Palm measure even for a process with infinite intensity. It is therefore natural to seek a version of Theorem 13.3.I valid even for processes with infinite mean rate; this is possible if Π is allowed to have infinite total mass. In fact, the proof of Theorem 13.3.I carries over with only notational changes as soon as we replace mΠ by the measure induced on T + by the Palm measure C˘P , which remains σ-finite but not necessarily totally finite. For brevity we state the theorem below in terms of a measure R on the space N0#∗ rather than a measure on the space T + of interval sequences.

304

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Theorem 13.3.II. There is a one-to-one correspondence between distributions P on B(NR#∗ ) of simple nonnull stationary point processes on R, and stationary σ-finite (but not necessarily totally finite) measures R on B(N0#∗ ) satisfying t1 (N0 ) R(dN0 ) = P(dN ) = 1. (13.3.5) N0#∗

NR#∗

The correspondence is effected via nonnegative B(N0#∗ )-measurable h and nonnegative B(NR#∗ )-measurable g in the equations

N0#∗

and

h(N0 ) R(dN0 ) =

N (0,1]

NR#∗

h(Sti N ) P(dN )

NR#∗

g(N ) P(dN ) =

N0#∗

(13.3.6a)

i=1

R(dN0 )

t1 (N0 )

g(St N0 ) dt.

(13.3.6b)

0

Proof. The normalization condition (13.3.5) follows from setting f (ξ) ≡ 1 in (13.2.11). The remaining results paraphrase those of Theorem 13.3.I; details of the proofs are left to the reader. We return now to the more general context of point processes on Rd . In the absence of a total ordering on Rd , it is not immediately apparent what should be the exact counterparts of the preceding results. Some initial progress can be made by replacing the role of τ1 above by the point of the realization, x∗ (N ) say, that is closest to the origin. We first check that this concept is well defined. Lemma 13.3.III. Let N be a simple nonnull stationary point process on X = Rd . Then the set {N : there exist x , x with x = x and N ({x }) ≥ 1, N ({x }) ≥ 1} is B(NX#∗ )-measurable and has P-measure zero. Proof. The set J ⊂ X (2) defined by J = {(x, y): x = y, x = y} is a measurable set in B(X (2) ) by inspection, and we can write IJ (x, y) N (dx) N (dy) = h(x, N ) N (dx), X (2)

X

where h(x, N ) =

1 if N ({y}) > 0 for some y with y = x, 0 otherwise,

13.3.


305

is measurable. Applying (13.2.5), we obtain ˘ E h(x, N ) N (dx) = h(x, S−x N0 ) dx. CP (dN0 ) N0#∗

X

X

The function h(x, S−x N0 ) equals 1 only on the at most countable set of surfaces {y: y + xi = xi , y = xi } obtained by letting xi run through the points of N0 . For d = 1, the surface consists of the single point y = −2xi ; for d > 1, it consists of a surface in Rd of dimension d − 1, and so is of zero Rd -Lebesgue measure. In either case, the inner integral vanishes for each N0 , and so the expectation is zero. It follows from Lemma 13.3.III that with probability 1 the distances from the origin to the points of a realization of a nonnull stationary simple point process in Rd can be set out in a strictly increasing sequence 0 < r1 (N ) < r2 (N ) < · · · . In this case the quantities ri (N ) are well-defined random variables because N Sa (0) ≥ i}, {ri (N ) < a} if and only if and for given i there is a.s. a unique point of the process, x∗i (N ) say, associated with a given distance ri . In the exceptional set (of probability 0) where there is no such unique point, we can put all the x∗i (N ) equal to zero. It follows that the x∗i (N ) form a measurable enumeration of the points of the realization (Definition 9.1.XI), for the measurability of sets such as {N : x∗i (N ) ∈ A} =

( k k+1 ≤ ri (N ) < ; N (A) > 0 n n n

(A ∈ B(Rd ))

k

implies that each x∗i (N ) is a well-defined random element of Rd . In the sequel we mostly use the point of a realization N that is closest to the origin, which we denote for brevity by (13.3.7) x∗ (N ) = x∗1 (N ). One immediate use for x∗ (N ) is to develop an inversion formula extending (13.3.2) to the case X = Rd . For this we need the concept of a Voronoi polygon: for any given realization N of a nonnull simple point process, and any point u ∈ Rd , the Voronoi polygon Vu (N ) with ‘centre’ u is the subset Vu (N ) = {x: x − u < x − xj , xj ∈ N and xj = u} of points x ∈ Rd that lie closer to u than to any point xj of N . Consequently, 0 if u ∈ / N, N (dx) = (13.3.8) 1 when u ∈ N and N is simple. Vu (N )

306

13. Palm Theory

In particular, if N0 ∈ N0#∗ , we write V0 (N0 ) for the Voronoi polygon about the origin (which is a point of N0 ), and note that N0 (dx) = N0 V0 (N0 ) = 1, V0 (N0 )

so that IV0 (N0 ) (x) = k(x, N0 ) where k(·) is a function as in (13.2.9) and used in the inversion formulae (13.2.11) and its variants. The inversion formula itself takes the form, for nonnegative B(NX#∗ )-measurable g(·), g(Sx N0 ) dx R(dN0 ), (13.3.9a) EP g(N ) = N0#∗

V0 (N0 )

where the measure P on B(NX#∗ ) is a probability measure provided R satisfies the normalizing condition dx R(dN0 ) = 1. (13.3.9b) N0#∗

V0 (N0 )

If R is totally finite with mass m, say R = mP0 , the right-hand side can be written as g(Sx N0 ) dx , (13.3.9c) m EP0 V0 (N (Sx N0 ))

in which case the left-hand side of (13.3.9a) corresponds to the probability distribution of a stationary simple point process with finite mean density m and Palm distribution P0 [the proof of this fact is similar to that of (13.3.2) and left to Exercise 13.3.4(a)]. The inversion formula (13.3.9a) is not as useful in Rd (d ≥ 2) as is (13.3.2) in R. This is chiefly a reflection of the increased structural complexity of the higher-dimensional Euclidean spaces, but a few simple results can be deduced from it, such as the intuitively obvious fact that the expected hypervolume (i.e., Lebesgue measure) of the Voronoi polygon about the origin equals m−1 [see Exercise 13.3.4(b)]. It does not supply a full counterpart in Rd of Theorems 13.3.I and 13.3.II, because it does not address the issue of defining a notion of stationarity for measures in N0#∗ extending that of interval-stationarity for point processes on the line; this extension must await the discussion of point-stationarity and Theorem 13.3.IX. A further use of x∗ (N ) is in the next theorem, which establishes the conditional probability interpretation of the Palm distribution for point processes in general Euclidean spaces Rd . Theorem 13.3.IV. Let N be a simple stationary point process in Rd with finite mean rate m, distribution P, and Palm distribution P0 , and let {An : n = 1, 2, . . .} be a nested sequence of bounded Borel sets with nonempty interiors satisfying (n → ∞). (13.3.10) diam(An ) → 0

13.3.


Then as n → ∞,

307

(An )−1 P{N (An ) > 0} → m.

(13.3.11)

If, furthermore, the sets {An } are spheres in R centred at the origin, then for bounded continuous nonnegative Borel functions f on NX#∗ , EP f (N ) | N (An ) > 0) → EP0 f (N0 ) . (13.3.12) d

Proof. The first assertion (13.3.11) is a corollary to the discussion on intensities in Chapter 9 [see, in particular, (9.3.22) and Exercise 9.3.11]. A further corollary of the same discussion, which we need in the sequel, is that ! (n → ∞) (13.3.13) P{N (An ) ≥ 2} (An ) → 0 (see Proposition 9.3.XV). We approach the assertion at (13.3.12) via the following result. Proposition 13.3.V. Let N , P, P0 , m, and {An } be as in Theorem 13.3.IV and x∗ (N ) as at (13.3.7). Then for bounded nonnegative B(NX#∗ )-measurable f (·), (13.3.14) EP f (Sx∗ (N ) N ) | N (An ) > 0 → EP0 [f (N0 )]. Proof. Note first that " " "EP f (Sx∗ (N ) N ) | N (An ) > 0 − EP0 f (N0 ) " −1 " " "EP [f (Sx∗ (N ) N )I{N (A )>0} ] − m(An )EP f (N0 ) " ≤ m(An ) 0 n " " −1 ! EP [f (Sx∗ (N ) N )I{N (An )>0} ]"1 − m(An ) P{N (An ) > 0}" + m(An ) ≡ J1 + J2

say.

In J2 , the modulus of the difference converges to zero as n → ∞ by (13.3.11), and the multiplier remains finite because P{N (A ) > 0} EP [f (Sx∗ (N ) N )I{N (An )>0} ] n ≤ supN ∈N #∗ f (N ) X m(An ) m(An ) by (13.3.11), → supN ∈N #∗ f (N ) X

and this supremum is finite by the boundedness assumption on f (·). Thus J2 → 0 as n → ∞. For J1 , we note first from the proof of Theorem 13.3.I and (13.2.8) that f (Sx N ) N (dx) = EP f (Sxi N ) . m(An ) EP0 f (N0 ) = EP An

xi ∈An

It is thus enough to consider the difference " " " 1 "" ", ∗ f (S N )I − f (S N ) E P xi x (N ) {N (An )>0} " " (An ) xi ∈An

308

13. Palm Theory

which certainly vanishes when N (An ) = 0. When N (An ) > 0, then we have x∗ (N ) ∈ An , implying that it can be identified with one of the xi ∈ An , so the first term cancels with one of the elements of the sum. Consequently, the difference is dominated by (supN ∈N #∗ f (N ))P{N (An ) ≥ 2}/(An ), which X tends to zero by (13.3.13). Resuming the proof of Theorem 13.3.IV, Proposition 13.3.V implies that it is enough to establish the convergence to zero of the difference " " "EP [f (N ) | N (An ) > 0] − EP [f (Sx∗ (N ) N ) | N (An ) > 0]" " ≤ EP |f (N ) − f (Sx∗ (N ) N )| " N (An ) > 0 (13.3.15) " ≤ EP supx∈A |f (Sx∗ (N ) N ) − f (Sx+x∗ (N ) N )| " N (An ) > 0 , n

because x∗ (N ) ∈ An under the condition N (An ) > 0. Fixing the set An for the supremum as An0 say, and letting n → ∞ for the conditioning, this last expression converges by Proposition 13.3.V to EP0 supx∈An |f (N0 ) − f (Sx N0 )| . Inasmuch as f is uniformly bounded and continuous, and the shift operation is continuous also, the argument of the supremum converges to zero pointwise as n0 → ∞, and then by dominated convergence the expectation must converge to zero. We note that in both Theorem 13.3.IV and Proposition 13.3.V the convergence results are sufficient to imply that both P{· | N (An ) > 0} and P{Sx∗ (·) | N (An ) > 0} converge weakly to P0 {·} as n → ∞. In fact the convergence in Proposition 13.3.V can be strengthened: see Exercise 13.3.6. The results just proved also provide some kind of analogue to the differential form of the Palm–Khinchin equations given at (3.4.11) of Chapter 3. Even in the one-dimensional case, however, it is not easy to provide a completely satisfactory account by this differential approach [see Slivnyak (1962, 1966), Leadbetter (1972), and Exercise 3.4.4]. It is a far more difficult exercise to extend to Rd the interval-stationarity interpretation of a stationary point process on R. Results in this direction centre around the concepts of point maps and point-stationarity, initially considered by Mecke (1975) (the term point-stationarity is generally used in Rd for what in R1 has often been called cycle stationarity). One way of looking at the intervals in a one-dimensional point process is as mappings, each of which links two points of a realization. The underlying idea is that, even in higher dimensions, we may be able to define a family of mappings that link points of the process in such a way that invariance under these mappings provides a characterization of the Palm measure. What distinguishes the two cases R1 and Rd is that the mappings in R1 can be restricted to points that are left or right neighbours, whereas in Rd they need not have any particular proximity relation.

13.3.


309

Extending the notation from the beginning of the section, with X = Rd , let {xi (N )} be a measurable enumeration of the points (of support) of N ∈ NX#∗ . We use x ∈ N as shorthand for x ∈ supp(N ), so x ∈ N ∈ NX#∗ implies that N ({x}) = 1. In view of Proposition 12.1.VI and the fact that the point process is stationary, we assume that P{N (X ) = ∞} = 1. Then we are in fact interested in that part of the subspace N0#∗ for which N (X ) = ∞, and the enumeration {xi (N )} is countably infinite. Let Ψ(N, x) be a measurable mapping from NX#∗ ×X to X . Call Ψ covariant when for all N ∈ NX#∗ and x, y ∈ X , Ψ(Sy N, Sy x) ≡ Ψ(N − y, x − y) = Ψ(N, x) − y ≡ Sy Ψ(N, x),

(13.3.16)

so for covariant Ψ, Sx Ψ(N, x) = Ψ(Sx N, 0). Definition 13.3.VI. A point map is a covariant mapping Ψ: NX#∗ × X → X such that ∈ N if x ∈ N , Ψ(N, x) (13.3.17) = x if x ∈ / N. For covariant Ψ(N, ·), Ψ(N, x) = S−x Ψ(Sx N, Sx x) = S−x Ψ(Sx N, 0), so Ψ being a point map and x ∈ N is the same as Sx N ∈ N0#∗ , and the first (and critical) case of (13.3.17) is equivalent to requiring that for N ∈ N0#∗ , Ψ(N, 0) is again a point of N.

(13.3.18)

[Earlier work in Thorisson (1995, 2000) and Heveling and Last (2005) introduced a point map as a mapping from NX#∗ → X satisfying (13.3.18). Their subsequent work starts from Definition 13.3.VI.] Take We define the composition of two point maps Ψ1 and Ψ2 as follows. x ∈ N ∈ N0#∗ and consider (Ψ2 ◦ Ψ1 )(N, x) ≡ Ψ2 N, Ψ1 (N, x) . Because y = Ψ1 (N, x) ∈ N by (13.3.18), it follows that z ∈ N also, that is, Ψ2 ◦ Ψ1 is well defined as a point map and, with N specified, it is indeed the usual composition of the two point maps. Consequently, starting from x0 ≡ x ∈ N , defining xn = Ψ(N0 , xn−1 ) for n = 1, 2, . . . yields a sequence of points in N , but they need not be distinct, nor need they enumerate the elements of N when N is countable. Now our interest is in countably infinite point sets N ∈ N0#∗ , so the sequence {xn } just defined by the nth point map iterates of x0 is well defined, and either constitutes an infinite chain or reduces to recurring cycles. Call a point map bijective if for any N ∈ NX#∗ , the function Ψ(N, ·) is a one-to-one mapping on X . It is evident that a bijective point map Ψ as at (13.3.17) maps N in a one-to-one manner onto itself, and for x ∈ / N it is just the identity. We can then define the inverse point map Ψ−1 for z ∈ N as the solution x of z = Ψ(N, x), and write x = Ψ−1 (N, z). We associate with every point map Ψ the point shift S Ψ on NX#∗ defined by (13.3.19) S Ψ N = SΨ(N,0) N.

310

13. Palm Theory

Because Ψ(N, 0) = 0 for N ∈ / N0#∗ , S Ψ N = N unless N ∈ N0#∗ , in which case Ψ the effect of S on N is to shift the points {xi (N )} of N to {xi (N )−Ψ(N, 0)}. Now by virtue of Ψ as a point map and 0 ∈ N ∈ N0#∗ , Ψ(N, 0) ∈ N , that is, it is one of the points enumerated as {xi (N )}, and this point therefore gives xi (N ) − Ψ(N, 0) = 0; that is, 0 ∈ SΨ(N,0) N when N ∈ N0#∗ . Thus, S Ψ N shifts the origin to Ψ(N, 0), and S Ψ defines an operator on NX#∗ that maps the space N0#∗ into itself (although not necessarily onto itself). For any N ∈ NX#∗ , the point sets N and its shifted version Sy N (any y ∈ X ) consist of points whose relative positions in X are not changed by the shift. However, the point shift S Ψ when operating on N ∈ N0#∗ yields a point set S Ψ N ∈ N0#∗ that in general differs from the original point set. The most familiar examples are for right- and left-shifts for sequences in R1 as in the next example. Example 13.3(b) Right- and left-shifts as bijective point maps in R1 . For X = R, enumerate the points of N ∈ NX#∗ as {xi } ≡ {xi (N ): i = 0, ±1, . . .} where xi < xi+1 with x0 ≤ 0 < x1 . For N ∈ N0#∗ with N (R− ) = N (R+ ) = ∞, define Ψ(N, 0) = x1 (N ) [i.e., for such N , Ψ(N, 0) is the first point of N to the right of the origin, x1 (N )], and for other N ∈ N0#∗ define Ψ(N, x) = x. Because Ψ is a point map and therefore covariant, Ψ(N, xi (N )) = Ψ(N − xi (N ), 0) + xi (N ) = xi+1 (N ). Again because Ψ is a point map, Ψ(N, x) = x whenever x ∈ / N , and therefore we have shown that this point map is indeed simply a right-shift. When N 0, the corresponding point shift S Ψ shifts the origin for S Ψ N to x1 (N ), thus subtracting x1 (N ) from each of the points of N , yielding xi (S Ψ N ) = xi+1 (N ) − x1 (N ). For example, we then have Ψ(S Ψ N, 0) = x2 (N ) − x1 (N ). For any x ∈ N , x (N ) say, we have Ψ(N, x ) = Ψ(N − x , 0) + x = x+1 (N ), so that Ψ−1 (N, x ) = x−1 (N ). It then follows that if we define n Ψn+1 = Ψ ◦ Ψn with Ψ1 = Ψ (and thus Ψ−n = Ψ−1 ), that {Ψn (N0 , x0 )} = {xn (N0 )}, i.e. this sequence enumerates the points of N0 . Observe that the point map Ψk ≡ Ψk above has the property that Ψk (N, x) maps x to the kth point of N to the right of x, and is again a bijective point map. The earlier theorems of this chapter can be reinterpreted as assertions that the Palm relations define a one-to-one correspondence between stationary point processes on NR#∗ and point processes on N0#∗ which are invariant under the bijective point maps taking one point of the realization to the next. When X = Rd with d > 1, however, the problem of finding a suitable family of bijective point maps becomes nontrivial. The next example, due to H¨ aggstr¨ om and quoted in Thorisson (2000, §9.2.8), underlies the extension of Theorem 13.3.II that we describe shortly.

13.3.


311

Example 13.3(c) Mutual nearest neighbour matching. Given distinct points x , x in N ∈ NX#∗ , x and x are mutual nearest neighbours in N when the element of N closest to x is x and the element of N closest to x is x . Define a point map for N ∈ N0#∗ by x if x and 0 are mutual nearest neighbours in N , Ψ(N, 0) = 0 otherwise. Then as x runs through all the elements of N , yx if yx ∈ N is mutual nearest neighbour to x, Ψ(N, x) = x otherwise,

(13.3.20)

so Ψ is a bijective point map. For Ψ defined in this way, we see that for x ∈ N , either x has a mutual nearest neighbour yx ∈ N or not, and Ψ(N, yx ) if the first case, Ψ(N, Ψ(N, x)) = = x in either case, Ψ(N, x) otherwise, so that Ψ−1 = Ψ; that is, this point map is self-inverse. A simple case of a systematic approach to the construction of bijective maps in Rd due to Heveling and Last starts from the following definition. Definition 13.3.VII. For any bounded Borel set B, a B-selective point map ΨB (N, x) is a mapping N0#∗ × X → X such that (1) if N ∩ [B ∪ (−B)] consists of a single point xB say, and if also 0 is the only point of N in (B + xB ) ∪ (−B + xB ), then ΨB (N, 0) = xB ; (2) for x ∈ N , ΨB (N, x) is defined from (1) and the covariant property; and (3) in all other cases, ΨB (N, x) = x. Property (2) here ensures that ΦB (N, x) is in fact a point map: we call it the B-selective point map. Observe that the conditions ensure that a unique point of Rd is defined for every N and x. Note, in particular, that ΨB (N, 0) = 0 whenever there are two or more points of N in B ∪ (−B). Because B ∪ (−B) is symmetric and xB = ΨB (N, 0) = x−B , the map ΨB is in fact bijective. To see this, if a point xB = 0 exists as in (1) for given N ∈ N0#∗ , the associated point shift S ΨB interchanges 0 and xB , leaving the other points of N unchanged. ΨB is also self-inverse in the sense that ΨB ◦ ΨB = I (identity). The aim now is to build up a comprehensive set of mappings via an exhaustive area search by letting B vary over a sufficiently wide class of testing sets. To this end, let {Tn } = {Bni } be a nested system of tilings of Rd , namely, a system of partitions of X like a dissecting system, each Tn consisting of a denumerable number of disjoint sets for which the norm Tn = supi {diam(Bni )} → 0 as n → ∞. As the sets get smaller, their ability to distinguish between points of the realization increases, until finally every

312

13. Palm Theory

point in the realization can be interchanged with the origin by one of the point shifts S ΨBni . It is now plausible that requiring a measure on N0#∗ to be invariant under the denumerable family of point shifts S ni ≡ S ΨBni ,

(13.3.21)

may be a condition comparable to interval stationarity in the one-dimensional case. This is a little too simple in general, because of the complexities of the possible configurations of points in a general element of NX#∗ . To avoid this difficulty, we impose on the point process a property similar to but stronger than simplicity. We say that the element N of N0#∗ in Rd is unaligned if it contains no equidistant pairs of points that are collinear. Exercise 13.3.11 shows that the set UX of all unaligned elements in X = Rd is measurable. Then the point process as a whole is unaligned if P(UX ) = 1, and we show that under this condition, invariance under the point shifts S ni is enough to imply stationarity of the original point process. In the general case treated by Heveling and Last (2005), this collection of point shifts has to be augmented by additional point shifts that deal with situations where the configurations may contain finite chains of equidistant collinear points. In any case, the aim is to establish the existence of a sufficiently comprehensive set of bijective point maps to justify the far-reaching generalization of the one-dimensional results outlined in the definition and theorem which follow. Definition 13.3.VIII. A σ-finite measure R on B(N0#∗ ) is point-stationary when it is invariant under all bijective point maps. Theorem 13.3.IX. If the measure P on B(NX#∗ ) corresponds to a stationary point process on X = Rd , then its associated Palm measure R on B(N0#∗ ) is point-stationary. Conversely, if the σ-finite measure R on B(N0#∗ ) is unaligned, satisfies the normalizing condition (13.3.5), and is invariant under the family of point shifts {S ni } associated with the sets Bni from a nested sequence of symmetric tilings {Tn } satisfying Tn → 0, then it is the Palm measure associated with an unaligned stationary point process. Proof. Start by appealing to the general relations between stationary random measures and their associated Palm measures that are embodied in equation (13.2.6) and its specializations (13.2.10) and (13.2.11). We have to verify that if R is the Palm measure for a stationary point process on X = Rd and #∗ h(·) a bounded, measurable, nonnegative function on NX , the quasi Palm expectation operator ER defined by ER [h(N )] ≡ N #∗ h(N ) R(dN ) satisfies 0

ER [h(S Ψ N )] = ER [h(N )] for any bijective point map Ψ.

(13.3.22)

13.3.


313

Writing j(x) = IUd (x) for the indicator function of the unit cube in Rd , (13.2.10b) yields j(x)h{S Ψ (Sx N )} N (dx) P(dN ) ER [h(S Ψ N )] = = = =

#∗ NX

#∗ NX

#∗ NX

=

#∗ NX

#∗ NX

X

X

j(x)h{SΨ(Sx N,0) (Sx N )} N (dx) P(dN )

by (13.3.19),

j(x)h{SΨ(N,x)−x (Sx N )} N (dx) P(dN )

by (13.3.16),

X

X

X

j(x)h{SΨ(N,x) N )} N (dx) P(dN )

because S−x Sx N = N,

j Ψ−1 (N, y) h(Sy N ) N (dy) P(dN )

setting y = Ψ(N, x).

Because Ψ is bijective, so is Ψ−1 . Returning to the basic relation (13.2.6), and using Fubini’s theorem, the last term in the above chain can be evaluated as h(N ) j Ψ−1 (N, y) − y dy R(dN ) = ER [h(N )]. N0#∗

X

Thus, (13.3.22) holds and the direct part is proved. To prove the converse part, call a family {Ψni } of bijective point maps distinctive if, for fixed n, N ∈ N0#∗ , and x ∈ N , the unions r Ψrni (N, x) are disjoint for distinct i = 1, 2, . . . , and exhaustive if δΨrni (N,0) (·) (every N ∈ N0#∗ ) (13.3.23) N (·) = lim n→∞

i,r

[here, Ψr denotes the r-fold product as below (13.3.18), else see Exercise 13.3.10]. The last condition implies that there are sufficiently many point maps to distinguish the points in any realization of a simple point process. The rest of the proof consists of two stages. (1◦ ) If R is σ-finite, and invariant under a family of bijective point shifts which is both distinctive and exhaustive, then it is the Palm measure of some stationary measure. Let Ψ be a bijective point map from the specified family, and define its cycle length relative to the realization N ∈ N0#∗ by CΨ (N ) = inf{r: Ψr (N, 0) = 0}. For any given value k of the cycle length, for the successive points within that cycle, invariance under Ψ implies that for any nonnegative measurable function f (N, x), f S Ψ N, −Ψ(N, 0) R(dN ) = f N, Ψ−1 (N, 0) R(dN ). {CΨ =k}

{CΨ =k}

(13.3.24)

314

13. Palm Theory

For it is easy to check that the length of the cycle is invariant under S Ψ , and −1 applying S Ψ to the terms in the left-hand side leads directly to the form in the right-hand side [see Exercise 13.3.10(d)]. The same equation holds also for each of the iterates of Ψ, because the length of the cycle is not affected, and the remaining features again follow from properties of Ψ−1 . Still continuing with Ψ fixed, let NΨ denote the (reduced) realization, derived from N ∈ N0#∗ , and comprising the points {Ψr (N, 0): r = 1, . . . , CΨ }. Applying (13.3.24) to each of the iterates in turn, yields for each k, f (Sx N, −x) NΨ (dx) R(dN ) {CΨ =k}

X

= {CΨ =k}

X

f (N, x) NΨ (dx) R(dN ).

This result holds for all values of the cycle length, including the case k = ∞, so we can amalgamate the above equations for all k > 0 (omitting the zero iterate to avoid duplication) and obtain f (Sx N, −x) (NΨ − δ0 )(dx) = E0 f (N, x) (NΨ − δ0 )(dx) . E0 X

X

(13.3.25) We next amalgamate these equations also over the mappings Ψni , holding n fixed, and summing over i. Because the family is distinctive, and the origin is excluded, there are no overlaps. Let Nn denote the point process obtained by amalgamating all the points in all the (NΨni − δ0 ) for i = 1, 2, . . . , and including the origin just once. Then (13.3.25) holds in the form f (Sx N, −x) Nn (dx) = E0 f (N, x) Nn (dx) . E0 X

X

Finally, because the family is exhaustive, we can let n → ∞, so that Nn → N0 for each N0 , leading to E0 f (Sx N, −x) N (dx) = E0 f (N, x) N (dx) . X

X

It now follows from Theorem 13.2.VIII that R is the Palm measure on N0#∗ for some stationary measure; R is a probability measure when the normalization condition at (13.3.6) holds. (2◦ ) The family of B-selective bijective point maps Ψni derived from the sets {Bni } of the symmetric tilings Tn is both distinctive and exhaustive over the set of unaligned elements of N0#∗ . Because the Ψni are self-inverse, they are also idempotent, so that in considering the powers Ψrni required by the distinctiveness condition it is sufficient to take r = 1. In this case the condition merely requires the points xB and xB identified by distinct B and B to be

13.3.


315

distinct; this follows trivially from the fact the subsets in a given level of the tiling are by definition disjoint. In considering exhaustiveness, the crucial point here is that when the point process is unaligned, S ni ≡ S Ψni either leaves the point at the origin invariant, or interchanges the point at the origin with the unique point in Bni . Under these circumstances, every point in the realization N0 will ultimately be exchanged with the origin by one of the S ni , without altering other points in the configuration. Thus it will ultimately appear in the sum at (13.3.23). Note that this will not be the case if the realization contains equi-spaced sequences of more than two collinear points, for then at least one point pair x1 , −x1 will be repeated in every set Bni containing at least one of the two points, sending the value of Ψni to 0, and hence ensuring that x1 never appears in the union. The converse is proved. The complete version of this theorem in Heveling and Last (i.e., without the restriction to unaligned processes) allows an equivalence statement for point processes in Rd analogous to that for point processes on the line given by Proposition 13.3.II as follows (see their paper for proof). Theorem 13.3.X. There is a one–to–one correspondence between the distributions of simple stationary point processes in Rd (i.e., on NX#∗ with X = Rd ) and point-stationary measures on N0#∗ satisfying the normalization conditions P{N = ∅} = 0 and (13.3.9b). In fact even the restriction to probability measures can be dropped, as in the discussion around Theorem 13.2.VIII, and the statement presented in symmetric form between σ-finite stationary measures on NX#∗ and σ-finite point-stationary measures on N0#∗ .

Exercises and Complements to Section 13.3 Note: Assume below that N is nonnull, that is, P{N = ∅} = 0. 13.3.1 Variants of Theorems 13.3.I–II. (a) [cf. (13.3.2) and (13.2.6)]. Substitute g(x, N ) = h(x, N )k(x, N ) with k(·) as in (13.3.3) to show that

t (N ) 1

EP (h(t1 (N ), St1 (N ) N )) = mEP0

0

h(x, N0 ) dx .

0

(b) Use (13.3.2) to show that m = EP0 (t1 (N0 )), and hence write (13.3.2) as

t (N ) 1

EP (g(N )) = EP0

0

g(St N0 ) dt 0

EP0 (t1 (N0 )).

13.3.2 For simple stationary N use (13.3.2) to find the joint distribution of X = t1 (N ) and τ1 ≡ t1 (N ) − t0 (N ) in terms of the d.f. F (·) of τ1 . In particular, show the following.

316

13. Palm Theory (a) The joint distribution for (X, τ1 ) has a density function representation f (u, v) du dv =

mv −1 (1 − F (v)) du dv

(0 ≤ u ≤ v),

0

(u > v).

(b) The conditional distribution of X given τ1 is uniform on (0, τ1 ). (c) The conditional distribution of X given the whole sequence {τn : n= 0, ±1, . . .} is uniform on (0, τ1 ). [Hint: Let A ∈ B(NR#∗ ) belong to the sub-σ-field generated by the τi and start from the definition of conditional expectation, so that for measurable h, τ E(h(X)IA | σ{τi }) = IA ({τi }) 0 1 h(x) dx.] 13.3.3 Use (13.3.2) and Exercises 13.3.1–2 to provide an alternative derivation of the formulae in Exercise 3.4.4 for Q(Bk ) and Pr(Bk ). 13.3.4 (a) Define k(x, N ) = 1 if x = x∗ (N ), = 0 otherwise. Verify that k(·) so defined satisfies (13.2.9) and establish (13.3.9) from (13.2.14) as in the derivation of (13.3.2). (b) For a simple stationary point process in Rd with mean rate m, use (13.3.9) to show that E[(V0 (N ))] = E[ V0 (N ) dx] = 1/m. 13.3.5 Supply an alternative derivation of (13.3.11) from (13.3.3) and (13.3.5). n − P0 → 0, 13.3.6 The conclusion of Proposition 13.3.V can be strengthened to P n is the measure on N #∗ induced by the conditional probabilities where P 0 EP (f (Sx∗ (N ) N ) | N (An ) > 0) and convergence is with respect to the variation norm. [Hint: The basic inequalities depend on f only through f and hence hold uniformly in f for f ≤ 1 say.]

13.3.7 Use the inversion equation (13.3.9) to deduce that under the conditions of Theorem 13.3.IV, P{N (An ) > 1} = o((An )) ≡ (An )o(1) and P{N (An ) = 1} = m(An )(1 + o(1)) [cf. Theorem 1.2.12 of Franken et al. (1981)]. For simple stationary point processes on R as in Theorem 13.3.I, the analogue of (13.3.9) is t1 (N0 )/2 g(St N0 ) dt . EP (g(N )) = mEP0 t−1 (N0 )/2

13.3.8 Under the assumptions that N is stationary, simple, and has finite mean rate m, show that in Theorem 13.3.IV there is a nested sequence {An } with (An ) → 0, not satisfying (13.3.10), for which (13.3.11) fails. Investigate whether (13.3.11) holds with (13.3.10) but without {An } being nested, and whether (13.3.12) holds for more general sets An than spheres. [Hint: Let the realizations span a lattice, and let An be the union of two small spheres with centres at two lattice points.] 13.3.9 Consider a Poisson process in R1 with a point at 0. Suppose that the origin is shifted to the point of the process closest to the origin. Show that the shifted point process cannot be Poisson. [Hint: The interval between the new origin and the old is shorter than the other interval with an endpoint at the old origin, so these intervals cannot be independently distributed (Thorisson, 2000, Example 9.2.1).]

13.4.

MPPs, Ergodic Theorems, and Convergence to Equilibrium

317

13.3.10 Properties of point maps. (a) Products. Given two point maps Ψ1 and Ψ2 , define the product map Ψ2 ◦ Ψ1 as below (13.3.18). Give a proof or counterexample to decide whether the product is (i) associative, and (ii) commutative. [Hint: Mecke (1975, §2) asserts that it is associative and distributive but not commutative.] If Ψ1 and Ψ2 are bijective, check that Ψ2 ◦ Ψ1 is bijective. When Ψ1 is bijective, can Ψ2 ◦ Ψ1 be bijective without Ψ2 being bijective? (b) Inverse. The inverse Ψ−1 of a bijective point map is well defined [see further below (13.3.18)]. More generally, given a point map Ψ, define Φ: NX#∗ ×X → X , written Φ(N, z) = x, to be any solution x of z = Ψ(N, x). If there is a unique solution, check whether Φ is (i) covariant, (ii) a point map, (iii) bijective. −1

(c) When Ψ−1 is well defined, show that S Ψ (S Ψ N ) = S0 N = N . (d) For bijective Ψ, show that Ψ(S Ψ N, 0) = −Ψ−1 (N, 0) [cf. equation (4.12) of Heveling and Last (2005)]. (e) When the cycle length CΨ (N ) as above (13.3.24) is finite and equal to k say, verify that Ψr (N, x) = (Ψ−1 )(k−r) (N, x), where Ψr = Ψ ◦ · · · ◦ Ψ denotes the r-fold product map of Ψ.

13.4. Marked Point Processes, Ergodic Theorems, and Convergence to Equilibrium In this section we further examine the role of the stationary Palm distribution P0 , especially in questions related to ergodicity and convergence to equilibrium. Before doing so we outline briefly the extensions of the previous theory to marked random measures and MPPs. Initially we state the results for general random measures; in the examples and development subsequent to Example 13.4(b) we focus on MPPs. For convenience we generally assume that the ground process of any MPP has finite mean rate, and is simple. We first examine the extension of the Campbell theory results to the marked case. As already mentioned in Section 13.1, application of the Radon– Nikodym derivative approach leads to families of local Palm distributions P(x,κ) indexed by an element (x, κ) in the product space X × K. The Campbell measure itself becomes a marked Campbell measure on Borel sets of the space X × K × Ω, where Ω = M# X ×K in the canonical framework. When the MPP is stationary, the arguments of Section 12.2 show that the local families of Palm measures are invariant under shifts in the location, but the dependence on the mark remains. Thus we obtain a family of stationary Palm distributions P(0,κ) on B(M# X ×K ) which in the point process case can be interpreted as the behaviour of the process conditional on the occurrence of a point at the origin with mark κ. The other important ingredient in the marked case is the stationary mark distribution. This appeared first as the nonnormalized measure ν(·) on K introduced in the discussion of marked random measures above Lemma 12.2.III.

318

13. Palm Theory

When the ground process has finite mean rate mg , ν is a totally finite measure, and we can write E[ξ(A × K)] = (A) ν(K) = mg (A) π(K), where π is the stationary mark distribution. In this situation the reduced Campbell measure itself takes the form C˘P (dξ × dκ) = ν(dκ) P(0,κ) (dξ) = mg π(dκ) P(0,κ) (dξ).

(13.4.1)

The arguments are outlined in Exercise 13.4.1. In the ergodic theorems of Sections 12.2 and 12.6, in particular Theorem 12.6.VI, the limits of products of random measures averaged over increasing sets are related to the reduced moment measures. Proposition 13.4.I below develops similar results for more general functionals of the random measure, with the limit identified as an integral against C˘P . The proposition is stated for marked random measures; the special case of MPPs is examined in detail in the discussion around Proposition 13.4.IV. Results for the unmarked case follow by letting the functional g be independent of κ and integrating over κ. Proposition 13.4.I. Let ξ be a strictly stationary, ergodic, marked random ), finite ground measure on Rd × K, with probability measure P on B(M# Rd ×K rate mg , stationary mark distribution π, and reduced Campbell measure (i.e., Palm measure) C˘P . Let g(ξ, κ) be a B(M# X × K)-measurable nonnegative # function on MX × K. Then for any convex averaging sequence {An }, as n → ∞, 1 g(Sx ξ, κ) ξ(dx × dκ) → g(ψ, κ) C˘P (dψ × dκ) P-a.s. (An ) An K M# X = mg g(ψ, κ) P(0,κ) (dψ) π(dκ). (13.4.2a) K

M# X

Proof. The result is an extension of the individual ergodic Theorem 12.2.II and the approximation arguments used in deriving Theorem 12.2.IV. As in the latter theorem, we give details mainly for the unmarked case. Suppose first that g is a nonnegative measurable function on M# X satisfying ˘ # g(ξ) C (dξ) < ∞ P-a.s., and introduce the function P MX g(Su ξ) g (u) ξ(du), f (ξ) = X

where g (·) is a continuous function ‘close’ to a δ-function and integrating to 1. As in the proof of Theorem 12.2.IV, we find f (Sx ξ) dx = g (y − x) g(Sy ξ) ξ(dy) dx An

≥

X

X

An

g (y − x)

A− n

g(Sy ξ) ξ(dy) dx =

A− n

f (Sx ξ) dx.

13.4.


319

From Theorem 12.2.II, using also (13.2.5), we have, as n → ∞, 1 f (Sx ξ) dx → E f (ξ) = g (u) du g(ψ) C˘P (dψ) (An ) An X M# X = g(ψ) C˘P (dψ), M# X

with similar results if An is replaced by An or A− n . Because is arbitrary, (13.4.2) follows when the limit is finite P-a.s. If not, then replace g(·) by an increasing sequence of functions {gr (·)} for which gr (ξ) ↑ g(ξ) (r → ∞), as, for example, gr (ξ) = min g(ξ), rα(ξ) , where α(·) is as in the discussion below (13.2.4). Then for every r, 1 g(Sx ξ) ξ(dx) ≥ gr (ξ) C˘P (dξ), lim inf n→∞ (An ) An M# X and the right-hand side → ∞ as r → ∞. For thegeneral marked case we start from a nonnegative function g(ξ, κ) satisfying M# g(ξ, κ) C˘P (dξ × dκ) < ∞ P-a.s., and introduce the function X ×K

f (ξ) = X ×K

g(Su ξ, κ) g (u) ξ(du × dκ).

The rest of the proof then proceeds much as for the unmarked case. Note that for g(ξ) a function of the realization ξ only, (13.4.1) yields 1 g(Sx ξ) ξ(dx) → g(ψ) C˘P (dψ × dκ) P-a.s. (An ) An K M# X = mg g(ψ) P 0 (dψ), (13.4.2b) M# X

where P 0 is not the Palm distribution of the ground process, but rather the averaged form K P(0,κ) (·) π(dκ) ≡ P 0 (·). The difference arises because the realizations ξ here are for a marked process, whereas the realizations of the ground process are unmarked. In the point process case the measure P 0 can be interpreted as the Palm distribution dependent on the occurrence of an arbitrary point (i.e., a point with an arbitrary mark) at the origin. The example below illustrates how marks can affect ergodic limits in some simple cases. Example 13.4(a) Ergodic limits for processes with independent and unpredictable marks. Consider first the case of a stationary marked Poisson process with nonnegative marks. Suppose the underlying Poisson process has intensity µ and the marks have distribution function F (·). Because of the total lack of memory, the Palm distributions P(0,κ) are independent of κ and all reduce to the distribution of the original marked Poisson process, with mg = µ and

320

13. Palm Theory

ν(dx) = µ π(dx) = µ dF (x). Let the function g be the indicator function of the event Γ that TM > τ , where TM is the time from the origin to the first point of the process with mark greater than M . Then the ergodic limit on the right-hand side of (13.4.2) becomes P(0,κ) (Γ) π(dκ) = µ exp{−µ[1 − F (M )]τ }. (13.4.3) µ K

Note that the function g here does not depend on the mark κ at the origin. Also, because, in approaching the ergodic limit, averages are taken over all points of the process, irrespective of the mark, the constant preceding the exponential in (13.4.3) is µ, and not µ[1−F (M )]. The latter rate could arise in situations that allow g to depend explicitly on κ. Suppose, for example, we required the mark at the origin to be greater than M before a contribution to Γ could be counted, so that g(ξ, κ) = IΓ (ξ) I(M,∞) (κ); the limit on the right-hand side of (13.4.3) then becomes µ[1 − F (M )] exp{−µ[1 − F (M )]τ }. In this case we are just looking at the time intervals between points where the mark exceeds M , and the limit has the form we would expect from a Poisson process with rate µ[1 − F (M )]. The situation changes significantly if we allow extensions to processes with unpredictable marks. For example, consider a process of independent exponential intervals, in which the length of the interval following a point with mark κ is exponentially distributed with mean κ. In this case the ground process is a renewal process in which successive intervals are i.i.d. with common d.f. given by the mixture distribution ∞ e−x/κ dF (κ) G(x) = 1 − ∞

0

with mean 0 κ F (dκ), which we again denote by 1/µ. The Palm distribution P(0,κ) now depends crucially on the mark κ at the origin, inasmuch as the time to the next point of the process (i.e., the first interval X1 ), is exponential with mean κ, and the remaining intervals X2 , X3 , . . . are i.i.d. with d.f. G. With no other constraints, and the same event Γ as before, P(0,κ) (Γ) can be evaluated as the sum of the probabilities Pr{X1 > τ, κ1 > M } + Pr{X1 < τ, X1 + X2 > τ, κ1 < M, κ2 > M } + · · · , where κ1 , κ2 , . . . are the marks of the successive points following the origin, and the special distribution of X1 must be observed. If we look at the ergodic limit, still with g = IΓ , the dependence on the value of the initial mark is lost. If the further constraint is added that the initial mark κ must exceed M before a contribution to Γ is counted, the initial factor µ on the right-hand side of (13.4.3) must first be multiplied by the rate factor 1 − F (M ), as in the previous case, but the calculation of P(0,κ) (Γ) must be modified to allow for the fact that the distribution of X1 is now constrained by the requirement κ > M . Some further details and examples are given in Exercise 13.4.2.

13.4.


321

Higher moment results analogous to Proposition 13.2.VI can be developed for the marked case as special cases of the above proposition, but are relatively more complex. Reducing the second moment measure leads to a measure on X × K × K which can also be described as a first-order moment ˚1 (du × dκ1 × dκ2 ) = measure of the Palm distribution, taking the form M −1 ˘ mg M2 (du × dκ1 × dκ2 ). The relation here is between an initial point of the process taken as origin and a second point of the process, (0, κ1 ) and (u, κ2 ) say. If we standardize the second-order reduced measure by dividing by the reduced second moment measure for the ground process, we obtain a bivariate distribution π2 (dκ1 × dκ2 | u) for the marks at 0, u respectively, conditioned by the occurrence of points at 0 and at u (see Lemma 8.3.III). Then we can write ˚1g (du) π2 (dκ1 × dκ2 | u), ˚1 (du × dκ1 × dκ2 ) = M (13.4.4) M ˚1g is the first moment measure for the Palm distribution of the ground where M process. As noted in the discussion of Lemma 8.3.III and Example 8.3(e), the bivariate distribution π2 need not have marginals which reduce to the stationary distribution π; indeed, the distribution need not even be symmetric (see Exercise 13.4.3). The reason for the discrepancy is that we are not merely conditioning on a point with a given mark at the origin, but on the occurrence of two points, one at a specified distance from the other, and this additional conditioning alters the distributions in general. Next we formulate extensions of the last proposition to the nonergodic case; these are summarized in the next two results, for which proofs are outlined in Exercise 13.4.5. To avoid notational confusion, we write ω for the element of the probability space even if this is the canonical space, and E(· | I)(ω) for conditional expectations with respect to the σ-algebra I of invariant events. Lemma 13.4.II. Let P be the distribution of a stationary marked random measure on Rd × K, C˘P its reduced (marked) Campbell measure, and I the σ-algebra of invariant events under shifts in X = Rd . Then there exists an invariant random measure ζ(· , · | ω), defined on sets in B(M# X ×K ), such that ˘ for nonnegative and CP -integrable functions g(x, κ, ξ), " " " g(x, κ, Sx ξ) ξ(dx × dκ) " I (ω) = dx E X

with

X

M# X

g(x, κ, ψ) ζ(dψ × dκ | ω) (13.4.5a)

E ζ(dκ × dψ) = C˘P (dκ × dψ) = ν(dκ) P(0,κ) (dψ).

(13.4.5b)

In particular, (13.4.5a) with g = IA×K×Γ , A ∈ BX , K ∈ K, Γ ∈ B(M# X ×K ) yields " " 1 E ζ(K × Γ | ω) = IΓ (Sx ξ) ξ(dx, K) "" I (ω) (A) A

((A) > 0).

322

13. Palm Theory

The next result establishes convergence of the sample path averages to the invariant random measure ζ. Note that ζ here is a refinement of the random measure ψ of Lemma 12.2.III. Both ζ and ψ are a.s. totally finite with random total mass Y as defined around (12.2.10 ). The measure ψ is the marginal measure of ζ on K after integrating out ξ, as in ζ(K × dξ) (K ∈ BK ; Y = ψ(K)). ψ(K) = M# X

Theorem 13.4.III. Let ξ be a strictly stationary marked random measure on Rd ×K, ζ the invariant random measure defined in Lemma 13.4.II, and with ˘ X = Rd , let h(·, ·) be a B(M# X ×K )-measurable nonnegative or CP -integrable # function on MX ×K . Then for any convex averaging sequence {An } and n → ∞, 1 h(Sx ξ, κ) ξ(dx × dκ, ω) → h(ψ, κ) ζ(dψ × dκ | ω) P-a.s. (An ) An M# X ×K (13.4.6a) ) and K ∈ BK , In particular, for h(ψ, κ) = IΓ×K (ψ, κ) with Γ ∈ B(M# X ×K 1 IΓ×K (Sx ξ, κ) ξ(dx × dκ, ω) → ζ(Γ × K | ω) P-a.s. (13.4.6b) (An ) An Notice that the random measure ζ(·) is associated with the reduced Campbell measure (i.e., the Palm measure) rather than the Palm distribution. Thus, in combining limits over different ergodic components, it is the conditional Palm measures, rather than the normalized Palm distributions, that combine linearly according to (13.4.5b). In considering ergodic results, it may seem more natural to combine the normalized limits in a linear manner, as suggested by Sigman (1995); this leads to the ‘alternative Palm distribution’ described in Exercise 13.4.6, where some consequences and elementary results are given. The next example illustrates the point. Example 13.4(b) A mixed Poisson process. Consider two stationary Poisson processes on R with parameters λ, µ (λ = µ), selected with probabilities p and q = 1 − p, respectively. The invariant σ-field contains two nontrivial events J1 , J2 say, corresponding to the choice of the λ and µ processes, respectively. Let Γ = {N (0, a] = 0} for some fixed a > 0. Then Sx Γ = {N (x, x + a] = 0}, and E IΓ (ξ) = p e−λa + q e−µa , whereas

ζ(Γ, ω) = λe−λa IJ1 (ω) + µe−µa IJ2 (ω),

in which the indicator variables have multipliers representing the average rate of occurrence of points followed by an empty interval of length a. The measure C˘P (Γ) represents the overall average rate of occurrence of such points, weighted according to the probabilities of the two components, and so it is

13.4.


given by

C˘P (Γ) = pλe−λa + qµe−µa ,

323

(13.4.7a)

corresponding to (13.4.5a). The overall Palm probability of Γ equals ! (13.4.7b) (pλe−λa + qµe−µa ) (pλ + qµ), the denominator representing the overall rate of occurrence of points. It can be interpreted as the limit of the conditional probability that the next point will occur after lapse of time a, given that a point occurs in a small interval about the origin. This expression is to be contrasted with the situation for the Palm distributions: the Palm probability of Γ on J1 is e−λa and on J2 it is e−µa , but the overall Palm probability of Γ is not p e−λa + q e−µa as might have been expected. See Exercise 13.4.6 for the latter probability. For simple ergodic point processes, the previous theorems, coupled with the interpretation of the Palm distribution as the distribution of the process at an ‘arbitrary point’, lead to a circle of important results on convergence to equilibrium. Specifically, starting from an ‘arbitrary point’ (i.e., with the Palm distribution for a simple point process) and translating through some x ∈ Rd , we seek conditions under which the translated distributions converge to the stationary distribution as x → ∞. Dually, starting from an ‘arbitrary location’ (i.e., the stationary distribution) and observing the process at the nth point nearest to that location, we seek conditions under which the distribution of the process, relative to that nth point as origin, converges to the Palm distribution as n → ∞. To approach these ideas for MPPs, we first establish some relevant notation. # For MPPs, the Palm measure P(0,κ) (·) is supported by the subspace, N(0,κ) say, of marked counting measures having a point with mark κ at the origin. The averaged form, P 0 (·) = P(0,κ) (·) π(dκ) (13.4.8) K

which we call the mean Palm distribution, occurs frequently in ergodic limits. # of It can be interpreted as a measure on the subspace N0K# = κ∈K N(0,κ) marked counting measures on Rd with a point at the origin whose mark κ ∈ K there is unspecified. We use N K to denote a generic marked counting measure with marks in K, NgK to denote its associated ground counting measure, N0K to denote a marked counting measure having a point with unspecified mark from K at the origin, and N(0,κ) to denote a marked counting measure with mark κ at the origin. Using this notation, inversion theorems for MPPs on Rd corresponding to the results in Section 13.3 can be developed. Thus (13.3.1) can be extended to MPPs on Rd : for bounded, nonnegative functions h(·) of N0K , 1 K EP(0,κ) [h(N(0,κ) )] π(dκ) = EP h(S N ) , EP [h(N0K )] = xi 0 i:xi ∈Ud mg K (13.4.9)

324

13. Palm Theory

where {xi ≡ xi (N K )} denotes some measurable enumeration of the points of the realization N K . The analogue of (13.3.2) for Rd is (13.3.9b), which for MPPs becomes, for bounded nonnegative functions g(·) of N K , EP(0,κ) g[Sx N(0,κ) ] dx π(dκ) EP [g(N K )] = mg K

= mg EP

0

V0 (N(0,κ) )

V0 (N0K )

g[Sx N0K ] dx

(13.4.10) ,

where, for example, V0 (N0K ) denotes the Voronoi polygon about the origin in Rd formed by the locations (points of the ground process) of the realization N0K . Exercise 13.4.1 sketches arguments that justify these inversion formulae. Using these representations and notations, we obtain the following combination of results from Section 12.2 and Proposition 13.4.I. Proposition 13.4.IV. Let P be the distribution of a stationary ergodic MPP on Rd with marks in K, finite ground rate mg , and stationary mark distribution π(·); let P(0,κ) be the associated family of stationary Palm distributions, and P 0 the mean Palm distribution defined at (13.4.8). If {An } is a convex averaging sequence in Rd satisfying (An )/(An+1 ) → 1 (n → ∞), and h(·) a bounded, measurable, nonnegative function of N K , then 1 h Sx N0K dx → EP h(N K ) (n → ∞) P 0 -a.s. (13.4.11) (An ) An Furthermore, if x∗j ≡ x∗j (N K ) as above (13.3.7) and g(·) is a bounded, measurable, nonnegative function of N0K , then

n 1 g[Sx∗j (N K )] → EP g(N0K ) (n → ∞) 0 n j=1

P-a.s.

(13.4.12)

Proof. We start by considering (13.4.12). In the present context of MPPs and with {An } a sequence of spheres, (13.4.2) of Proposition 13.4.I can be put in the form 1 (An )

NgK (An )

g[Sx∗j (N K )] → mg EP g(N0K ) 0

P-a.s.,

(13.4.13)

j=1

because the locations of points in An are precisely those of NgK with modulus rj satisfying rj = x∗j ≤ radius(An ). Taking g ≡ 1, it follows that NgK (An )/(An ) → mg , so in place of (13.4.13) we can write 1 NgK (An )

NgK (An )

g(Sx∗j N K ) → EP g(N0 ) 0

j=1

P-a.s.

(13.4.14)

13.4.


325

We can suppose that the {An } have been so chosen that (An )/(An+1 )→ 1 as n → ∞, in which case we have ! P-a.s. NgK (An ) NgK (An+1 ) → 1 For brevity, write gj = g(Sx∗j N ), and Nk = NgK (An(k) ), where n(k) is chosen to satisfy x∗k ∈ (An(k)+1 \ An(k) ). Then the inequalities 1

Nk

Nk+1

j=1

gj ≤

Nk+1 Nk 1 1 gj ≤ gj Nk j=1 Nk j=1

show that the limit relation (13.4.14) implies (13.4.12). Equation (13.4.11) has the same form as (12.2.6) in the ergodic case, with one notable exception: (12.2.6) holds outside a set of counting measures with P-measure zero, whereas in (13.4.11) the corresponding result is required to hold outside a subset of N0K , which itself has P-measure zero. Thus we cannot immediately draw conclusions from (12.2.6). To get over this difficulty, let Γ be the subset of NX# (X = Rd × K) on which (13.4.11) holds and Γ0 its restriction to N0K . From (13.4.9), taking h as the indicator of Γ0 , 1 P 0 (Γ0 ) = EP mg

K (Ud ) Ng

IΓ0 Sx∗i (N K )

i=1

1 EP IΓ0 Sx (N ) N (dx) = mg d U 1 IΓ (Sx N ) N (dx) P(dN ) = mg Γ U d 0

(13.4.15)

because Γ has P-measure one. Now Γ is invariant, so Sx N ∈ Γ whenever N ∈ Γ. But if Sx N ∈ Γ0 , as required by the indicator function IΓ0 in the integrand, we must have N ({x}) = 1, and so the last line in (13.4.15) can be rewritten EP N (Ud ) 1 d = 1. P 0 (Γ0 ) = N (U ) P(dN ) = m Γ m This establishes (13.4.11), the set of P 0 -measure zero being taken as the relative complement of Γ0 in P 0 . In the nonergodic case, the argument leading to (13.4.15) can be extended to show that P and P 0 induce the same measures on the invariant σ-algebra I; see Exercise 13.4.7. Equation (13.4.11) of Proposition 13.4.IV can be interpreted as an a.s. statement about convergence to equilibrium. To put the result in the context of Section 12.5, take expectations with respect to P 0 and P in equations (13.4.11) and (13.4.12), respectively, and apply the dominated convergence theorem. Then conclude as follows.

326

13. Palm Theory

Corollary 13.4.V. Suppose that the conditions of Proposition 13.4.IV hold, with g and h bounded. Then 1 (An ) and

An

EP h(Sx N0 ) dx → EP h(N ) 0

k 1 EP g(Sx∗j N ) → EP g(N0 ) 0 k j=1

(n → ∞),

(k → ∞).

(13.4.16)

(13.4.17)

Notice that although the left-hand side of (13.4.17) converges to the Palm distribution in the ergodic case, this is not so in the nonergodic case, when it converges rather to the ‘alternative Palm distribution’ of Exercise 13.4.6. Equation (13.4.16) can be interpreted as asserting weak convergence of the measures (1/(An )) An S+x P 0 dx to the limit measure P. In the onedimensional context, if we consider intervals An = [0, n) and project onto the half-line R+ (see Exercise 13.4.8 for details), this is nothing other than the weak (C, 1)-asymptotic stationarity of the measure P0 . Corollary 12.6.VIII then implies that P0 is also strongly (C, 1)-asymptotically stationary, with the same limit measure P. Because the weak and strong versions coincide, we call such a process simply (C, 1)-asymptotically stationary. To develop a similar interpretation of equation (13.4.17), first recall that, in the one-dimensional case, every stationary MPP can be associated with a stationary marked interval process, and vice versa, where by a marked interval process we mean a sequence of pairs {(τi , κi−1 )}, with τi the length of the ith interval and κi−1 the mark associated with its left-hand endpoint. As in the discussion in Section 13.3, the probability distribution of a marked interval process can be treated either as a distribution on the space of sequences {(τi , κi−1 )}, or as a distribution on the subspace N0# of MPPs with a point at the origin. Note that if the sequence of pairs is stationary, then its distribution on N0# is the mean Palm distribution P 0 for some stationary MPP with measure P related to P 0 by (13.4.9–10). Note also that for a stationary MPP, the initial interval (0, t1 (N )) has the stationary interval distribution for the ground process, and the initial mark κ1 has a form of stationary mark distribution which is different in general from the stationary mark distribution π. There is also a stationary joint distribution for the initial interval and the initial mark; see Exercise 13.4.9, which also sketches out a proof of the stationarity results summarized above. The concepts of coupling and shift-coupling, as of weak and strong (C, 1)asymptotic stationarity, can be developed for an interval process, whether simple or marked, just as easily as, and in a parallel fashion to, those for a point process. As for point processes, the concepts of weak and strong (C, 1)asymptotic stationarity coincide, and can be referred to simply as (C, 1)asymptotic stationarity. Now (13.4.17) can be interpreted as asserting that, if we start an interval process from the first (or indeed the kth) point following

13.4.


327

the origin in a stationary MPP, that interval process is (C, 1)-asymptotically stationary with limit process equal to the stationary interval process associated with the original MPP. Sigman (1995) calls a point process N ‘event-stationary’ when the sequence of intervals initiated by t1 (N ) is interval-stationary. In this terminology, the MPP corresponding to P 0 is event-stationary. In general, a stationary point process will not be event-stationary in this sense, but Corollary 13.4.V implies that on R+ , a stationary MPP is (C, 1)-asymptotically event-stationary, and conversely that an event-stationary MPP is (C, 1)-asymptotically stationary. Finally in this circle of ideas, these last conclusions can be extended to processes that are not themselves stationary, but only asymptotically stationary, as summarized below. Proposition 13.4.VI. Suppose that the MPP N K is either stationary or (C, 1)-asymptotically stationary with limit measure P. Then the interval process, starting from t1 (N ) as origin, is (C, 1)-asymptotically stationary with limit measure P 0 associated with P through (13.4.9–10). Conversely, suppose that an MPP N0K defined on N0# represents an interval process which is either stationary or (C, 1)-asymptotically stationary, with limit measure P 0 . Then N0K is (C, 1)-asymptotically stationary with limit measure P associated with P 0 through (13.4.9–10). Proof. We start from the observation that if two MPPs N and N shiftcouple, then the same is true of the associated interval processes N0 and N0 , started at t1 (N ) and t1 (N ), respectively, and vice versa. Indeed, if , N of the MPPs such that there are stopping times T, T and versions N N (t + T ), N (t + T ) are a.s. equal for t ≥ 0, then the corresponding interval 0 and N are equal after discrete times J = N (T ) + 1, J = processes, say N 0 + 1 and so the interval processes shift-couple. Conversely, if the two inN terval processes, corresponding to MPPs N0 , N0 , shift-couple, with coupling times J, J respectively, then N0 and N0 shift-couple as point processes with coupling times T = tJ (N ), T = tJ (N ), respectively. Suppose now that N is (C, 1)-asymptotically stationary with limit process N having distribution P, so that N and N shift-couple. Then the corresponding interval processes, which we may associate with point processes N0 and N0 , also shift-couple, so that N0 is (C, 1)-asymptotically stationary with limit process N0 . This last process is not itself stationary, but corresponds to the process on the left-hand side of (13.4.16), which by Corollary 13.4.V is (C, 1)-asymptotically interval stationary with limit the stationary interval process, N0 say, associated with the stationary process N ; thus N0 shiftcouples to N0 . The transitivity of shift-coupling now implies that, as interval processes, N0 shift-couples to N0 and is therefore (C, 1)-asymptotically stationary with limit process N0 . But we already know that the distribution of a stationary point process is associated with the distribution of the corresponding stationary interval process through equations (13.4.9–10), and so the first statement of the

328

13. Palm Theory

proposition follows. The second statement follows by an analogous argument with the roles of the point and interval processes reversed. The last proposition finds applications in queueing theory and related fields, where it provides a starting point for a systematic approach to results on convergence to equilibrium, and the forms of the resulting stationary distributions. Accounts are given in Franken et al. (1981), Baccelli and Brémaud (1994), and Sigman (1995), and we do not attempt to repeat the material here. However, to give the flavour of the applications, we indicate how they apply to regenerative processes. Example 13.4(c) Regenerative processes via embeddings. Call an MPP N K on R+ , with internal history H, regenerative if the sequence of event times {ti : i ≥ 1} includes an embedded renewal process. More precisely, we require the existence of a subsequence {τj = tij : j = 1, 2, . . .} (the regeneration points) such that (1) for j ≥ 1, the intervals τj+1 −τj form an i.i.d. family with proper d.f. F (x) (and in the sequel, we suppose that this distribution has a finite mean, so that the corresponding renewal process has a stationary version); and (2) between regeneration points, the successive families of marked points N (j) = {(0, κij ), (t1+ij − tij ), κ1+ij , (t2+ij − tij ), κ2+ij , . . . , (tij+1 − tij ), κij+1 } are i.i.d. versions of a finite MPP on X = [0, ∞) × K. To bring such processes within the purview of the previous theory, we first regard the sequence {(τj , N (j) )} as a marked renewal process with i.i.d. marks N (j) , treating the latter as random variables on the portmanteau space X ∪ of Chapter 5 [equation (5.3.10)]. If the intervals in the renewal process have finite mean length, the resultant renewal process is stationary, and (having i.i.d. marks) so is the associated marked renewal process. The first interval may be exceptional (i.e., the renewal process may be a delayed renewal process) but in any case the process is asymptotically stationary, hence (C, 1)asymptotically stationary. The stationary mark distribution is nothing other than the common distribution, say Pf , of the finite MPPs N (j) . These remarks imply not only that the embedded marked renewal process is asympotically stationary, but also that the original MPP is asymptotically stationary. The underlying reason is that any bounded functional of the MPP can be represented, in terms of the sequence of renewal times and the i.i.d. sequence N (j) , as a bounded functional of the marked renewal process. Hence the original MPP is stationary (expectations of bounded functionals invariant under shifts) if and only if the latter is stationary. It follows also that if the original MPP is started from a regeneration point, it is asymptotically stationary; that is, there is a shift-coupling with the asymptotic form of the original MPP. Moreover, if we take any stopping time T , defined in terms of the initial finite process N (0) , the process started at T also shift-couples to

13.4.


329

the limit process (modify the shift). Thus the original MPP is asymptotically stationary if started from any such T , in particular, from any of the points of the MPP in the initial cycle. A more explicit argument, exhibiting the form of the ergodic limit in terms of the N (j) is outlined by Sigman (1995, Section 2.6). For another approach use the ideas of Lemma 12.5.III, taking the initial condition to be the history of the current process N (j) up to the time-point chosen as origin. As in other situations of this kind, although these general results are useful for discussing the convergence to stationarity and existence of stationary versions of the particular process in view, the hard step is the evaluation of the form of the stationary distribution, which in the present case means evaluating the explicit form of the distribution of the i.i.d. components N (j) . If, for example, the regenerative process is taken to be the arrival of a customer at an empty queue in an M/G/1 queueing system, and the original MPP is the number of customers waiting in the queue at the arrival of a new customer, explicit arguments are still needed to evaluate the form of the stationary distribution of the queue size. Although the results have been stated for regenerative processes, the independence of the cycles has been used only tangentially. The essential points of the argument are the existence of an embedded sequence of points which are (C, 1)-asymptotically stationary, and a law of large numbers for the behaviour within cycles. For extensions in this direction see Sigman for references. A stronger form of convergence holds when the process is mixing, and we can drop the (C, 1)-averaging that stems from the ergodic theorem. For a brief statement of such results, we return to unmarked processes, leaving the reader to formulate extensions. Proposition 13.4.VII. Let P be the distribution of a simple stationary mixing point process on Rd , with finite density, and let P0 be the corresponding Palm distribution. Then S+x P0 → P

weakly

(x → ∞).

(13.4.18)

Proof. To establish weak convergence we need to show that for all bounded continuous f on M# X, EP0 f (Sx N0 ) → EP f (N ) as x → ∞. Proposition 13.3.V is a convenient starting point for the proof. From that result, for any given > 0 and sufficiently small sphere An , " " "EP f (Sx N0 ) − EP f (Sx Sx∗ (N ) N ) | N (An ) > 0 " < 1 (13.4.19) 0 2 for each fixed x ∈ X [replace f (·) by f (Sx ·) in (13.3.14)]. Inspection of the proof of (13.3.14) shows that the inequality is uniform in x (see Exercise 13.3.6) because the two critical inequalities used in its proof depend only on

330

13. Palm Theory

(13.3.11) and sup |f (·)|, both of which are independent of x. Consequently, we can fix n from (13.4.19) and proceed by simply evaluating the difference " " "EP f (Sx Sx∗ (N ) N ) | N (An ) > 0 − EP f (N ) " " " "EP f (Sx Sx∗ (N ) N )I{N (A )>0} (N ) − EP f (N ) EP I{N (A )>0} (N ) " n n . = P{N (An ) > 0} Now it is enough to apply the result of Exercise 12.3.5 with X(N ) = f (Sx∗ (N ) N ),

Y (N ) = I{N (An )>0} (N )

to deduce that this difference can also be made less than sufficiently large.

1 2

by taking x

To illustrate the close connection between these results and the classical renewal theorems, we prove a result for mixing second-order processes that Delasnerie (1977) attributes to Neveu. It asserts that for large x, the reduced second moment measure approximates its form under a Poisson process. Theorem 13.4.VIII. Let N be a stationary second-order point process in ˘ 2 (·). If N is mixing Rd with density m and reduced second moment measure M then as x → ∞, ˘ 2 (·) →w m2 (·). (13.4.20) Sx M Proof. The formal connection here lies with the representation of the reduced moment measures as moments of the Palm distribution. However, we do not need to call on the Palm theory as such; it is enough to use the definition of the reduced moment measure, which yields, with b∗ (x) = b(−x) as in Chapter 12 and functions a(·), b(·) that vanish outside a bounded set, ˘ 2 (dv) a(u − x)b(v) N (du) N (dv) = (a ∗ b∗ )(v − x) M E X X X ˘ 2 (dv). (a ∗ b∗ )(v) Sx M = X

Now when N is mixing, the first expectation converges as x → ∞ to 2 2 m a(u) du b(v) dv = m (a ∗ b∗ )(v) dv, X

X

X

which by letting a(·) run through boundedcontinuous functions ensures the ˘ 2 )(·) to b ∗ m2 (·) , from which a standard weak convergence of b ∗ (Sx M sandwich argument yields (13.4.20). This result assumes a more familiar form in the case that d = 1 (i.e., X = R) when expressed in terms of the expectation function U (·) introduced in Theorem 3.5.III. Then, for a second-order stationary simple point process on R we have " U (x) = 1 + lim EP N (0, x] " N (−h, 0] > 0 h↓0 ! ˘ 2 (0, x] m, = 1 + EP0 N (0, x] = 1 + M leading to the following corollary to Theorem 13.4.VIII.

13.4.


331

Corollary 13.4.IX (Generalized Blackwell Theorem). Let N be a simple stationary mixing point process on R with finite second moment measure. For all h > 0, U (x + h) − U (x) → mh (x → ∞). Because a renewal process with m < ∞ has finite second moment measure and is mixing if its lifetime distribution F is nonlattice, this corollary includes the standard version of Blackwell’s theorem (Theorem 4.4.1) as a special case. However, the simplicity of the above argument as compared with the intricacies of Chapter 4 is misleading, because what is obscured here is the fact that to prove that a renewal process is mixing, a result close to Blackwell’s theorem must be assumed. For this case, therefore, the corollary would then become the conclusion of a somewhat circular argument. More generally, there is no very simple relation between mixing of the basic process and mixing of the sequence of intervals, in contrast to the case of ergodicity for which the concepts coincide (see Exercise 13.4.9). For the Wold process, similar questions regarding the lattice structure have to be overcome as in Chapter 4, and, additionally, the function U (·) refers to expectations when the initial interval has the stationary distribution. Consequently, further extensions are needed to cover the case of a process starting with an arbitrary distribution for the initial interval.

Exercises and Complements to Section 13.4 13.4.1 Reduced Campbell measure for marked random measures and MPPs. (a) The marked Campbell measure on X × K × M# X ×K is defined for A ∈ BX , K ∈ BK , and U ∈ B(M# X ×K ) by CP (A × K × U ) =

ξ(dx × dκ) P(dξ). U

A

K

Show that if P is stationary under shifts on X = Rd , CP is invariant under the actions of the transformations Θu (x, κ, ξ) = (x − u, κ, Su ξ). (b) Find a mapping on the triple product space analogous to D in (13.2.4), and show that when P corresponds to a stationary random marked measure ξ, the marked Campbell measure factorizes into a product of ˘P (dκ × dψ) Lebesgue measure on X and a reduced Campbell measure C defined by the following extension of (13.2.6),

g(x, κ, Sx ξ) ξ(dx×dκ)

E

=

dx

X ×K

X

#

K×MX ×K

˘P (dκ×dψ). g(x, κ, ψ) C

(c) If the ground process has finite mean rate mg , with associated stationary mark distribution π, show (using a disintegration argument) that the right-hand side of the above equation can be written

dx

mg X

π(dκ) K

#

MX ×K

g(x, κ, ψ) P(0,κ) (dψ),

where {P(0,κ) } is a family of stationary Palm distributions conditional on the mark κ.

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13. Palm Theory

13.4.2 MPP with exponentially distributed intervals. Consider the second situation in Example 13.4(a), where successive intervals are exponentially distributed with mean length equal to the mark κ of the point initiating the interval. Find a series expansion for P(0,κ) (Γ) as a function of M and κ. Is the averaged form P0∗ (Γ) smaller or larger than the its value in the first situation, where the intervals are i.i.d. exponential with the same mean µ as in the mixed case. Does a version of the renewal theorem hold for the mixed process? 13.4.3 A marked Gauss–Poisson process. Consider a stationary Gauss–Poisson process [see, e.g., Exercise 7.1.9(a)], in which the parent point has mark A and the offspring point, following it after a distance u with d.f. dF (u) = f (u) du (u ≥ 0), has mark a. If parent points occur as a stationary Poisson process with rate λ, find the reduced second-order moment densities (2) (2) (2) (2) mA,a (u), ma,A (u), mA,A (u), ma,a (u), and the bivariate (2 × 2) distribution (2)

(2)

(2)

π2 (κ1 , κ2 ). Show in particular that mA,a (u) = ma,A (−u) = ma,A (u). 13.4.4 Markov renewal process. Let X(t) be a stationary semi-Markov process on countable state space X, with counting function Ng (·), as in Example 10.3(a), and Hg (x) = E[Ng [0, x] | transition in X(t) at t = 0], = EP 0 (Ng [0, x]) = i πi j Hij (x) [cf. (13.4.8–9) and Example 10.3(a) for notation]. Use πk = x ∞ π i i j 0 Hij (du) x−u Gjk (dz) (all x > 0) to show that Hg (·) is subadditive; that is, Hg (x + y) ≤ Hg (x) + Hg (y) for x, y > 0. [Hint: Daley, Rolski, and Vesilo (2007) gives a proof.]

13.4.5 Ergodic properties of nonergodic MPPs. Mimic the steps in the proofs of Lemma 12.2.VI and Theorem 13.2.III to establish Lemma 13.4.II and Theorem 13.4.III. 13.4.6 Alternative version of Palm distribution [cf. Sigman (1995, Section 4.4)]. (i) Establish a form of Corollary 13.4.V for the nonergodic case, and hence verify the comment below the corollary. (ii) Apply this form to Example 13.4(b) and verify the comment at the end of the example. (iii) Find a Radon–Nikodym interpretation of this distribution [Sigman p. 65 quotes Nieuwenhuis (1994) and Thorisson (1995)]. [Hint: The basic link comes from the r.v. Y controlling the invariant distribution. For (i), use Exercise 13.4.4 to express the result of the corollary in terms of the invariant random measure. In particular, argue from (13.4.11) (in the unmarked case) that k 1 1 g(Stj N ) → ζ(dN | I), k j=1 Y where we have written ζ(· | I) to emphasize the dependence on the invariant σ-algebra. Now, following Sigman, write k 1 g(Stj N ) → E[g(N0 ) | I] = g(N0 ) P0 (dN | I) k j=1

for an ergodic measure relative to discrete shifts. Deduce that E[ζ(dN | I)] = (1/m)E[Y P0 (dN | I)]. ]

13.4.


333

13.4.7 Equivalence of invariant σ-algebras for P and P0 . If P is stationary and Γ is an invariant set in I with P(Γ) = p, then Γ0 = Γ ∩ N # (X0 ) is invariant in I0 with P0 (Γ0 ) = p. Deduce that a stationary point process N is ergodic if and only if the associated Palm distribution P0 is ergodic (e.g., when X = R, N is ergodic if and only if the corresponding interval process is ergodic). [Hint: Replace the expectation in (13.4.15) by the conditional expectation with respect to I; then the integral on the right-hand side of (13.4.15) is zero outside Γ. A similar argument holds inside Γ. To prove the converse, if Γ0 ∈ I0 , first put Γ = {N : Sx1 (N ) N ∈ Γ0 }, then use (13.2.9) with h the indicator of Γ. Otherwise put, ergodicity is equivalent to the requirement that all invariant sets have probability 0 or 1; now use Lemma 13.4.II, except that a converse to the lemma is needed: P0 (Γ0 ) = 1 implies the existence of Γ in B(NX# ), such that Γ is invariant with P(Γ) = 1 and Γ0 = Γ ∩ N0∗ .] 13.4.8 Let P be the distribution of a stationary point process on R+ for which P{N (R) = ∞} = 1. Restate (13.4.16–17) in terms of integrals on (0, tn ] with tn → ∞, where tn is the nth point of the process in R+ . 13.4.9 Stationary marked interval process. Consider a stationary sequence of pairs {(τi , κi−1 )}, as below (13.4.17), with stationary mean ground rate mg . Identify its distribution with the distribution of an MPP with a point at the origin with mark distributed according to the stationary distribution of marks in the original sequence. Then find a one-dimensional version of (13.4.10) to define the distribution of an MPP P and verify that it is stationary. Find the distribution of the time to the first point occurring after the origin, and of the mark distribution of that point. Construct an example to illustrate that this may differ from the stationary mark distribution. [Hint: Condition on the value of the mark of the point at the origin, which by assumption has the stationary mark distribution. For a counterexample, consider, e.g., an alternating renewal process. See also Sigman (1995, Section 3.4).] 13.4.10 Let the multivariate point process N (·) = (N0 (t), N1 (t), . . . , Nk (t)) in t > 0 be such that the successive points t00 ≤ 0 < t01 < t02 < · · · of the component process N0 are regeneration epochs for N (·) in the sense that, for ui ≥ 0, the conditional distributions of N (t + ui ) given t0j = t for some j ≥ 1 are independent of t, and the interval lengths {t0,j+1 − t0j } are i.i.d. r.v.s with density function f (·). Define H(t) = E(N1 (t) | t00 = 0, t01 = t),

V (t) = var (N1 (t) | t00 = 0, t01 = t).

Assuming the moments are finite, show that Ê∞ V (t)f (t) dt var (N1 (t)) , lim ≥ β ≡ Ê 0∞ t→∞ E(N1 (t)) H(t)f (t) dt 0 with equality holding if and only if H(t)/t is constant a.e. on the support of f (·). [See Berman (1978) for this and other results concerning such multivariate point processes that are regenerative in the sense of Smith (1955). Results concerning ergodicity and Palm distributions have been extended from regenerative processes to the context of marked point processes; there are details and several examples in Franken et al. (1981).]

334

13. Palm Theory

13.5. Cluster Iterates We turn next to the operation of iterated cluster formation, already mooted in Section 11.4, but postponed then because it makes essential use of Palm theory concepts. We consider clusters with mean size unity, but exclude the case P{N (X | x) = 1} = 1 (all x) inasmuch as this is the case of random translations already considered in Section 11.4. Standard results on branching processes [e.g., Harris (1963, Chapter 1)] imply that for critical branching processes (mean cluster size m = 1) the offspring from a given ancestor eventually become extinct, or in other words the iterated clusters eventually collapse to the zero measures. In such circumstances it may seem surprising that stable limit behaviour can occur. The explanation is to be found in the infinite character of the initial distribution, which allows local depletion to be perpetually replenished by immigration from more successful clusters in distant parts of the state space. The higher the dimension of the space, the greater the opportunities for such replenishment become, so that stable behaviour is the norm for d ≥ 3, whereas it is the exception for d = 1 or even d = 2. An earlier account may be found in Liemant, Matthes, and Wakolbinger (1988); Wakolbinger has several subsequent publications with various co-workers. The nature of the limiting behaviour is most easily understood by studying first the situation where the initial process is Poisson, with mean density equal to unity say. Any limit, although no longer Poisson, is still infinitely divisible, so the discussion can be phrased in terms of the convergence of the n to their limit (Proposition 11.2.II). Because associated KLM measures Q the successive Poisson cluster processes formed by iterating the clustering operation are also stationary, the discussion can be further reduced to the (n) associated with these KLM measures [see study of the Palm measures Q 0 Example 13.2(c), especially the discussion leading to Proposition 13.2.IX, and Exercise 13.2.10]. The essential question can now be phrased in terms of the (n) }, namely, find conditions on the cluster mechanism such Palm measures {Q 0 (n) (∞) say. that Q converges to some boundedly finite limit measure Q 0 0 The cluster mechanism itself can be conveniently specified in terms of (A) a distribution {πk : k ≥ 0} for the size of the cluster; and (B) a family of symmetric d.f.s Pk (dx1 × · · · × dxk ) (k ≥ 1) specifying the locations of the cluster members relative to the cluster centre at the origin. The assumptions m ≡ E{N (X | x)} = 1 and P{N (X | x) = 1} < 1 imply both ∞ ∞ πk = kπk and π0 > 0. 1= k=0

k=0

Also, because by assumption the cluster mechanisms are homogeneous in space, the locations relative to a cluster centre not at 0 are specified by the appropriately shifted versions of the Pk (·).

13.5.

Cluster Iterates

335

The KLM measure corresponding to the Poisson cluster process formed at the first stage of clustering is concentrated on the set of totally finite counting measures and allocates mass πk to those trajectories containing just k points. Because m = 1, it may be considered as a probability distribution on N0 (X ) in its own right. As in Example 13.2(c), the associated Palm measure is then defined in terms of a modified cluster structure, in which the cluster size is distributed according to {πk } = {kπk : k ≥ 0} (note that π0 = 0), and the locations of the cluster members are specified by placing one cluster member at the origin and distributing the remaining k − 1 points about it according to the symmetrized measures Pk dy × (y + A2 ) × · · · × (y + Ak ) . Pk−1 (A2 × · · · × Ak ) = X

Note that the Palm clusters considered here differ from the clusters arising in the regular representation (Propositions 12.1.V and 12.4.II) only through the relative weightings given to the different cluster sizes. Note also that the intensity measure for the underlying cluster process, given by ρ(dx) =

∞

kπk

k=1

X (k−1)

Pk (dx × dy2 × · · · × dyk ),

(13.5.1)

is here a probability measure on X , whereas the intensity measure for the Palm cluster process is given by ρ˜(dx) = δ0 (dx) +

∞ k=2

k(k − 1)πk

X (k−1)

Pk (y2 + dx) × dy2 × · · · × dyk .

Now consider the Palm cluster resulting from two stages of clustering. To ease the notation only, we use here density notation, with corresponding lower case symbols. First note that the quantity kπk pk (y, x2 + y, . . . , xk + y)

(13.5.2)

can be interpreted as the joint density of locating the parent (cluster centre) at −y and k − 1 siblings at x2 , . . . , xk , given one point of the cluster at the origin (cf. Exercise 1.2.5). The marginal density for the parent, given a point at the origin, is thus g(y) =

∞

kπk

· · · pk (y, x2 + y, . . . , xk + y) dx2 . . . dxk = ρ(y),

k=1

where we here write ρ(y) for the density of the intensity measure (13.5.1). The members of the two-stage Palm cluster can now be classified into three groups: first, the point located at the origin; second, its immediate siblings,

336

13. Palm Theory

jointly located with the cluster centre according to (13.5.2); and third, its ‘cousins’, found by locating a grandparent and a set of ‘uncles’ by (13.5.2), but given the parent at −y rather than at the origin, and then superposing the clusters generated by each of the uncles. In symbols, we may write briefly 2 = δ0 + N1 + N2 . N Evidently, this process, introduced by Kallenberg (1977a) and called by him the ‘method of backward trees’, can be continued. At each stage we move one generation further back, taking the location of what was previously the oldest ancestor as origin, locating the ancestor of next order and the siblings of the previously oldest ancestor by (13.5.2), then moving forward to add in to the current generation the superposition of clusters of appropriate order deriving from the siblings of the previously oldest ancestor. n developed in this way have a monotonic character, beThe processes N cause we can imagine them as defined on a common probability space and embedded into an indefinitely continued process of superposition of this kind. (n) (∞) Whether the Palm measures Q0 converge to some limit Q0 thus reduces to the question of whether this process of superposition produces in the limit an a.s. boundedly finite limit measure. Because each stage is formed from its predecessor by an independent operation representing a shift of locations and corresponding augmentation of the number of branches by the distribution (13.5.2), it follows from the Hewitt–Savage zero–one law that the limit is boundedly finite either with probability 1 or with probability 0. This dichotomy allows us to make the following definition. Definition 13.5.I. The cluster mechanism described by (A) and (B) is stan described above ble or unstable according as the sequence of processes N converges a.s. to a boundedly finite limit or diverges a.s. In complete generality, the problem of determining conditions that are necessary and sufficient for the stability of a given cluster mechanism appears to be still open. What is known is that the conditions are closely linked to the behaviour of the random walk with step-length distribution governed by the symmetrized form (13.5.3) σ = ρ ∗ ρ− , where ρ− (A) = ρ(−A), of the intensity measure for the clusters. To see how this measure arises, suppose that the mean square cluster size ∞ 2 k=1 k πk is finite; this ensures that the Palm clusters also have finite intensity, which we write in the form ρ˜ = δ0 + ρˆ, where ρˆ(A) =

∞

k−1 (A × X × · · · × X ), (k − 1)P

k=2

and we note ρˆ(X ) =

∞ k=2

k(k − 1)πk .

13.5.

Cluster Iterates

337

n+1 − N n between the Palm clusters at the Consider now the differences N (n + 1)th and nth stages of clustering. To obtain the intensity measure of this increment, we should start from ρn∗ − , representing the n steps taken to the left (i.e., according to ρ− ) to locate the position of the nth generation ancestor, convolve this with ρˆ to obtain the locations of that ancestor’s siblings, and finally convolve again with ρn∗ + to obtain the intensity measure of the superposition of the nth stage clusters generated by those siblings. Thus, n , we have if ρñ denotes the intensity measure for N ρñ+1 = ρñ + ρn∗ ˆ ∗ ρn∗ ñ + σ n∗ ∗ ρˆ; − ∗ρ + =ρ hence, ρñ = δ0 + ρˆ ∗ (δ0 + σ + σ 2∗ + · · · + σ (n−1)∗ ). The series on the right-hand side converges toward the renewal measure (boundedly finite or infinite) of the random walk with step-length distributions σ. It is boundedly finite if and only if the random walk is transient (see Exercise 9.1.11). On the other hand, it follows from the monotonic character n that they converge to a boundedly finite limit with boundedly finite of the N first moment measure if and only if the sequence ρñ so converges. We are thus led to the following result [see Liemant (1969, 1975)]. Proposition 13.5.II. A critical cluster member process with finite mean square cluster size is stable if and only if the random walk generated by the symmetrized intensity measure (13.5.3) is transient. Recall that random walks in three or more dimensions are necessarily transient, so that a properly three-dimensional cluster process with finite mean square cluster size is always stable. In one or two dimensions, however, a random walk is not necessarily transient: it is if the step distribution has nonzero mean (d = 1 or 2) or infinite variance (d = 2), so it is only under particular conditions that the associated cluster process can be stable. Kallenberg (1977a) gives further results concerning stability. In particular, for a process of Neyman–Scott type, transience of the random walk alone is necessary and sufficient for stability, but necessary and sufficient conditions for stability cannot be formulated in full generality solely in terms of conditions on the cluster size distribution and the cluster intensity measure. Granted that the cluster mechanism is stable, convergence to a limiting process can be established by arguments similar to those used in discussing (n) random translations. As above, write Q0 for the Palm measure corresponding to the Poisson cluster process formed after n stages of clustering from a Poisson process with unit rate, and suppose the clusters are stable, so that by hypothesis (n) (∞) (n → ∞) (weakly in N0# ) Q0 → Q0 (∞)

for some limit distribution Q0 . This convergence implies the corresponding assertion for the associated KLM measures (see Exercise 13.5.3), namely, ∞ n → Q Q

(n → ∞)

(weakly).

(13.5.4)

338

13. Palm Theory

(n) (∞) ∞ ) allots zero mass to the totally finite Unlike Q0 , Q0 (and hence also Q n+1 − counting measures. To see this, observe that the successive increments N Nn are independent and nonnegative, and that for positive constants c, c ,

n ≥ 1} ≥ cP{Zn > 0} ≥ c /n, n+1 − N P{N where {Zn } is a Galton–Watson branching process governed by the cluster size distribution {πk } (see Harris, 1963, Chapter 1). Because the sum of these terms diverges, it follows from the Borel–Cantelli lemmas that with probability 1, an infinite number of the events on the left-hand side occur. n (X ) is infinite a.s., which is equivalent to the assertion that Thus, limn→∞ N (∞) Q0 allots zero mass to the counting measures with finite total mass. Now the various Poisson cluster processes formed from the initial Poisson process all have unit rate, so their distributions on NX# are weakly relatively compact, and from (13.5.4) above and Proposition 11.2.II it follows that the limit of any weakly converging subsequence must be infinitely divisible with ∞ . This limit process must therefore be the overall weak KLM measure Q limit. Recalling that an infinitely divisible point process is singular if its KLM measure is supported by the counting measures with infinite total mass, we can assert the following result. Lemma 13.5.III. In the stable case, the Poisson cluster processes derived from an initial Poisson process of unit rate converge weakly to a limit point process that is stationary, singular infinitely divisible, and has KLM measure ∞ . Q It can be shown further that the limit process is actually mixing and therefore weakly singular (Fleischmann, 1978). Also, if we start from an initial Poisson process of rate λ, the Poisson cluster processes converge to a limit ∞ . point process that is infinitely divisible with KLM measure λQ Granted the Palm versions of the cluster iterates converge to the limit ∞ , we may raise more generally the question of convergence of the processes Q formed by successive clusterings from a general initial distribution. Because stability implies that the intensity measure ρ(·) of the cluster member process has at least two points in its support, we obtain an estimate for the p.g.fl. Gn [h] for the nth cluster iterate with cluster centre at the origin, namely, sup |Gn [Tx h] − 1| ≤ θn ≡ sup x

x

X

ρn∗ (x + dy) 1 − h(y)

(h ∈ V),

and the last quantity → 0 as n → ∞ by Lemma 11.4.I. Using this estimate, we can start the discussion along much the same lines as that of Theorem 11.4.II. As in that proof, the above estimate implies " " " " E "− log Gn [Tx h] N0 (dx) − (1 − Gn [Tx h]) N0 (dx) " → 0. X

X

13.5.

Cluster Iterates

339

If now Y is given by (11.4.10) and we write G∞ [h] for the putative limit of the terms in the exponent, namely, ∞ (dN # ), exp G∞ [h] = log h(x)N (dx) − 1 Q N0# (X )

X

then to complete the proof along the lines of Theorem 11.4.II we need to show that "2 " " " E "" (1 − Gn [Tx h]) N0 (dx) − Y G∞ [h]"" → 0. X

Now we have already established that the KLM measures of the Poisson clus ∞ , so we have further that ter process converge to Q (1 − Gn [Tx h]) dx → G∞ [h], X

and hence it is in fact sufficient to show that "2 " " " (1 − Gn [Tx h]) dx"" E "" (1 − Gn [Tx h]) N0 (dx) − Y X

X

→ 0.

(13.5.5)

At this point we meet a difficulty, because we do not have the detailed information concerning the asymptotic behaviour of the cluster p.g.fl.s 1 − Gn [tx h], in x and n, which would correspond to the information concerning the convolution powers 1 − X h(y) ρn∗ (x + dy) used to complete the proof of Theorem 11.4.II. Such information can in fact be obtained by the ‘method of reduced trees’ introduced by Fleischmann and Prehn (1974, 1975). The underlying idea here is that if the clustering survives a large number of generations, the offspring in the current generation come with high probability from a single line in the family tree, other lines having become extinct. In other words, the current offspring have a single common ancestor a few generations back, so that for all generations preceding that it is enough to track the positions of this single ancestor and its forebears. Each backward step in this reduced part of the tree corresponds to a step in a random walk governed by the distribution ρ(·), as discussed below (13.5.2). Hence, we may approximate the p.g.fl.s Gn [Tx h] in (13.5.5) by the corresponding p.g.fl.s X h(y) ρn∗ (x + dy) for the random translations process governed by the distribution ρ. Then we may refer to the proof of Theorem 11.4.II again to deduce that for this process, assuming ρ is nonlattice, the terms corresponding to those in (13.5.5) are asymptotically equal. These considerations lead to the limit result set out in Theorem 13.5.IV below, representing an extension of Theorem 11.4.II. For details of the argument, as well as extensions and strengthenings of the theorem and analogous results for subcritical branching mechanisms, see MKM (1978, Chapter 12) and the extended and updated version in MKM (1982, Chapter 10).

340

13. Palm Theory

Theorem 13.5.IV. Let N0 be a second-order stationary point process on X =Rd , and let {πk ,Pk } be a stable cluster mechanism in the sense of Definition ∞ denote the limiting KLM measure (13.5.4) asso13.5.I. Furthermore, let Q ciated with the iterates of the cluster mechanism and {Nn } the sequence of point processes derived from N0 by successive independent clusterings according to {πk , Pk }. If the intensity measure ρ for {πk , Pk } is nonlattice, then the sequence {Nn } converges weakly to the limit point process with p.g.fl. exp X log h(x) N (dx) − 1 Q∞ (dN ) , G[h] ≡ E exp Y N0# (X )

where Y = E[N0 (Ud ) | I].

Exercises and Complements to Section 13.5 13.5.1 Find the class of point processes invariant under the cluster operation of Theorem 13.5.IV (i.e., find an analogue of Corollary 11.4.III to Theorem 11.4.II for Theorem 13.5.IV). 13.5.2 Let the critical branching mechanism {πk , Pk } be neither stable nor a random translation. Show that the cluster iterates {Nn } starting from a stationary second-order point process N0 converge weakly to the zero point process. [Hint: Consider first the case where N0 is a stationary Poisson process.] 13.5.3 Complete the convergence results relating KLM measures and the Palm measures of n-stage Poisson cluster processes asserted in the text following Proposition 13.5.III. [Hint: See the discussion of weak convergence of KLM measures preceding Proposition 11.2.I, and link to Exercise 13.5.2.]

13.6. Fractal Dimensions Although the concept of a fractal set, and many related concepts associated with the names fractals and multifractals, were introduced principally in the work of Mandelbrot (1982), the fractal dimensions we consider here have their origins in earlier work of Rényi (1959), who at that time was studying generalizations of the concept of entropy. They are characteristics of a measure rather than a set. Their link to point processes might seem obscure, because a point process, as a measure, has the dimension of a single point, namely, zero. The connection arises through estimation procedures for fractal dimensions, for example, box-counting and Grassberger–Procaccia estimates, which can be and often are applied to a wide range of point process data. In this case it turns out, as we describe below, that the estimates can be related to the moment measures of the Palm distribution of the underlying point process. Indeed, in such contexts, the estimation procedures might well be better directed toward a careful study of these moment measures than to limit properties which have relatively limited application and are fraught with practical difficulties.

13.6.

Fractal Dimensions

341

The material for this section is based on Vere-Jones (1999). Falconer (1990) and Cutler (1991) provide basic references for the Rényi dimensions and the multifractal formalism, and Harte (2001) gives a broad overview of both concepts and statistical issues. The topic now comprises a major field in its own right, and has generated a huge literature; we consider here only the specific issues that arise in interpreting estimates of multifractal dimensions when the estimates are derived from point process data. The Rényi or multifractal dimensions Dq are defined for a measure µ on Rd as the limits q−1

log X µ Sδ (x) µ(dx) (13.6.1a) Dq (µ) = lim δ→0 (q − 1) log δ for q = 1, where Sδ (x) is a sphere radius δ, centre x. In the special case q = 1, D1 (µ) = lim

δ→0

X

log µ Sδ (x) µ(dx) , log δ

(13.6.1b)

a quantity sometimes called the entropy dimension. For measures that have a bounded density with respect to Lebesgue measure over some bounded set A, and vanish outside that set, the Rényi dimensions are all equal to the dimension d of the space on which the measure is defined. For singular measures, the situation can be more complicated. In the case of the Cantor measure, for example, the Rényi dimensions are still equal, but to a value less than the dimension of the space. Such a measure is unifractal. In still other examples, the growth rates of the measure may vary from point to point, resulting in different weights being given to the increment µ(dx) for different x; in such examples, the Rényi dimensions will vary with q, and the measure is described as multifractal. Simple variants on the Cantor measure which possess this property are outlined in Exercise 13.6.1. When µ is a probability measure over a bounded observation region A, one might attempt to ascertain the values of the Rényi dimensions empirically by generating i.i.d. observations according to µ, and replacing µ in (13.6.1) by the corresponding empirical distribution µ +(B) = N (B)/N (A). In this context, integrals of the type appearing in the numerators of (13.6.1) are referred to as correlation integrals [cf. Harte (2001)], although strictly speaking the term refers to the particular case q = 2. To keep to quantities that can be related to point process moment measures, we restrict our discussion to multifractal dimensions of positive integral order q = k ≥ 2. In such cases, a correlation integral which we denote by Ck (·) has a particularly simple interpretation, namely, k−1 µ Sδ (x) µ(dx) = Pr{Mk ≤ δ}, (13.6.2) Ck (δ, A, µ) = A

where

Mk = max X1 − Xk , X2 − Xk , . . . , Xk−1 − Xk

342

13. Palm Theory

and the Xj are i.i.d. with common distribution µ which vanishes outside A. It then follows (see Exercise 13.6.2) that the relation Dk (µ) = η is equivalent to the statement that, at the origin, the distribution of Mk has power-law growth ν of order ν = (k − 1)η, so that Pr{Mk ≤ δ} = φ(y) y for some function φ(y) with | log φ(y) | = o | log y| . In applications, empirical correlation integrals of the above kind may be calculated for many different types of data, and used to obtain estimates of some quantity purporting to be a fractal dimension. In circumstances when there is no obvious generating measure µ, however, it is not clear whether the quantities obtained empirically have any meaningful interpretation. The discussion which follows has the aim of elucidating this point in situations where the data are generated by a point process. We start by writing the correlation integral in (13.6.1a) in a form more convenient for calculations with general point processes. Let Ik,δ (·) denote the indicator function of the set Uk,δ = {x: max |xi − xk | ≤ δ}, i

so that

k−1 µ Sδ (x) µ(dx) = A

A(k)

Ik,δ (x) µ(k) (dx),

(13.6.3)

where x denotes a k-vector (x1 , . . . , xk ) with each component xi ∈ Rd . Replacing µ by the empirical measure µ + formed from the counting measure N (·) from a finite set of observations {x1 , . . . , xN (A) } over some bounded region A, +) which we write as we obtain Ck (δ, A, µ k−1 +k (δ, A) = µ + Sδ (u) C µ +(du) A 1 = Ik,δ (u1 , . . . , uk ) N (k) (du1 × · · · × duk ) [N (A)]k A(k) 1 Ik,δ (x∗1 , . . . , x∗k ), (13.6.4) = k [N (A)] perm the last sum being taken over all permutations of the points x1 , . . . , xN (A) of the realization in A taken k at a time. Points lying outside A are ignored, a convention which can lead to bias if the measure of interest has continued support outside A. Because we are concerned with the limits for small testing sets, such edge effects are generally of smaller order than the main terms, but their possible existence needs to be borne in mind. In practice, they can be important even when the size of the testing set is as small as 10% of the observation region; see Harte (2001), especially Chapters 9 and 10, for discussion and illustration of these and related aspects. For computations, it is convenient to consider contributions only from distinct pairs, triplets, and so on, taking advantage of the symmetries in the

13.6.

Fractal Dimensions

343

counting measure to reduce the number of terms to be considered. In the present case, the function Ik,δ (x1 , . . . , xk ) is symmetric in the first k − 1 arguments, so that each combination of one point of the realization for xk , and a set of k − 1 points for x1 , . . . , xk−1 , will be repeated (k − 1)! times, leading to the alternative estimate, a quasi factorial moment estimator, . N (A) N (A) − 1 ∗ ∗ + , (13.6.5) Ik,δ (x1 , . . . , xk ) N (A) C[k] (δ, A) = k−1 j=1 comb

where the inner sum is taken over distinct combinations of one term x∗j and k − 1 different terms {x∗1 , . . . , x∗k } \ {x∗j } from x1 , . . . , xN (A) . No combinations with repeated points from the realization appear in this representation, so that in taking expectations it should be written as an integral against the modified product counting measure N [k] used in defining the factorial moments (see Section 5.2). Another way of writing this last formula, which may help to show up its link to power-law growth, is outlined in Exercise 13.6.3. We describe two general situations where the estimates based on the correlation dimension estimates (13.6.4) and (13.6.5) do lead to consistent estimates of a multifractal dimension. Both situations relate to a space–time point process, both use the same estimates, but the estimates are embedded in different limit processes, and the quantities that they estimate are likewise different. In one case the process is stationary in time and observations accumulate over a fixed spatial region, whereas in the other, the process is homogeneous in space, and observations are considered over an expanding sequence of spatial sets, the time span being held fixed. In the first situation, observations accumulate ever more densely over a bounded spatial region, until in the limit their spatial distribution approximates that of the first spatial moment measure over that region. In this situation the estimates can be related to the fractal dimensions of the first spatial moment measure. In the second situation, although we do not accumulate information about density variations in any particular spatial subregion, we do start to collect information about the behaviour of groups of points at various relative distances from each other, leading to the possibility of estimating the power-law growth of the reduced moment measures when spatial homogeneity is assumed. The proposition below establishes consistency of the correlation integral estimates in these two cases. It provides a starting point for considering the limit behaviour of the multifractal dimension estimates themselves. We describe the process as space–time (cf. Section 15.4 below), this being a convenient nomenclature for the two components of the state space in which the point process exists. In part (b) the ‘time’ variable plays no role so mention of it has been omitted. Proposition 13.6.I. Let N be a simple space–time point process, with state space X = R × Rd , and k > 1 a positive integer. (a) Suppose that N is stationary and ergodic in time, and that its first spatial moment measure exists and has stationary spatial distribution µ over a

344

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given spatial set A. Then, for A fixed and time interval T → ∞, both +k (δ, A) → Ck (δ, A, µ) C

+[k] (δ, A) → Ck (δ, A, µ) and C

a.s. and in L1 -norm. (b) Suppose that N is stationary on X = Rd , and that its moment measures ˘ [k] (·) denote its reduced ordinary and fac˘ k (·), M exist up to order k. Let M ˚k−1 (·) and M ˚[k−1] (·) the corresponding Palm torial measures of order k, M moment measures, m the mean spatial density, {An ; n = 1, 2, . . .} a con(k) vex averaging sequence of sets in Rd , Sδ the set {x1 , . . . , xk : max |xi | ≤ [r] δ}, and with Nn = N (An ), Nn = Nn ! /(Nn − r)! . Then as n → ∞, +k∗ (δ, An ) ≡ Nnk−1 C +k (δ, An ) → m−1 M ˘ k (S (k−1) ) = M ˚k−1 (Sδ(k−1) ), C δ + ∗ (δ, An ) ≡ Nn[k−1] C +[k] (δ, An ) → m−1 M ˘ [k] (S (k−1) ) C [k] δ a.s. and in L1 norm. Proof. The proofs are exercises in using the ergodic theorems of Sections 12.2 and 12.6, and the link between reduced moment measures and the moment measures of the Palm distribution established in Section 13.4 and quoted in the proposition. For case (a) we need the extension of Proposition 12.2.IV to product integrals [see below (12.2.15)]. Write x for the k-vector (x1 . . . , xk ) as earlier, NT (·) for the projection of N (· × (0, T )) onto the spatial component Rd of X , and NT (A) for the total number of points in the observation region A. The proof of the following lemma is sketched in Exercise 13.6.4. Lemma 13.6.II. With the notation just given, suppose that the assumptions of Proposition 13.6.I(a) are satisfied and that h(·) is a µ(k) -integrable function on (Rd )(k) . Then as T → ∞ with region A fixed, 1 (k) k h(x) NT (dx) → m h(x) µ(k) (dx). (13.6.6) T k A(k) A(k) If in addition h is symmetric, then . NT (A) → h(xi ) h(x) µ(k) (dx), k (k) A

(13.6.7)

comb

where the sum is taken over all combinations xi of k distinct elements from the realization (x1 , . . . , xNT (A) ). To establish part (a) of the proposition, apply (13.6.6) twice, the first time with h = Ik,δ and the second time with h ≡ 1, and take the ratio. Comparing the resulting equations with the expressions (13.6.3) and (13.6.4) yields the first statement from part (a) of the proposition. The second statement from part (a) follows similarly from (13.6.5) by applying the variant of (13.6.2) which holds for functions with symmetry only in the first k − 1 arguments.

13.6.

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345

It is noteworthy here that the same limit is obtained whether or not we allow repeated indices in the sums over permutations of the sample elements. This is because the contributions from terms with multiple points, corresponding to the diagonal concentrations in the moment measures, are of lower order in N (or T ) than the terms from sets of distinct points. The argument for (b) rests on the higher-order ergodic Theorem 12.6.VI. Replacing the sets Bi in (12.6.10) by spheres Sδ ≡ Sδ (0) ⊂ Rd we obtain 1 (An )

An

k−1

N (x + Sδ ) N (dx) =

j=1

n 1 N (k−1) (xi + Sδ )(k−1) (An ) i=1

N

(k−1)

˘ k (S →M δ

).

(13.6.8)

This form also neglects edge effects, for it assumes that the process is observed not only within An but also within any parts of the translated spheres Txi Sδ which happen to fall outside An even for xi ∈ An . A sandwiching argument shows that such edge effects are asymptotically negligible provided () (An )/(An ) → 1, which we assume. in the middle term of (13.6.8) is just another way of expressing Now the sum ∗ ∗ perm Ik,δ (x1 , . . . , xk ) from (13.6.4). Rewriting this in terms of Ck (δ, An ), adjusting the scaling factor [only a single integral over An is involved in (13.6.8), whereas a multiple integral over (0, T )(k) is implicit in (13.6.4) and Lemma 13.6.II], and recalling that N (An )/(An ) → m, we obtain the first statement in part (b) of the proposition. For the second statement, we omit repeated points, thereby obtaining a representation in terms of the factorial product counting measures N ∗[r] whose expectation defines the factorial moments. In place of (13.6.8) above we start from Nn 1 ˘ [k] (S (k−1) ). N ∗[k−1] [(xi + Sδ )(k−1) ] → M (13.6.9) δ (An ) i=1 The left-hand side of (13.6.9) can be rewritten as Nn (k − 1)! I[k,δ] (x∗1 , . . . , x∗k ). (An ) i=1 comb

Replacing (An ) by Nn , the result is of the same form as (13.6.5) except for [k−1] . Incorporating this factor and rewriting the expression in the factor Nn terms of the sum in (13.6.5) completes the proof2 of (b). As already noted, the limits in (b) can equally well be written in terms of ˘ k (S (k−1) ), repre˚k−1 (Sδ(k−1) ) = m−1 M moments of the Palm probabilities M δ senting the moment measures (of one order lower) for the process conditioned 2 Note two errata in Vere-Jones (1999): (a) the last term in equation (35), should start with 1/(q − 1)! not (q − 1)! , and (b) the scaling factor in equations (37) and (41) should read (Nn − 1) . . . (Nn − k + 1) and not (Nn − 1)! or [N (X ) − 1]! .

346

13. Palm Theory

on a point at the origin. This interpretation is a natural one in the present context because the construction is based on the maxima Mk [see below (13.6.3)] which already singles out one point of the group as a local origin. A range of different estimates based on the correlation integrals have been proposed and their behaviour analyzed in different contexts. For example, the simplest, na¨ıve, estimate is based directly on the definition (13.6.1a) and takes the form, for k = 2, 3, . . . , + + [k] (δ, A, T ) = log C[k] (δ, A) . D (k − 1) log δ

(13.6.10)

Its major drawback in practice is that the behaviour is unreliable when δ is small because of measurement error and lack of data within very small spheres; on the other hand for large δ it is likely to be significantly biased by edge effects, which we have ignored in the discussion above, and by any departures from simple power-law growth as the test region expands. To avoid at least some of these difficulties, Grassberger and Procaccia (1983) suggested replacing (13.6.10) by + + + ∗ (δ, A, T ) = log C[k] (δ1 , A) − log C[k] (δ2 , A) , D [k] (k − 1)[log δ1 − log δ2 ]

(13.6.11)

where the interval (δ1 , δ2 ) is chosen essentially by inspection, to obtain a portion of the graph of (13.6.10) linear in δ, and taking δ1 as small as seems reasonable. Let us note in passing, however, that although the Grassberger–Procaccia and similar procedures may establish the existence of power-law growth over a certain distance range, something which may well be of importance in its own right, it is another matter to assert that the power-law index for this range necessarily coincides with the limiting value at vanishingly small distances. In applications, the difficulty of distinguishing the two situations has been one factor leading to confusion over whether fractal behaviour refers generally to power-law growth, or specifically to the limiting behaviour near zero. Another approach, developed by Mikosch and Wang (1995), is based on the Hill (1975) estimate; it uses extreme quantiles to estimate the limiting powerlaw growth. Mikosch and Wang also advocate use of a bootstrap method to estimate the dimension estimates and their confidence bounds. The Hill method is broadly similar to the approach of Takens (1985) which also bases the estimate on the behaviour in the extreme tail. The methods are reviewed and compared in Harte (1998, 2001) where the use of the Hill method is illustrated in practical situations in which bias from measurement error and finite boundary effects cannot be ignored. Whatever form of estimate is adopted, it is a further nontrivial exercise to establish conditions for consistency of the estimates. Because the fractal dimension is itself defined as a limit, a double limit problem is involved: as

13.6.

Fractal Dimensions

347

either T or N (T ) → ∞, and as δ → 0. In general, the maximum rate at which δ → 0 will be constrained by the rate at which the study region expands in time or space. A slightly unusual form of limit process is required, in which one dimension shrinks to zero and the other expands to infinity. Results for the na¨ıve estimate (13.6.10) are summarized in the theorem below, and in Exercises 13.6.5–6. Consistency of a Grassberger–Procaccia type of estimate, for the index of power-law growth over a predetermined range, is more easily established on the basis of Proposition 13.6.I; an outline is given in Exercise 13.6.7. Theorem 13.6.III. (a) Suppose that the conditions of Proposition 13.6.I(a) hold, that the moment measures for the space–time process N exist up to order 2k, and that the stationary space distribution µ has kth fractal dimension Dk (µ). Defining + [k] (δ, A, T ) , d}, + [k] (δ, A, T )+ = min{D D if the controlled diagonal growth conditions of Exercise 13.6.5 hold, then + [k] (δ, A, T )+ is a mean-square consistent estimate of Dk (µ). D (b) Suppose that the conditions of Proposition 13.6.I(b) hold, that the moment measures of N exist up to order 2k, and that the reduced factorial ˘ [k] is such that the limit moment measure M (k−1)

˘ [k] (S log M ) δ δ→0 (k − 1) log δ

∗ D[k] ≡ lim

(13.6.12)

exists and is finite. Setting + ∗ (δ, A, T )+ = min D [k]

+ ∗ (δ, A) log C [k] (k − 1) log δ

, d ,

if the bounded growth conditions of Exercise 13.6.6 hold, then + ∗ (δ, A, T )+ is a mean-square consistent estimate of D∗ . D [k] [k] Proof. We make a few comments only, referring to Vere-Jones (1999) and the Exercises 13.6.5–6 for details. The nature of the problems which arise is most easily illustrated by considering the expected value of the quantity which appears in the numerator of (13.6.5). Written out directly in terms of the modified product counting measure N [k] (·), which omits repeated points, it takes the form Ik,δ (x∗1 , . . . , x∗k ) N (k) (dx∗1 , . . . , dx∗k ), A(k)

where A is the region over which points are observed. In part (a), the spatial process in view is the ground process of the space– time process over the region A×(0, T ), time being treated as a mark and then

348

13. Palm Theory

ignored. Its expectation therefore reduces to the integral over [A × (0, T )](k) of the kth order factorial moment M[k] (·) of the space-time process: N (A) ∗ ∗ Ik,δ (x1 , . . . , xk ) E j=1 comb

= [A×(0,T )](k)

Ik,δ (x1 , . . . , xk ) M[k] (dx1 × dt1 × · · · × dxk × dtk ).

In case (a), we write M[k] (dx1 × dt1 × · · · × dxk × dtk ) =

k

M1 (dxi × dti ) + ∆(dx1 × dt1 × · · · × dxk × dtk ),

(13.6.13)

i=1

where in view of stationarity, M1 (dx × dt) = m µ(dx) dt, m being the mean rate of occurrence of points over the whole region A and µ being normalized to a probability measure. The product term is what we would expect if we were looking for dimension estimates of µ, whereas the term ∆(·), which in general consists of an amalgam of lower order factorial moment and cumulant measures, defines the error which must be controlled if consistent estimates are to be obtained. Indeed, the ‘controlled diagonal growth’ condition of Exercise 13.6.5 puts a bound on the growth of the expected value of the integral against ∆, and the ‘controlled bi-diagonal growth’ condition puts a bound on the growth of its variance. In part (b), the time variable plays no significant role, but the process is homogeneous in space. In this case, omitting the dt terms, we can write Ik,δ (x1 , . . . , xk ) M[k] (dx1 × · · · × dxk ) A(k) ˘ [k] (du1 × · · · × duk−1 ) = dxk I(max(|u1 |,...,|uk−1 | 0} with finite factorial moments ν[j] for j = 1, . . . , k. We consider the behaviour of the correlation integral (k = 2) and its estimates under various assumptions concerning the components. Consider first the behaviour under assumptions of Theorem 13.6.III(a), so that Λ(dx × dt) = λ dt θ(dx), where θ is a probability measure. After integrating out the time factor, we find for the ground (spatial) process (13.6.14) M1g (dx) = λT ν1 B(dx − y) θ(dy). Also for k = 2, the expansion (13.6.13) takes the particularly simple form [cf. equations (6.3.5), (6.3.17)], g g M[2] (dx1 × dx2 ) = M1 (dx1 ) M2 (dx2 ) + C[2] (dx1 × dx2 ) = (λT ν1 )2 θ(dx1 ) θ(dx2 ) + λT ν2 B(dx1 − y) B(dx2 − y) θ(dy). (13.6.15)

350

13. Palm Theory

Integrating against I2,δ we obtain, denoting the convolution of θ and b by θ ∗ B, g I2,δ (x1 , x2 ) M[2] (dx1 × dx2 ) = (λT ν1 )2 (θ ∗ B)[Sδ (x)] (θ ∗ B)(dx) A×A A B(dx1 − y) B(dx2 − y) θ(dy). (13.6.16) + λT ν2

A

Bearing in mind that θ is a probability distribution, the second term on the right-hand side can be written in the form

λT ν2 1 − (δ, A)

B[Sδ (x)] B(dx),

0 < (δ, A) < 1,

A

where the correction term becomes negligible as δ → 0. The order of growth of the terms in the right-hand side of (13.6.16) is T 2 δ D2 (θ∗B) for the first, and T δ D2 (B) for the second. Their relative behaviour can be very different under different assumptions. If both cluster centre and cluster member processes have bounded densities, then all terms grow like small areas (i.e., proportional to δ 2 ), so that the correlation dimension is 2. Moreover the second term in (13.6.16) is then O(T ) and the first term is O(T 2 ) so the first term dominates and the controlled growth conditions are satisfied even for δ = O(1/T ). If the clusters are dispersed, so that B has a bounded density b say, but the cluster centres are concentrated along a set of lines in two-dimensional space, then the correlation dimension of θ is 1, but the first moment measure of the point process still has a smooth density m1 (x) = λT ν1 b(x − y) θ(dy) corresponding to a correlation dimension of 2. This is typically the situation when observations are contaminated by spatial measurement errors, so the correlation integral grows as O(δ 2 ) until δ reaches the same order of magnitude as the limits of measurement error, or in our case the effective range of the distribution B, say δ0 . When δ is increased beyond δ0 , the linear concentration of θ starts to tell, and the correlation integral starts to grow as O(δ × δ0 ) [i.e., as O(δ)], corresponding to a fractal dimension estimate of 1 rather than 2. If the converse situation obtains, so that cluster centres are smoothly (say uniformly) distributed over the observation area, but the cluster members are distributed along a line, then the first moment measure is now uniform, corresponding to growth rate O(δ 2 ), but the first term may no longer dominate the expression (13.6.16). Its overall contribution is O(T 2 δ 2 ), whereas that of the second term is O(T δ). For fixed T , the correlation integral grows initially as O(δ) and then as O(δ 2 ), so it shows two regions of power-law growth. If we look for a consistent estimate, and take δ = O(T −(1+η) ), then the first term will be O(T −2η ) and the second term O(T −η ). In this case the bounded growth conditions break down, and the dimension estimate is dominated by the behaviour of the local clustering. If we take δ = O(T −(1−η) ), then the first

13.6.

Fractal Dimensions

351

term is O(T 2η ) and the second is O(T η ), so the first-order term dominates; in this case the clusters accumulate sufficiently thickly over the observation region that their local linear structure is no longer the dominating feature, and the dimension estimates revert to estimates of the first moment measure. In general, the more extreme the concentrations of the local clusters are, the more slowly δ will need to approach zero before the conditions of the theorem are satisfied. In such situations, the correlation integral may show a sequence of ranges, all with different power-law growths: which of these is picked out as the dimension estimate will depend on the rate at which δ approaches 0 as T → ∞ (see Exercise 13.6.8 for a more elaborate example). Finally consider the behaviour under the assumptions of part (b). The same basic equations hold, and the estimates are still controlled by the growth behaviour of the second factorial moment in (13.6.16). Ignoring the time coordinate, and recalling that the process is assumed spatially homogeneous, ˘ [2] (Sδ (0)), with the (13.6.16) directly gives the reduced moment measure M additional simplification that the first moment measure here averages out to a multiple of Lebegue measure; θ in the second term is again proportional to Lebesgue measure. Then the same issues arise, but in a slightly different form, because the question now is to determine the correlation dimension of (13.6.16) as a whole, meaning therefore the initial growth rate, from whichever term that rate happens to derive. In the examples we have just been considering, when B has a bounded density both terms are O(δ 2 ) so the correlation dimension is again 2. With locally linear clusters, it is the second term which will dominate for δ → 0 and so the dimension here will be 1. We see that for this particular example, the effect of changing from the assumptions in (a) to those in (b) is to switch attention from the growth rate of the first moment measure to the growth rate of whatever feature of the cluster structure dominates the behaviour in the initial range of power-law growth. Another commonly occurring and widely studied situation relates to processes generated by a dynamical system, where the measure µ of interest is an invariant measure for the process. The common examples are determinsistic in character, but can be randomized by introducing a random starting point. We do not look at this example in great detail, referring the reader rather to Cutler (1991), Serinko (1994), and the review in Harte (2001) for further discussion and references. Example 13.6(e) Point processes generated by a dynamical system [Serinko (1994)]. Let Θ be a measurable mapping taking the closed, bounded set A ∈ Rd into itself, and let µ be a totally finite invariant measure for Θ; we suppose µ normalized to form a probability measure. We consider the application of Theorem 13.6.III(a) to the point process formed by the sequence {xn } = {Θn x0 : n = 0, 1, . . . }. If the initial value x0 itself has distribution µ, then the sequence {xn } is stationary; we suppose that it is also ergodic. The process may also be regarded as a space–time point process with discrete time variable. The results of Proposition 13.6.I and Theorem 13.6.III(a) are not

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affected by the character of the time variable, because they all concern limits for large values of T . The task is then to find the moment measures of the resulting point process, and to verify the conditions of Theorem 13.6.III(a). For bounded measurable functions h(x) on A and h2 (x, y) on A(2) we can characterize the actions of the first and second moment measures of the process on A, up to time T , by means of the equations h(x) M1 (dx) = A

T k=1

k

h(Θ x) µ(dx) = T

A

h(x) µ(dx) A

and A(2)

h(x, y) M2 (dx × dy) =

k=1 =1

=T

T T

h2 (x, x) µ(dx) + A

T −1

h(Θk x, Θ x) µ(dx)

A

(T − r)

[h2 (x, Θr x) + h2 (Θr x, x)] µ(dx). A

r=1

To verify the bounded diagonal growth condition for M2 we should set h2 in the second equation to be the indicator I||x−y||≤δ and examine the behaviour of the integral as T → ∞ , δ → 0. Evidently, the critical feature will be the rate at which Θr x − x increases with r. If the rate is fast enough, the bounded growth condition will hold. For large r, we expect the distribution of Θr x to approximate µ, for µ-almost all x. The rate at which this occurs is governed by mixing conditions on Θ. When appropriate mixing conditions are satisfied, therefore, we expect the estimates to be consistent. Serinko (1994) gives details from a somewhat different point of view.

Exercises and Complements to Section 13.6 13.6.1 Multinomial measures. Let b be a positive integer, and consider a division of the unit interval into b successive subintervals of length 1/b, then a further subdivision of each such subinterval into b equal sub-subintervals, and so on. Starting with unit mass for the whole interval, at each stage of this process, divide the mass of any given interval among its component subintervals in b proportions {p1 , p2 , . . . , pb }, with r=1 pr = 1. At the nth stage of this process, the subinterval corresponding to the b-adic expansion 0.ω1 ω2 . . . ωn will have mass n j=1 pωj . Denote the corresponding probability distribution by µn . By considering the values of the distribution function at points with finite b-adic expansions, or otherwise, show that the Rényi dimension of order q of µn is approximately equal to − logb [ br=1 pqr ]/(q − 1), and converges to this value as n → ∞. Show also that as n → ∞, the measures µn converge weakly to a limit µ, and that the fractal dimensions converge to the fractal dimensions of the limit measure. Investigate conditions under which the Dq are equal. [See, e.g., Harte (2001, Chapter 3).]

13.6.

Fractal Dimensions

353

13.6.2 Prove the equivalence of the relations [see below (13.6.2)] Dk (µ) = η and Pr{Mk < y} = φ(y) y (k−1)η for some function φ(y) such that log φ(y)/ log y → 0 as y → 0. [Hint: To establish the basic link between Ck (δ, A, µ) and the distribution of Mk , condition on Xk and then take expectations.] 13.6.3 For a given set {x1 , . . . , xk } of k distinct points, let the function k

Nδ (x1 , . . . , xk ) = i=1

ISδ (xi − xj )

j=i

count the number of points of the set with the property that the remaining k − 1 points of the set lie within distance δ of the selected point. Show that in this notation, (13.6.5) can be written as

C[k] (δ, A) =

comb

Nδ (x∗1 , . . . , x∗k ) k

N (A) , k

the sum being taken over all distinct combinations (x∗1 , . . . , x∗k ) of observation points.

13.6.4 Prove Lemma 13.6.II, assuming first that h is of product form k1 hr (xr ) with each hr measurable and bounded on A, and extending via linear combinations to general h. [Hint: When h is a product, (13.6.6) is just the product of limits from the one-dimensional case. To derive (13.6.7) take ratios of (13.6.5) first for the given h and then for h ≡ 1. Because of the symmetry of h, each combination of distinct terms appears k! times, and contributions from repeated arguments are of lower order and can be neglected in the limit.] 13.6.5 Suppose N (· × ·) is an orderly space–time point process, stationary and ergodic with respect to time, observed over the bounded spatial region A, and with moment measures existing up to order 2k. In Theorem 13.6.III(a) let ∆k denote the signed measure ∆k = Mk − mk (µ × )(k) , ∆+ k its total variation, and let Vk,δ denote the restriction of the set Uk,δ defined in (13.6.3) to A(k) . Say that N has controlled diagonal growth of order k, if ∆k satisfies the condition (controlling the bias in replacing the kth moment measure by its first-order approximation) k (k−α) β δ ∆+ k [Vk,δ × (0, T ) ] ≤ CT

for some positive constants C, α, β and sufficiently small δ, 1/T . Furthermore, say that N has controlled bi-diagonal growth of order k if ∆2k satisfies the condition (controlling the growth of the variance) (2)

2k ∆+ 2k [Vk,δ × (0, T ) ] ≤ C T

(2k−2η) 2ν

δ

for positive constants C , η, ν and sufficiently small δ, 1/T . Show that if (i) the stationary distribution µ over A has kth order Rényi dimension Dk (µ), (ii) N has controlled diagonal and bi-diagonal growth of order k, with constants as above, and

354

13. Palm Theory (iii) δ is chosen to vary with T in such a way that δ → 0 but T r δ → ∞, where α η , r > max , (k − 1)Dk (µ) − β (k − 1)Dk (µ) − ν then the consistency result of Theorem 13.6.III(a) holds. [Hint: See Vere-Jones (1999, Proposition 2) for details.]

13.6.6 Let N (· × ·) be orderly, stationary (homogeneous), and ergodic with respect to space; let N be observed over a convex averaging sequence {An } of sets in X = Rd , and suppose its moment measures up to order 2k exist. In Theorem 13.6.III(b), let ∆∗2k denote the signed measure ∆∗2k = M[2k] −M[k] ×M[k] , and n the restriction of Uk,n defined in (13.6.18) to the set An . Say that N has Vk,δ controlled Palm growth of order k if ∆∗2k satisfies the condition (controlling the growth of the variance) (2)

(∆∗2k )+ (Vk,δ ) ≤ K(An )2k−2η δ 2ν for positive constants (K, η, ν) and sufficiently small δ, 1/n. Show that if ∗ of (13.6.12) exists, (i) the limit D[k] (ii) N has controlled Palm growth of order k, with constants as above, and (iii) δ is chosen to vary with T in such a way that δ → 0 but T r δ → ∞, where η , r> ∗ (k − 1)D[k] −ν then the consistency result of Theorem 13.6.III(b) holds. [Hint: See Vere-Jones (1999, Proposition 4) for details.] 13.6.7 Grassberger–Procaccia estimates. (a) Suppose the correlation integral C(k, δ, µ) of (13.6.2), where µ is the stationary spatial distribution under assumptions (a) of Proposition 13.6.I, shows power-law growth over a given interval (a, b), so that log C(k, b, µ) − log C(k, a, µ) = η, log b − log a say. Show that under assumptions (a), replacing C in the above expression by its sample counterpart (13.6.5) yields a consistent estimate of η. (b) Formulate and prove a similar result under assumptions (b). 13.6.8 Further Neyman–Scott examples. (a) Consider Example 13.6(d) but with homogeneous cluster centres and cluster structure determined by a spatial component B that has gamma distribution on a line with shape parameter α < 12 , so that the density has a singularity at 0. Show that if X, Y both have such a distribution, then X − Y also has a singularity at 0, with initial power-law growth 2α. Show that the correlation integral (13.6.31) has three ranges of power law growth, initially with δ = 2α, then with δ = 1, and finally with δ = 2. Investigate the consequences for the correlation dimension estimates. (b) Extend to R3 , and investigate also the behaviour when both cluster centre and cluster structure components have linear or planar concentrations.

CHAPTER 14

Evolutionary Processes and Predictability

14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8

Compensators and Martingales Campbell Measure and Predictability Conditional Intensities Filters and Likelihood Ratios A Central Limit Theorem Random Time Change Poisson Embedding and Existence Theorems Point Process Entropy and a Shannon–MacMillan Theorem

356 376 390 400 412 418 426 440

The ideas discussed in this chapter have transformed the study of point processes over the last few decades. They provide the background not only for the results on conditional intensities and likelihoods summarized in Chapter 7, but also for general theories of estimation, prediction and control that have been influential as much in the engineering as in the statistical communities. The introduction to Chapter 7 provides some references to the early literature. Last and Brandt (1995) give a thorough study of the theory for marked point processes; other recent texts covering related material include Asmussen (1987, 2003), Baccelli and Brémaud (1994, 2003) and Andersen et al. (1993). Broadly speaking, the chapter provides a setting for the functionals that arise in describing the evolution, or ‘dynamics’, of a simple or marked point process evolving in time. This setting embraces certain broad structural features of a point process embodied in the Doob–Meyer decomposition, Theorem A3.4.IX, which is a basic tool. A point process N (·) on R+ is equivalent to the nondecreasing function N (t) = N ((0, t]) for which it is plausible that we should be able to separate N (·) into a ‘generally’ increasing part A(·), its compensator (here, A comes from accroissement or ‘growing’ part), and the ‘unpredictable’ variable part M (·); that is, N (t) = A(t) + M (t). The essence of the Doob–Meyer result is that such a decomposition is possible with A(·) ‘predictable’ and M (·) a martingale. The notion of predictability which arises here is crucial for the development of a rigorous theory; it forms one part of the so-called ‘general theory of 355

356

14. Evolutionary Processes and Predictability

processes’, as set out in Dellacherie and Meyer (1978), for example. A brief outline of some key points is provided in Appendix 3. Section 14.1 gives a general introduction to the notions of compensator and point-process martingale, and their links to the Doob–Meyer theorem. In Section 14.2 we extend these concepts to random measures and marked point processes. Section 14.3 re-introduces the conditional intensity, which appears here as a Radon–Nikodym derivative of Campbell measure on Ω × R, when a certain absolute continuity condition is satisfied. Later sections discuss various applications, including the likelihood and time-change theorems treated more informally in Sections 7.2 and 7.4, a martingale-type central limit theorem, and the notion of entropy rate which lies behind the discussion in Section 7.6.

14.1. Compensators and Martingales The aim of this section is to introduce the basic ideas of the martingale approach to the study of point processes on the open half-line1 R+ 0 ≡ (0, ∞). Many of the technicalities are summarized in Appendix 3, so as to allow scope here to stress the connections with other aspects of point process theory. In much of the discussion we are concerned with a random measure on R+ 0 , for although the case of point processes is of paramount importance, the more general theory can be covered with little extra effort. Thus, general results are stated in terms of the cumulative process ξ(t) ≡ ξ(t, ω) ≡ ξ((0, t], ω) for some random measure ξ(·, ω) (we use ξ for both, but the abuse of notation should not lead to difficulties). Observe that such processes have trajectories that are a.s. monotonic increasing and right-continuous, as is true in particular for the counting processes N (t) ≡ N (t, ω) ≡ N ((0, t], ω) of a point process N (·, ω) on R+ . A more important extension, taken up in Section 14.2, is to multivariate and marked point processes. Because the mark may include a spatial location as well as a size or indicator variable, this extension also includes space–time processes. In such cases, it is necessary to consider, not just a single cumulative process, but a family of cumulative processes indexed by the bounded Borel sets of the mark space. To facilitate the discussion of martingale properties, we suppose throughout the chapter that, unless otherwise stated, the point processes and random measures in view have boundedly finite first moment measures, or finite mean rates in the case of a stationary process. The ‘information available at time t’ is represented mathematically by a σ-algebra Ft of sets from the underlying probability space (Ω, E, P). The 1

We use R+ = [0, ∞) and R+ 0 = (0, ∞) to distinguish the closed and open half-lines.

14.1.

Compensators and Martingales

357

accumulation of information with time is reflected in Ft being a member of an increasing family F = {Fs : 0 ≤ s < ∞} of σ-algebras; that is, Fs ⊆ Ft for 0 ≤ s ≤ t < ∞. F is called a history for the process ξ provided ξ(t) is Ft -measurable for all 0 ≤ t < ∞ (i.e., ξ is F -adapted). F0 plays the special role of subsuming all information / available before observations commence at 0+, and the σ-algebra F∞ ≡ t≥0 Ft subsumes all information in the history F . For the rest of this section ‘the pair (ξ, F )’ or ‘the process (ξ, F )’ always means a cumulative process ξ and a history F such that ξ is F -adapted. The history H consisting of the σ-algebras Ht generated for each t by {ξ(s): 0 < s ≤ t} plays a special role: we call it the internal history (it is also called the natural or minimal history, reflecting the fact that H is the smallest family of nested σ-algebras to which the observed values of ξ are adapted). Note that H0 = {∅, Ω}. Histories with the particular structure Ft = F0 ∨ Ht , with F0 in general nontrivial, are called by Brémaud intrinsic histories; among other uses, they are important in the analysis of doubly stochastic processes. The special considerations which are associated with a stationary process observed over a finite or infinite past are examined as part of the discussion of complete conditional intensitiesin Sections 14.3 and 14.7. A history F is called right-continuous if Ft = s>t Fs ≡ Ft+ . Counting processes and other cumulative processes, being right-continuous and boundedly finite by assumption, necessarily yield internal histories that are rightcontinuous. In general, right-continuity represents a mild constraint on the admissible forms of conditioning information; in any case, whenever the process is adapted to the history F , it is adapted also to the right-continuous history F(+) ≡ {Ft+ : 0 ≤ t < ∞} (see Exercise 14.1.3). It is part of our basic framework that the realizations of the cumulative process are a.s. finite for finite t, for this reflects our assumption that, as random measures, the trajectories are a.s. boundedly finite (elements of M# R+ ). This assumption rules out the possibility of explosions, and imposes a certain requirement on the sequence of F -stopping times defined for n = 0, 1, . . . (see Definition A3.3.II and Lemma A3.3.III for background) by Tn ≡ Tn (ω) = sup{t: ξ(t, ω) < n} ∞ if ξ(t, ω) < n for all 0 < t < ∞, = (14.1.1) inf{t: ξ(t, ω) ≥ n} otherwise, namely, that Tn → ∞ a.s. as n → ∞. Exercise 14.1.5 addresses the case that the sequence is finite; otherwise we suppose the sequence continues indefinitely. A feature of the general theory of processes is that the family ξ(t, ω) is regarded as a single real-valued mapping ξ: R+ × Ω → R+ rather than as an indexed family of r.v.s. The product σ-algebra B(R+ ) ⊗ E of sets from the product space R+ × Ω contains a hierarchy of important sub-σ-algebras, each of which is associated with a corresponding class of processes, namely, the measurable, progressively measurable, and predictable processes; these are defined and discussed briefly in Section A3.3.

358


In this section we need especially the concept of an F -predictable process X, which is a process measurable with respect to the F -predictable σ-algebra, which in turn is the sub-σ-algebra of B(R+ ) ⊗ E generated by all product sets of the form (s, t] × U for U ∈ Fs and 0 ≤ s < t < ∞ (see above Lemma A3.3.I). The main result of this section is a theorem which asserts that, for every pair (ξ, F ), where ξ is F -adapted, there exists an integrated form of the conditional intensity function described in Section 7.2 that is predictable, and is the key to a martingale property extending the result of Lemma 7.2.V. In general, what must be subtracted from an increasing process ξ to yield a martingale is called a compensator; it is formally defined as follows. Definition 14.1.I. Let ξ(t) be an F -adapted cumulative process on R+ . An F -compensator for ξ is a monotonic nondecreasing right-continuous predictable process A(·) such that for each n and F -stopping time Tn at (14.1.1), the stopped process {ξ(t ∧ Tn ) − A(t ∧ Tn ): 0 ≤ t < ∞} is an F -martingale. In passing, note that a process {D(t): 0 ≤ t < ∞} such that {D(t ∧ Tn ): 0 ≤ t < ∞} is a martingale for some sequence of stopping times {Tn } for which E|D(t ∧ Tn )| < ∞ (all t ≥ 0, n = 1, 2, . . .), is often called a local martingale. The notion occurs repeatedly in more general treatments of point processes [e.g., Liptser and Shiryaev (1974, 1977, 1978, 2000)]. See also Exercise 14.1.7. Example 14.1(a) below, although trivial, illustrates the fact that the compensator is effectively of interest only for processes with jumps: indeed, as subsequent examples illustrate, the compensator may be regarded as a device for smoothing out jumps and producing an a.s. diffuse random measure from a random measure that may have atoms, but has no fixed atoms. Example 14.1(a) Cumulative process with density. Suppose ξ(·) is the cumulative process of an absolutely continuous random measure, with density x(t, ω) some F -progressively measurable nonnegative process. Then ξ(t) = t x(u) du is its own compensator, for because x(·, ω) ≥ 0, ξ is monotonic 0 nondecreasing and continuous, and for this reason predictable, inasmuch as it is both F -adapted and left-continuous. Given a pair (ξ, F ) as in Definition 14.1.I, the first problem is to give conditions that ensure that a compensator for ξ exists. We start with the simplest example, a one-point process consisting of a single point whose location is defined by a positive r.v. X with d.f. F [see Example 7.4(b)]. The associated counting process is defined by N (t, ω) = I(0,t) X(ω)

(0 < t < ∞, ω ∈ Ω).

(14.1.2)

If we let F coincide with the internal history of the process, that is,

Ht ∈ H is the σ-algebra generated by the sets {ω: X(ω) ≤ s}: 0 < s ≤ t , we can give a direct construction of the H-compensator without any need to appeal to the deeper theorems of the general theory of processes.

14.1.


359

Observe first that N is monotonic nondecreasing, right-continuous, and even uniformly bounded so there is no problem about the existence of moments. Next, because N (t, ω) = 1 implies N (t , ω) = 1 for all t ≥ t, the compensator, for the same ω, must also be constant for such t ≥ t. On the other hand, if N (t, ω) = 0, then we know that X(ω) > t, and thus, in a small interval (t, t + dt), we can expect E[dN (t, ω)] ≈

dF (t) , 1 − F (t)

(14.1.3)

which equals h(t) dt if the d.f. has a density and hence a hazard function h. These heuristics are approximately correct; the key to obtaining a precise statement is the integrated hazard function (IHF) of Definition 4.6.IV. Lemma 14.1.II. The one-point process N at (14.1.2) generated by the positive r.v. X has H-compensator A(t, ω) = H(t ∧ X(ω))

(0 < t < ∞, ω ∈ Ω),

where H is the IHF of X,

H(t) = 0

t

(14.1.4a)

dF (x) . 1 − F (x−)

(14.1.4b)

The compensator A(t) so defined is continuous except at jumps ui of F , where in terms of ∆F (u) = F (u) − F (u − 0), A(·) has jumps of height ai = ∆F (ui )/[1 − F (ui −)] ≤ 1,

(14.1.5)

with equality if and only if ∆F (ui ) = 1 − F (ui −). Proof. Note that for each Ht the set {ω: X(ω) > t} constitutes a large ‘atom’ (i.e., a subset of Ω that cannot be decomposed by the σ-algebra). Of course, H0 = {∅, Ω}, whereas H∞ is the σ-algebra generated by the r.v. X. Like H itself (see Definition 4.6.IV), H(t ∧ X(ω)) is monotonic increasing and right-continuous in t. To verify that it is predictable, we first check that X(t, ω) ≡ t ∧ X(ω) is predictable, so we study {(t, ω): X(t, ω) > x} = {t > x} × {ω: X(ω) > x}. Now {ω: X(ω) > x} ∈ Hx , so the set in (t, ω) has the form of a generating set for the predictable σ-algebra. Thus, X(t, ω) is predictable. The IHF H(x) is monotonic increasing and right-continuous in x and thus has a uniquely defined inverse H −1 for which H(x) ≥ y if and only if x ≥ H −1 (y). In particular, {X(t, ω) ≥ H −1 (y)} is a predictable set, so H(X(t, ω)) is a predictable process. It remains to verify the martingale property that for fixed s and t with 0 ≤ s < t, E[N (t ∧ X) − H(t ∧ X) | Hs ] = N (s ∧ X) − H(s ∧ X)

a.s.

(14.1.6)

360


Note first that, because of the special structure of Ht here, we have for any bounded function g(·), ⎧ ⎨ g(X) E[g(X) | Ht ] = E[g(X)I{X>t} ] ⎩ E(I{X>t} )

on {X(ω) ≤ t}, on {X(ω) > t},

(14.1.7)

because when X > t, E[g(X) | Ht ] = E[g(X) | X > t], so the second case of (14.1.7) can be written in terms of the d.f. F (·) of X as ∞ 1 g(u) F (du) on {t < X(ω)}. E[g(X) | Ht ] = 1 − F (t) t On {s ≥ X(ω)}, N (t∧X) = N (s∧X) = 1 and H(t∧X) = H(s∧X) = H(X), which for X ≤ s is Hs -measurable, so (14.1.6) holds in this case. On the complement where {s < X(ω)}, using (14.1.7), 1 E[N (t ∧ X) | Hs ] = 1 − F (s)

t

F (du) = s

F (t) − F (s) , 1 − F (s)

so that from Lemma 4.6.I we obtain [1 − F (s)][E(H(t ∧ X) | Hs ) − H(s)] t [H(u) − H(s)] F (du) + [H(t) − H(s)][1 − F (t)] = F (t) − F (s). = s

Thus, E[N (t ∧ X) − H(t ∧ X) | Hs ] = −H(s) on {s < X(ω)}, so (14.1.6) holds generally as asserted. It is a standard property of the distribution function F of an honest positive r.v. that F (x) = Fa (x) + Fc (x) for a purely atomic function Fa and a continuous function Fc (see above Lemma A1.6.II). Similarly, the IHF H of F , being a monotonic nondecreasing function, can be decomposed as H(x) = Ha (x) + Hc (x), with x x dFa (u) + dFc (u) dFc (u) = , ai + Ha (x) + Hc (x) = 1 − F (u−) 1 − F (u−) 0 0 i:ui ≤x

where on the right-hand side the sum is atomic and the integral is a continuous function of x. The asserted nature of A(·) follows. Example 14.1(b) One-point process with absolutely continuous or discontinuous H-compensator. Suppose first X above has an exponential distribution, say F (x) = 1 − e−λx , so that its IHF H(t) = λt. Then the corresponding onepoint process has A(t) = λ min(t, X) which is differentiable except at X, and in any case is absolutely continuous with density λ∗ (t) = λ (t ≤ X), = 0 (t > X).

14.1.


361

Now suppose that X = 1 with probability p, and with probability 1 − p is exponential as above, so that X has survivor function p + (1 − p)e−λt (t < 1), S(t) = (t ≥ 1). (1 − p)e−λt Then from (4.6.4) X has IHF λt − log 1 + p(eλt − 1) (t < 1), λt (t ≥ 1), with a jump of size ∆H = log 1 + p(eλ − 1) at t = 1. The compensator is given by A(t) = H(t∧X), and thus has a discontinuity at x = 1 if X ≥ 1. The risk is reduced in the interval (0, 1) because there is a positive probability that the event will occur at the end of the interval rather than randomly during the interval. Typically, discontinuities in the compensator are associated with the occurrence of deterministic elements such as fixed atoms as around (14.1.5). See Exercise 14.1.10 for further examples.

H(t) =

An important extension of Example 14.1(b) can be given when F is an intrinsic history for the one-point process and so consists of σ-algebras of the form Ft = F0 ∨ Ht . At least in the case that X has a regular conditional probability distribution given F0 , a version of which we denote F (· | F0 ), the influence of F0 can be described very simply: all we need to do in Lemma 14.1.II is to replace the distribution of X and its IHF by this conditional distribution F (· | F0 ) and its associated IHF, H(t | F0 ) = 0

t

dF (u | F0 ) . 1 − F (u− | F0 )

Lemma 14.1.III. A one-point process with prior σ-algebra F0 and regular conditional distribution F (· | F0 ) for X has compensator H(t | F0 ) relative to the intrinsic history Ft = F0 ∨ Ht . Proof. Note first that because Ft ⊇ F0 , E( · | F0 ) = E E(· | Ft ) | F0 . We now claim that for nonnegative measurable functions g: R → R, ⎧ on {X(ω) ≤ t}, ⎨ g(X) ∞ (14.1.8) E[g(X) | Ft ] = g(u) F (du | F0 ) ⎩ t on {X(ω) > t}. 1 − F (t | F0 ) The first part of (14.1.8) is obvious, while on {X(ω) > t}, Ft consists entirely of sets of the form U ∩ {X(ω) > t} for some U ∈ F0 . In this case we can write g(X(ω)) P(dω) = I{X>t} g(X(ω)) P(dω) U ∩{X>t}

U

362


E[g(X)I{X>t} | F0 ] P(dω)

= U U

I{X>t} E[g(X)I{X>t} | F0 ] P(dω) E(I{X>t} )

U

E[g(X)I{X>t} | F0 ] P(dω). E(I{X>t} | F0 )

= =

(14.1.9)

The first expression in this chain reduces to the left-hand side of (14.1.8) and, from the assumption that F (· | F0 ) is a version of the regular conditional distribution, the last expression reduces to the right-hand side of (14.1.8) on {X(ω) > t} as asserted. This result can now be used in place of (14.1.6) to establish the compensator property of the conditional IHF, provided at least that we can manipulate the conditional distributions in the same way as unconditional distributions: this is certainly the case when we can choose a regular version of the conditional distribution. Example 14.1(c) One-point process with randomized hazard function. To take a specific example, suppose that X has a negative exponential distribution with parameter λ, where λ itself is a positive r.v. determined by F0 (i.e., λ is F0 -measurable). Then the F -compensator, AF (t, ω) say, can be represented in terms of the IHF of the exponential (λ) distribution, namely, AF (t, ω) = λ(t ∧ X(ω)). On the other hand, to find the H-compensator we must first evaluate the survivor function for the resultant mixed exponential distribution. If, for example, λ itself has a unit exponential distribution with density e−λ dλ, then the unconditional survivor function is ∞ 1 . e−λt e−λ dλ = H(t) = E[H(t) | F0 ] = 1 + t 0 The IHF is therefore equal to log(1+t), and for the H-compensator we obtain AH (t, ω) = log(1 + t ∧ X(ω)). Such examples show that the choice of prior σ-algebra can drastically affect the form of the compensator. We can now construct the compensator for a simple point process with respect to the intrinsic history F = {F0 ∨Ht : 0 < t < ∞}; that is, we allow some initial conditioning as in the last example. Such a history F is completely described by the initial σ-algebra F0 and the family of stopping times {Tn } as at (14.1.1): in view of the assumed simplicity, {Tn } is a.s. a strictly increasing sequence. Given F(n−1) ≡ FTn−1 ,

14.1.


363

which means we are given F0 and T1 , . . . , Tn−1 , choose a family of regular conditional distributions Gn (x | F(n−1) ) for the distributions of the successive differences (n = 1, 2, . . . , T0 ≡ 0). τn = Tn − Tn−1 Writing N (t) =

∞

[N (t ∧ Tn ) − N (t ∧ Tn−1 )] =

n=1

∞

N (n) (t) say,

n=1

each N (n) (·) is a one-point process with a single point of increase at Tn . Defining now the IHFs Hn (·) ≡ Hn (· | F(n−1) ) from the conditional d.f.s Gn (· | F(n−1) ) by x Gn (du | F(n−1) ) , Hn (x | F(n−1) ) = 0 1 − Gn (u− | F(n−1) ) we assert that the F -compensator for N (n) (·) has the form ⎧ on t < Tn−1 (ω), ⎪ ⎨0 on Tn−1 (ω) ≤ t < Tn (ω), A(n) (t, ω) = Hn (t − Tn−1 ) ⎪ ⎩ Hn (Tn − Tn−1 ) on Tn (ω) ≤ t.

(14.1.10)

Then by additivity, N (·) has the F -compensator A(t, ω) =

∞

A(n) (t, ω).

n=1

To establish (14.1.10), note that predictability of A(n) (·) is established as in Lemma 14.1.II, so it remains to show that each difference Z (n) (t, ω) ≡ N (n) (t, ω) − A(n) (t, ω) is an F -martingale. We establish the requisite equality E[Z (n) (t) | Fs ] = Z (n) (s)

(14.1.11)

for 0 < s ≤ t separately on the sets Bn = {ω: Tn−1 ≤ s} and Bnc , observing that Bn and Bnc ∈ F(n−1) . Considering first the subsets of Bn , we have Fs ∩ {Tn−1 (ω) ≤ s < Tn (ω)} = F(n−1) ∩ {Tn−1 (ω) ≤ s < Tn (ω)}, (14.1.12) which means that, given any C ∈ Fs , there exists C ∈ F(n−1) such that C ∩ Bn = C ∩ Bn and conversely: that this is so is clear from the structure of the σ-algebra Fs (because Fs ⊃ Hs ) and a consideration of the basic sets such as {ω: N (s, ω) = k}. Now on Bn , the stopping time τn plays the same role for N (n) (·) as X plays for the one-point process of Example 14.1(b), with

364


F(n−1) here playing the role of F0 there. In particular, on Bn we have for any bounded measurable function f (·) E[f (τn )I{τn >xn } | F(n−1) ] = E[f (τn ) | Fs ] = E[I{τn >xn } | F(n−1) ]

∞

xn

f (u) Gn (du | F(n−1) ) , 1 − Gn (xn | F(n−1) )

where xn = s − Tn−1 , necessarily ≥ 0 on Bn . In principle, this evaluation of the conditional expectation involves the extension of (14.1.8) to the case where t there (equals xn here) is a r.v. measurable with respect to the prior σalgebra F0 there (which is F(n−1) here). However, scrutiny of (14.1.9) and the surrounding argument shows that nothing need be altered there, with (14.1.8) remaining F0 -measurable, so (14.1.9) is still valid. Thus, on Bn , proof of the martingale equality (14.1.11) follows as in Lemma 14.1.III. On the sets {s < t < Tn−1 (ω)} and {s ≥ Tn (ω)} the equality is trivial because all terms are zero. There remains the case {s < Tn−1 (ω) ≤ t}. Here we proceed by conditioning first on F(n−1) , when equality follows as a special case of the above, because this equality is not affected by further conditioning on Fs . We summarize this discussion as follows. Theorem 14.1.IV. Let N be the counting process of a simple point process on (0, ∞), F a history for N of the form {F0 ∨ Ht }, and {Tn } the sequence of stopping times at (14.1.1). Suppose there exist regular versions Gn (· | F(n−1) ) of the conditional d.f.s of the intervals τn = Tn − Tn−1 , given F(n−1) , such that 1 − Gn (x−) > 0 for x > 0. Then a version of the F -compensator for N is given by ∞ A(n) (t, ω), (14.1.13a) A(t, ω) = n=1

where ⎧ ⎪ ⎨0 (n) A (t, ω) = ⎪ ⎩

0

(t ≤ Tn−1 (ω)), τn ∨(t−Tn−1 )+

Gn (du | F(n−1) ) 1 − Gn (u− | F(n−1) )

(t > Tn−1 (ω)). (14.1.13b)

The following special case ties in the result above with the earlier discussion of Section 7.2, in particular with Proposition 7.2.I and Definition 7.2.II. Corollary 14.1.V. The F -compensator A(·) at (14.1.13) is absolutely continuous a.s. if and only if the conditional d.f.s Gn (· | F(n−1) ) have absolutely continuous versions, with densities gn (· | F(n−1) ) say, in which case one version of the F -compensator is given by

t

A(t, ω) = 0

λ∗ (u, ω) du,

14.1.


365

where λ∗ (t, ω) =

∞ n=1

λ∗n (t, ω) ≡

∞ gn (t ∧ Tn − Tn−1 | F(n−1) )I{Tn−1 0 and ψ(x) = 1 otherwise, and that h(x) = λ, corresponding to exponential interarrival times in the absence of the modulating factor. Then for any measurable X(·) we would have A(t) = λ(t + Yt ), where the random variable Yt is the length of time for which X(s) > 0 during the interval (0 < s < t). This assumes that the process X(t) is observable; when this is not the case a filtering problem is involved, requiring averaging of the F -intensity over the coarser σ-algebras of the internal history, as discussed further in Sections 14.3 and 14.4. For more general processes ξ(·) and histories F , even if we cannot establish explicit representations, many important results can still be derived from the Doob–Meyer decomposition, as, for example, below where we show both the existence and uniqueness of the compensator for a cumulative process ξ with general history F and in discussing quadratic variation.. Theorem 14.1.VII. Let {ξ(t): t > 0} be a cumulative process adapted to the right-continuous history F . Then ξ(·) admits an F -compensator A(t, ω), which is uniquely defined P-a.e. in the sense that for any other compensator ˜ ω), P{A(t, ω) = A(t, ˜ ω) (all t)} = 1. A(t, Proof. We again use the stopped process ξn (t) = ξ(t ∧ Tn ), where the stopping times {Tn } are as at (14.1.1). Because ξn (t, ω) ≤ n, each ξn (·) is uniformly bounded in (t, ω), and also has bounded first moment, so that in addition it is uniformly integrable in t. Also, each ξn (t) has its trajectory a.s. nondecreasing in t, so for 0 < s < t, E[ξn (t) | Fs ] = E[ξn (t) − ξn (s) | Fs ] + E[ξn (s) | Fs ] ≥ ξn (s)

a.s. on Fs .

Thus, {ξn (t): 0 < t < ∞} is a right-continuous, bounded submartingale with respect to the history F , and the Doob–Meyer decomposition (Theorem A3.4.IX) can be applied. It implies that there exists a right-continuous nondecreasing F(n) -predictable process An (·) and an F -martingale Mn (·) such that ξn (t) = An (t) + Mn (t).

368


Moreover, the processes An , Mn are uniquely defined P-a.s. as functions on (0, ∞), which implies that the functions {An (t)} are a.s. nested in the sense that for m > n, An (t, ω) ≤ Am (t, ω) a.s. for t ≤ Tn (ω). Now the definition of a cumulative process in terms of a boundedly finite random measure requires Tn → ∞ a.s.; letting n → ∞ it follows that, a.s. for all t, a well-defined limit A(t, ω) exists and defines an F -adapted process such that for every n > 0, A(t, ω) = An (t, ω)

t ≤ Tn (ω) .

Clearly, A(t, ω) inherits the monotonicity and right-continuity properties from each member of the sequence {An (t, ω)}. For predictability, observe that because Tn → ∞, the left-continuous F(n) adapted processes generate left-continuous F -adapted processes, so A(t, ω), which is F(n) -predictable for each n, is also F -predictable. Finally, uniqueness of the overall decomposition ξ(t) = A(t) + M (t) follows from the uniqueness of the Doob–Meyer decomposition on each of the sets t ≤ Tn . The results do not address directly the question of which predictable cumulative processes can be compensators, nor which nonnegative predictable processes can be conditional intensities. To make sense of this question we suppose first that the processes are defined on the canonical space X × NX# , with X = R+ , where a predictable cumulative process A(t, N ) takes the form of a function Ψ{N (t), T1 (N ), . . . , TN (t) (N )} of the points of the realization occurring before time t. It is then a matter of finding conditions on the function Ψ that allow a consistent set of integrated hazard functions to be defined from it. These conditions must ensure that the hazard functions satisfy the requirements set out in Exercise 14.1.8. A brief outline of the argument is sketched in Exercises 14.1.13–14; a thorough discussion, incorporating also the marked case, is given by Last and Brandt (1995, Chapter 8). Another important application of the Doob–Meyer decomposition is in proving the existence of the quadratic variation of a martingale when the martingale itself is square integrable. Let ξ(t) be an F -adapted cumulative process on R+ with finite second moments E [ξ(t)]2 < ∞

(0 < t < ∞),

write A(t) for its F -compensator (which then necessarily exists), and M (t) for the F -martingale ξ(t)−A(t). Then [M (t)]2 is again an F -adapted process whose expected increments are nonnegative because E[(M (t))2 − (M (s))2 | Fs ] = E[(M (t) − M (s))2 | Fs ] + 2E[(M (t) − M (s))M (s) | Fs ] = E[(M (t) − M (s))2 | Fs ],

(14.1.15)

14.1.


369

using the fact that M (·) is an F -martingale. Thus [M (t)]2 is an F -submartingale, which therefore has a Doob–Meyer decomposition, say [M (t)]2 = Q(t) + M2 (t),

(14.1.16)

where M2 (t) is the F -martingale component, and the F -compensator Q(t) is called the quadratic variation process. The name stems from the fact, as follows from (14.1.16) on taking expectations and using the martingale property of M2 , that E Q(t) − Q(s) | Fs = E [M (t)]2 − [M (s)]2 | Fs − E[M2 (t) − M2 (s) | Fs ] = E [M (t) − M (s)]2 | Fs (14.1.17) (cf. the last equation of A3.4, where, however, the conditional expectation on the left-hand side has been omitted). But E M (t) − M (s) | Fs = 0, so (14.1.17) shows that the increments in Q are the conditional variances of the increments in the martingale M , and hence the terminology. The right-hand side of (14.1.17) also provides one approach to evaluating Q: write the argument as t t 2 M (du) M (dv) = (M × M )(du × dv), [M (t) − M (s)] = s

(s,t]×(s,t]

s

and consider the integral on the three regions D1 = {s < u < v ≤ t}, D2 = {s < v < u ≤ t}, and D3 = {s < u = v ≤ t}. On D1 the martingale property implies that " E[M (du) M (dv) | Fs ] = E E[M (du) M (dv) | Fu ] " Fs " = E M (du) E[M (dv) | Fu ] " Fs = 0, and similarly for the integral over D2 . This leaves only the conditional expectation of the integral over D3 , hence " " (14.1.18) (M × M )(du × dv) " Fs . E Q(t) − Q(s) | Fs = E s 0, then the randomly normed integrals XT /BT converge F0 -stably in distribution to the unit normal random variable U . Proof. We use the result from Proposition A3.2.IV that if Xn → X (Fstably) then g(Xn , Y ) → g(X, Y ) (F-stably) for bounded continuous functions g(·). Supposing first that Y is bounded away from zero and X is essentially bounded, we can take g(x, y) = x/y so that Xn /Y → X/Y (F-stably). The constraint on X is immaterial in that it is given that P{|X| < ∞} = 1 (because X is a well-defined r.v.). Now suppose also that Yn → Y in probability, where each Yn is a.s. positive and F-measurable. Then (Xn , Yn ) → (X, Y ) in distribution and thus, again, Xn /Yn → X/Y (F-stably). Finally, by approximating Y by a sequence of r.v.s bounded away from zero, the result extends to the case Y > 0 a.s. Taking Yn = BTn and Xn = XTn for some sequence Tn → ∞ and F = F0 , the result follows. The form of condition (iii) is not the most general possible. Kutoyants noted that it may be replaced by a Lindeberg type of condition, although the Liapounov type of condition suffices for most applications. Versions of the theorem for multivariate and MPPs can be given [see Kutoyants (1984a, b)]. The major application of the theorem is to the proof of the asymptotic normality of parameter estimates. This application is discussed and illustrated at length in Kutoyants (1980) for the case of inhomogeneous Poisson processes, and in Kutoyants (1984b) for more general processes. The next two examples may also help illustrate the range of applications for the theorem. Example 14.5(a) Poisson and mixed Poisson processes. As the simplest possible example, let N be a simple Poisson process with rate µ. Successful application of the theorem relies on identifying the appropriate norming function fT (·) for the quantity of interest. Here, to study N (t), recall first that its H-compensator is µt. Thus, we need fT (·) to satisfy T [fT (u)]2 µ dt → const. (T → ∞), 0

416


so fT (u) = T −1/2 is the simplest choice here, with the constant = µ and nonrandom. Then the left-hand side of (iii) reduces to µT −δ/2 → 0 (T → ∞) as required. Thus, N (T ) − µT → µ1/2 U in distribution, T 1/2 with U a standard normal r.v. as in the theorem. If in fact the process is mixed Poisson with µ a r.v. as in Example 14.3(a), the same conclusion holds provided we use the F -compensator with Ft = F0 ∨ Ht . Indeed, from Corollary 14.5.III we should have N (T ) − µT →U (µT )1/2

in distribution.

If we want to devise a result concerning an estimate of µ, it is preferable to express the left-hand side here as [N (T )/T − µ]/(µ/T )1/2 and then observe that, as T → ∞, we have N (T )/T → µ a.s. As a result of this, we can replace µ in the denominator and deduce further that N (T )/T − µ →U [N (T )]1/2 /T

in distribution,

with µ the only quantity on the left-hand side that is unknown at T . A final possibility would be to use the H-compensator, which in the special case given in Example 14.3(a) has the form dA(t) =

N (t−) + 1 dt, t+α

and leads to virtually the same conclusions. In examples of this kind, where F0 is either trivial or very simple, there is little advantage in using the extensions to random norming. Only weak convergence is asserted and the theorem sheds no light on whether the estimates converge H∞ -stably, for example. For detail on this question see Jarupskin (1984). It also underlies the next example. Example 14.5(b) Simple birth process. This is a standard example [see, e.g., Keiding (1975); Basawa and Scott (1983)] for showing ‘nonergodic’ behaviour in the sense that the asymptotic distribution of the maximum likelihood estimate is not normal but a mixture of normals. If the probability of an individual producing offspring in time (t, t + dt) is λ dt, and all individuals reproduce independently, it is known that with N (t) denoting the sum of the initial number n0 and the number of individuals born in (0, t] and qt = 1 − pt = e−λt , P{N (t) = n} =

0 qtn0 pn−n t

n−1 , n − n0

(14.5.4)

14.5.

A Central Limit Theorem

417

that N (t)e−λt → W

a.s.,

(14.5.5)

where W is a r.v. which, if n0 = 1, has the unit exponential distribution, and that ˆ t = N (t) − n0 λ t N (u) du 0 is the maximum likelihood estimate of λ. Clearly, the process may be treated as a point process, and it is then of interest to see what light the present methods shed on the behaviour of the likelihood estimate. The conditional intensity of the process with respect to the internal history H generated by the N (t) themselves is just equal to λN (t−). If we use this history, the first derivative of the likelihood of the process on (0, T ) is proportional to T N (t−) dt, N (T ) − λ 0

which has variance function

T

N (t) dt ∼ E(eλT ).

λ 0

This suggests that the norming factor k(T ) = e−λT /2 is appropriate, but because W is not F0 -measurable with this choice of history, further discussion is required. In fact, what is needed is the F -intensity when F0 = σ{W } and (N ) Ft = F0 ∨ Ht . The history F is a refinement of the internal history, and to find the F -compensator we have to discuss the behaviour of the process conditional on the value of W . This can be computed by writing down from (14.5.4) the joint distribution of N (s) and N (t) for s > t, conditioning on N (s), and letting s → ∞, taking into account (14.5.5) and using Stirling’s formula [cf. Keiding (1975)]. The result can be stated as follows. Given N (t) and W , the conditional distribution of N (s) − N (t) is Poisson with parameter λ(s | t, W ) = W eλt (eλ(s−t) − 1). Hence the F -intensity is

λF (t) = λW eλt .

Note that E[λF (t) | Ht ] = λeλt E[W | N (t−)] = λeλt N (t−)e−λt = λN (t−), which is just the H-compensator if a predictable version of N (t) is taken. We now consider the asymptotic behaviour of the scaled difference −λT /2

∆(T ) = e

T e−λT /2 N (T ) − λ 0 N (u) du ˆ . (λT − λ) = T λe−λT 0 N (u) du

(14.5.6)

418


Applying the theorem, we find after simple computations that the pair e

−λT /2

T

[dN (u) − λW eλu du] = e−λT /2 [N (T ) − n0 − W (eλT − 1)]

0

and e−λT /2

T

λ(T − u) [dN (u) − λW eλu du] T = e−λT /2 N (u) du − no T − W (eλT − 1 − λT ) 0

0

converges F0 -stably to the pair (Z1 W 1/2 , Z2 W 1/2 ), where Z1 , Z2 are independent of F0 and jointly normally distributed with covariance matrix

1 1

1 , 2

so that in fact we can write Z2 = Z1 − Z1 where Z1 , Z1 are independent unit normal r.v.s. Thus, the numerator in the term on the right-hand side of (14.5.6) converges F0 -stably to Z1 W 1/2 and from (14.5.5) the denominator converges a.s. to W . Hence, ∆(t) ∼ Z W −1/2

(F0 -stably).

but an exponential r.v. W has just the form 12 χ2(2) , where χ2(2) denotes a chisquare r.v. on two degrees of freedom, and so the ratio has a t-distribution on two degrees of freedom [again, see Keiding (1975)].

Exercises and Complements to Section 14.5 14.5.1 For the self-correcting or stress-release model of Isham and Westcott (1979) for which λ(t) = exp (α + β[t − ρN (t)]), show that conditions for estimators of β and ρ to have central limit theorem properties hold for β > 0 and ρ > 0 but fail when β = 0. [Hint: When β > 0, ρ > 0, the process X(t) = t − ρN (t) is Markovian and the law of large numbers implies that condition (ii) of Theorem 14.5.I holds, but this fails when β = 0. See also Vere-Jones and Ogata (1984).]

14.6. Random Time Change The topics of both this section and the next describe methods for reducing more general point processes to Poisson processes, emphasizing yet again the fundamental role played by Poisson processes in point process theory. We start by recapitulating some introductory material, including Watanabe’s (1964) characterization of the Poisson process as a process with deterministic compensator. The time-change theorems in this section were introduced in Section 7.4 and linked there to Ogata’s (1988) residual analysis for

14.6.

Random Time Change

419

checking the goodness-of-fit for a point process model [and now extended to the wider range of residual methods introduced for spatial point processes by Baddeley and co-workers, reviewed in Baddeley et al. (2005) and discussed briefly in Sections 15.4–5 below]. Our main goal here is the extension of the time-change theorem to MPPs, first proved for multivariate point processes by Meyer (1971) [but see also Doléans-Dade (1970)] using orthogonal martingale arguments, and later extended and simplified by Brown and Nair (1988). The proof we give is for general MPPs, and appears to be new, combining generating functional arguments with the use of the exponential formula much as in Brémaud (1981) and Brown and Nair, but avoiding the second-order theory used in the orthogonal martingale arguments. To illustrate the technique we first use it to give an extension of Watanabe’s theorem to Cox processes; in essence the proof is a minor variation of those in Brown (1978) and Brémaud. Random time-change results for spatial processes are much more problematic: see Nair (1990) and Schoenberg (1999). Theorem 14.6.I. Let N be a simple point process on R+ adapted to the history F . If the F -compensator A of N is continuous and F0 -measurable, then N is a Cox process directed by A. Proof. With a view to characterizing the process via its p.g.fl., take a fixed continuous nonnegative h ∈ V(R+ ) (cf. Definition 9.4.IV), with h(0) = 1 and h(u) = 1 outside a finite interval [0, T ), and for a given realization {ti } of the process consider the expression

Φ(h) =

∞ − [h(u)−1] dA(u) h(ti ) e 0 .

(14.6.1)

0≤ti 0). Both his result and its generalization are closely related to a theorem in Papangelou (1972), already stated in Theorem 7.4.I, that any simple point process with continuous compensator is locally Poisson in character, in the sense that there exists a local transformation of the time axis that converts the process into a Poisson process. We develop a proof based on a use of the exponential formula similar to that of Theorem 14.6.I. Consider then a simple point process that is F -adapted for some general history F for which A is the F -compensator, and consider the time-change defined by τ = A(t) (t ∈ R+ ), equivalently, t = A−1 (τ ) = inf{t: A(t) ≥ τ }. Note that if the compensator A is continuous, then A−1 is right-continuous (and monotonic, like A), with jumps at the at most countable set of values of τ of constancy of A, and A(A−1 (τ )) = τ for all τ > 0. It follows (see Lemma A3.3.III) that for every τ , A−1 (τ ) is an F -stopping time. Moreover, the σ-algebra Fτ ≡ FA−1 (τ ) is well defined (Definition A3.4.V), with Fτ ⊆ Fυ for τ ≤ υ (Theorem ≡ {Fτ : 0 < τ < ∞} constitutes a history for the process A3.4.VII), and F −1 (τ ) = N A (τ ) . N We now imitate the proof of Proposition 14.6.I, but take the p.g.fl. in the transformed space rather than the original space. Write τi = A(ti ), and again suppose h(·) ∈ V(R+ ) is continuous and equal to unity outside some finite interval (0, T ). In place of (14.6.1), consider ∞ − [h(τ )−1] dτ h(τi ) e 0 , (14.6.5) Φ(h) = 0≤τi 0, k = 1, . . . , K, k=1 πk = 1, simple ground process, and t the F -conditional intensity λk (t). Let a(t, k) = 0 λk (s) ds and denote by N rescaled MPP defined to have a point at {a(t, k), k} if and only if the kth is a stationary compound component of N contains a point at t. Then N Poisson process with unit intensity and stationary mark distribution π. has a point at Equivalently, if the rescaling is performed so that N (a(t, k)πk , k) whenever the original process has a point at (t, k) (this is the same as the reference measure π assigning unit mass to each component), then the resultant process consists of K independent, unit-rate Poisson processes, one for each mark. Suppose that the mark space is the real line; let FK (m) ≡ κ<m K (dκ) be the cumulative measure corresponding to the probability measure K , and suppose that FK (m) is continuous as a function of m. Then a simple rescaling of the mark space, taking m∗ = FK (m), converts the stationary mark distribution for the transformed process into the uniform distribution on [0, 1]. Now a compound Poisson process with constant rate and uniform mark distribution can equally be interpreted as a two-dimensional Poisson process on a strip; hence we obtain the following result. Corollary 14.6.VI. Suppose that the MPP N has real marks, that the conditions of Theorem 14.6.IV hold, and that the reference probability measure K admits a continuous cumulative version FK . Then the doubly transformed ∗ , defined to have a point at a(ti , mi ), F (mi ) when the original process N K process has a point at (ti , mi ), is a two-dimensional Poisson process with unit rate over the half-strip R+ × [0, 1]. The role of absolute continuity in Proposition 14.6.IV and its corollaries, in particular the requirement that the compensator A(t, K) have a density with respect to the mark κ as well as time t, is illustrated in the next example. Example 14.6(a) Poisson process with alternating marks. Let {ti } be a realization of a Poisson process at unit rate on R+ , and attach to points {t2i−1 } the mark 1, and to the points {t2i } the mark 2. Then, writing t0 = 0 and using the notation of Corollary 14.6.V, for i = 1, 2, . . . , (1, 0) for t2i−2 ≤ u < t2i−1 , λ1 (u), λ2 (u) = (0, 1) for t2i−1 ≤ u < t2i . Let Ui = ti − ti−1 (i = 1, 2, . . .), so that for t2i ≤ t < t2i+2 , t U1 + · · · + U2i−1 + min((t − t2i )+ , U2i+1 ) (k = 1), a(t, k) = λk (u) du = U 2 + · · · + U2i + min((t − t2i+1 )+ , U2i+2 ) (k = 2). 0 It is immediately evident that each case on the right-hand side here is a sum of independent unit exponential random variables, and so corresponds to a

14.6.

Random Time Change

425

Poisson process with unit rate; furthermore, the two cases are independent. It follows that the conditional intensity of the initial MPP can be written in the form λ∗ (t, κ) = λ∗g (t)f (κ | t), where λ∗g (t) = 1 and f (0 | t) = i(t) = 1 − f (1 | t), where i(t) is the parity of N (t), so that i(t) = 1 if N (t) is odd, = 0 otherwise. The mark space here has just two points, and the natural reference measure has unit atoms at each point, so that the function f described above can be regarded as a density with respect to this measure. So far, this example has illustrated the behaviour to be expected when the conditions of Corollary 14.6.V are satisfied. Now suppose, as a variant on this example where the conditions are not satisfied, that the sequence {ti } is as before, but that t2i−1 has mark κ2i−1 = Xi that is uniformly distributed in (0, 1), independently of the other random variables, and κ2i = 2 − Xi . With Ui = ti − ti−1 as before, we can define the marked point process compensator for K ∈ B((0, 2)) in the form $ U1 + · · · + U2i−1 + (t − t2i ) IK (Xi ), t2i ≤ t < t2i+1 , A(t, K) = U2 + · · · + U2i−2 + (t − t2i−1 ) IK (2 − Xi ), t2i−1 ≤ t < t2i . Here, in the first case, F (· | t) is just the uniform distribution on (0, 1), but in the second case, F (· | t) has an atom in 2 − Xi , because the value of Xi is now known at time t. For this model, therefore, the absolute continuity requirement is not satisfied, and, as the reader may check, the conclusion of Theorem 14.6.IV fails.

Exercises and Complements to Section 14.6 14.6.1 In Example 14.6(a), show that each of the four terms on the right-hand side of (14.6.15) converges in probability to zero as γ → 0 on the assumption that (γ) N (t)/t converges a.s. to some limit λ ≡ λ(ω) ∈ F0 for all γ. [Hint: The convergence is shown directly for the first three terms; for the last, investigate supt 0, of a ρ-subinvariant function for a positive kernel such as H(η, κ) is a mild constraint only on the regularity of the kernel. When the mark space is finite, it is a direct consequence of the Perron– Frobenius theorem, when ρ can be taken to be the maximum eigenvalue of the matrix, and equality rather than inequality holds in (14.7.6b). A similar result holds whenever H defines a compact operator (Lerch’s theorem). More general conditions can be established using generating function arguments, and are outlined, for example, in Vere-Jones (1968) for the denumerable case and more generally in Liggett (1985). The requirement H(η, κ) > 0 for all η, κ, which implies also r(κ) > 0 (all κ), is a strict irreducibility condition, imposed for convenience in order to avoid having to detail the possibilities when the kernel is reducible. Note that the kernel h can be interpreted as the analogue of the matrix density function for a Markov renewal process [see Example 10.3(a)] for which the renewal distributions are defective (hence, the processes are transient). The major constraint is the scaling requirement ρ < 1, which is analogous to the subcriticality requirement in the branching process interpretation of a Hawkes process. The main difference between our approach and that in Massoulié (1998) is in the controlling role we give to the subinvariant function r(·), including the assumption that it is K -integrable. This last assumption is associated with the requirement for the process to have finite ground intensity and could be relaxed if this requirement were dropped. These assumptions increase the transparency of the arguments at the loss of some generality. The existence of the subinvariant function r(·) in (14.7.6b) implies that the kernel H(η, κ) defines a bounded linear operator H on the space K1 , say, of measurable functions f (κ) satisfying K f (κ)r(κ) K (dκ) < ∞, and that its transpose H ∗ defines a bounded linear operator on the space K∞ , say, of measurable functions g(η) satisfying ess sup[g(η)/r(η)] < ∞, through the respective actions H(η, κ)f (κ) K (dκ), (14.7.7a) (Hf )(η) = K (H ∗ g)(κ) = H(η, κ)g(η) K (dη). (14.7.7b) K

Moreover, under the given conditions, each of the operators H and H ∗ has its norm bounded by ρ, as indicated in Exercise 14.7.4. Among other consequences, the contractive condition implies that the sum ∞ n n=0 H converges geometrically fast and defines a bounded limit operator, R say, so that for functions f ∈ K1 and g ∈ K∞ , ∞ g(η)H n (η, κ)f (κ) K (dη) K (dκ) n=0

K

K

= K

K

g(η)R(η, κ)f (κ) K (dη) K (dκ) < ∞.

Our basic result can now be stated as follows.

(14.7.8)

432


Proposition 14.7.III. Suppose that the functional ψ and kernel h satisfy Conditions 14.7.II for some K -integrable function r(κ), and that for some C < ∞, ψ(∅, κ) ≤ Cr(κ). (14.7.9) Suppose also that an initial MPP N− is given on R− such that for some D < ∞, h(t − s, η, κ) N− (ds × dη) ≤ D r(κ). (14.7.10)

(t, κ) ≡ E R− ×K

Then there exists a unique MPP N on R, with N = N− on R− and conditional intensity (14.7.3) on R+ , and with finite, bounded mean ground rate on R+ . Proof. Without loss of generality we suppose here and throughout that r(κ) K (dκ) = 1. K

Then the subinvariant function r(·) plays the role of the density of a dominating stationary mark distribution. We return to the sequence of approximations (14.7.4). Writing λn∆ (t, κ) = n+1 (t, κ) − λn (t, κ)|, we have λn∆ (t, κ) = 0 for t < 0 because all approxi|λ mations coincide with the initial condition N− on R− , and for t ≥ 0 (14.7.5) implies that for n = 1, 2, . . . , E λn∆ (t, κ) = E |ψ(St N n , κ) − ψ(St N n−1 , κ)| n n−1 h(t − s, η, κ) N ∆ N (ds × dη) ≤E (−∞,t)×K t

=E

−∞

K

h(t − s, η, κ)λn−1 (s, η) ds (dη) , K ∆

(14.7.11)

the last equality following from the H-martingale property applied to the n n−1 , which has conditional intensity |λn − λn−1 |. point process N ∆ N Writing φn (κ) = supt∈R+ E[λn∆ (t, κ)], for n = 0 we have from (14.7.9) and (14.7.10a), φ0 (κ) = ψ(St ∅) + [ψ(St N0 ) − ψ(St ∅)] ≤ Cr(κ) + supt (t, κ) ≤ (C + D)r(κ). For n = 1, 2, . . . , (14.7.11) now implies n φn−1 (η)H(η, κ) K (dη) φ (κ) ≤ K

so that on using the inequalities for H ∗ following from Conditions 14.7.II, Mn ≡ sup[φn (κ)/r(κ)] ≤ ρMn−1 ≤ ρn M0 , κ

14.7.

Poisson Embedding and Existence Theorems

433

∞ where M0 ≤ C + D < ∞. The series n=0 E[λn∆ (t, κ)] therefore converges uniformly and absolutely in (t, κ). Considering the ground processes Ngn , it then follows that for any finite T > 0, ∞ ∞ n n−1 E |Ng (0, T ) − Ng (0, T )| = E

T

(C + D)T . 1−ρ 0 n=0 n=0 (14.7.12) Now the ground processes Ngn can differ by at most positive integer values, so at most a finite number of the differences Ngn (0, T ) − Ngn−1 (0, T ) can be nonzero, a.s. It follows that the ground processes Ngn , and hence also the full processes N n , must be a.s. all equal after a finite number of terms, and so must converge almost surely and in expectation to some limit process N . Such convergence implies in turn that the fidi distributions of the processes N n (·) converge for all finite intervals, implying finally that the processes N n converge weakly to the limit process N (Theorem 11.1.VI). ∞ n Moreover, from the convergence of the series n=0 φ (κ), we have also that (14.7.13) sup E λ(t, κ)] ≤ (C + D)r(κ)/(1 − ρ), λn∆ (t, κ) dt dκ

≤

t>0

implying that for t > 0 the limit process has finite mean ground rate and mark distribution dominated by a multiple of r(κ). The limit point process N starts from the same distribution of initial condias the approximants N n , and on R+ , N has conditional intensity tions on NR#∗ − λ which, using Fatou’s lemma and condition (14.7.6b), satisfies E |λ(t, κ) − ψ(St N, κ)| ≤ lim E |λn (t, κ) − ψ(St N, κ)| n→∞ = lim E |ψ(St N n−1 ) − ψ(St N )| n→∞ t h(t − s, η, κ)(N n ∆ N n−1 )(dt × dη) . ≤ lim E n→∞

−∞

K

But h(·, ·) is ( × κ )-integrable and N n ∆ N n−1 converges weakly to the zero process, so the last limit is zero, showing that λ(t, κ) = ψ(St N, κ) a.s. Finally, uniqueness follows from Propositions 14.1.VI and 14.2.IV, because we may regard the realization on R− as defining a prior σ-algebra F0 , and the compensator defines the process uniquely when the history is an intrinsic history Ht ∨ F0 . We turn now to stationary processes and convergence to equilibrium, recalling from Section 12.5 the terms weak and strong asymptotic stationarity to describe MPPs converging weakly or strongly (i.e. in variation norm) from given initial conditions to a stationary process. The next result is again a variant of the corresponding results in Brémaud and Massoulié (1996) and Massoulié (1998), to which we refer for further discussion and extensions.

434


Theorem 14.7.IV. Suppose that the functional ψ and kernel h of (14.7.5) satisfy Conditions 14.7.II for some K -integrable function r(κ), and that 0 < ψ(∅, κ) < Cr(κ) for some C > 0. Then there exists a unique stationary MPP N † whose complete intensity is given by (14.7.3), which has finite mean ground rate ψ(S0 N− , κ) K (dκ) , (14.7.14) m†g = E K

and whose stationary mark distribution is dominated by r(κ). Furthermore, given initial conditions satisfying (14.7.10), the unique MPP N specified by Proposition 14.7.III is weakly asymptotically stationary with limit process N † . If in addition the function h in (14.7.5) satisfies, for some D < ∞, the condition ∞

0

K

t h(t, η, κ)r(η) dt K (dη) < D r(κ),

(14.7.15)

then N is strongly asymptotically stationary with the limit N † . Proof. Again we start the existence proof by constructing a sequence of approximating processes, but in the present situation the processes are defined over the whole line R, starting from N 0 which is taken to be the empty process over the whole line. It means that if ψ(∅, κ) ≡ 0, then all subsequent approximations are also empty, and the construction fails to lead to any nontrivial stationary process. Otherwise, N 1 is a stationary compound Poisson process with intensity λ1 (t, κ) = λg f (κ) for some finite, nonzero constant λg and stationary probability density f dominated by some multiple of r. The arguments leading to (14.7.12) now carry over with only minor changes, the supremum in the definition of φn (κ) now being taken taken over the whole real line. They lead as before to the existence of a well-defined limit MPP N † . In this case, however, it follows from the stationarity of N 0 and the time invariant character of ψ, that N 1 , and by induction all subsequent approximants, are also stationary, and hence that the limit process N † is stationary. Moreover, the boundedness conditions on E[λ(t, κ)], which follow as in (14.7.13), imply here that N † has finite, constant ground rate mg , and stationary mark distribution dominated by some multiple of r(κ). Uniqueness of the stationary solution will follow if we can establish the asymptotic stationarity results, for any stationary solution N ∗ of (14.7.3), satisfying the conditions of the theorem, also satisfies the conditions of Proposition 14.7.III, (14.7.10) following here from the assumption that the stationary mark distribution is bounded by a multiple of r(·). If we assume the weak asymptotic stationarity results, therefore, this second solution should be weakly asymptotically stationary with limit N † , which is possible only if N ∗ and N † coincide. It remains to prove the assertions concerning asymptotic stationarity. We suppose N and N † are defined as in the theorem. From the Lipschitz condition we obtain for the difference between the corresponding intensities, λ∆ (t, κ) ≡

14.7.


435

|λ(t, κ) − λ† (t, κ)| say, E λ∆ (t, κ) = E |ψ(St N, κ) − ψ(St N † , κ)| † h(t − s, η, κ) (N ∆ N )(ds × dη) ≤E (−∞,t)×K t

=E

−∞

K

h(t − s, η, κ)λ∆ (s, η) ds K (dη)

t h(t − s, η, κ)λ∆ (s, η) ds K (dη) , = a(t, κ) + E where

a(t, κ) = E

0

(14.7.16)

K

0

−∞

K

h(t − s, η, κ)λ∆ (s, η) ds K (dη) .

(14.7.17)

If we set g(t, κ) = 0 for t < 0, g(t, κ) = E[λ∆ (t, κ)] for t ≥ 0, (14.7.16) has the form of a Markov renewal equation for the joint density with respect to ds K (dκ), namely, t g(t, κ) = a(t, κ) + K

0

h(t − s, η, κ)g(s, η) ds K (dη);

(14.7.18)

formally this has the solution t g(t, κ) = 0

K

a(t − s, η)R∗ (s, η, κ) ds K (dκ),

(14.7.19)

∞ in which R∗ (t, η, κ) = n=0 hn∗ (t, η, κ) is the sum of the iterates of h under the Markov convolution operation t (h ∗ g)(t, η, κ) = 0

K

h(t − s, η, ν)g(s, ν, κ) ds K (dν).

∞ Note that, in the notation of (14.7.8), 0 R∗ (t, η, κ) dt = R(η, κ) < ∞ for all t ≥ 0, and moreover that as a sum of iterates of the contraction operator H, R also satisfies R(η, κ)r(η) K (dη) = R∗ (t, η, κ)r(η) dt K (dκ) ≤ (1 − ρ)−1 r(κ). K

R+ ×K

(14.7.20) Because both intensities λ(t, κ) and λ† (t, κ) are dominated uniformly in t by multiples of r(κ), it follows easily from (14.7.17) that a(t, κ) is dominated by a multiple of r(·). Hence the integral in (14.7.19) converges, and from (14.7.18) and (14.7.20) we see that g(t, κ) is also dominated by a multiple of r(·).

436


We now consider the ground process of N∆ , N∆,g say, over the finite interval (t, t + T ), with the aim of proving that it converges weakly to the empty measure. We have t+T g(u, κ) du K (dκ). Pr{N∆,g (t, t + T ) > 0)} ≤ E[N∆,g (t, t + T )] = t

K

Substituting for g(·, ·) from (14.7.19) and using Fubini’s theorem, leads to the estimates t+T g(u, κ) du K (dκ) t K t+T u ∗ = a(u − s, η)R (s, η, κ) ds K (dη) du K (dκ) t

≤T 0

≤T

0

K t+T

0

K×K t+T K×K

K

a(u − s, η)R∗ (s, η, κ) ds K (dη) K (dκ) a(u − s, η) ∗ R (s, η, κ)r(η) ds K (dη) K (dκ). r(η)

(14.7.21)

Now a(t, κ)/r(κ) is bounded, and as t → ∞, using (14.7.17) and the inequality f (t, κ) ≤ C r(κ), we see that 0 h(t − s, η, κ)f (s, η) ds K (dη) a(t, κ) = −∞ K ∞ h(u, η, κ)r(η) du K (dη) → 0 ≤ C t

K

from the integrability of h. Also, as a function of (s, η) after integrating out κ, the second integral in (14.7.21) converges from (14.7.20), and so (14.7.21) as a whole represents a moving average of the function a(t, κ)/r(κ) which is bounded and converges to zero as t → ∞. Thus Pr{N∆,g (t, t + T ) > 0} → 0, from which we deduce that the fidi distributions of St N∆,g , and hence those of St (N ∆ N † ) itself, converge to the fidi distributions of the empty process, and hence that the fidi distributions of St N converge weakly to those St N † . But this implies (by Theorem 11.1.VII again) the weak convergence of St N to St N † , and hence the weak asymptotic stationarity of N . This argument fails if T = ∞, but under the additional condition (14.7.15), the expected value of the total number of points in N∆ is finite. Indeed, from the convolution representation (14.7.18), we have ∞ f (t, κ) dt K (dκ) E[N∆,g (0, ∞)] = 0 ∞ K ∞ ∗ a(u, η)R (s, η, κ) ds K (dη) du K (dκ) = K 0 ∞ K 0 = a(u, η)R(η, κ) K (dη) K (dκ) du. 0

K×K

14.7.


437

Substituting for a(u, η) from (14.7.17), using the inequality f (t, κ) ≤ C r(κ), and changing the order of integration, the last expression becomes ∞ ∞ h(v, ν, η)f (u − v, ν) dv K (dν) R(η, κ) du K (dη) K (dκ) 0 u K×K K ∞ ∞ ≤ C h(v, ν, η)r(ν)R(η, κ) du dv K (dν) K (dη) K (dκ) 0 u K×K×K ∞ v h(v, ν, η)r(ν) dv K (dν)R(η, κ) K (dη) K (dκ). = C 0

K×K×K

Incorporating the condition (14.7.15) we obtain for the last integral, J say, the telescoping sequence of reductions r(η)R(η, κ) K (dη) K (dκ) ≤ ρD r(κ) K (dκ) = ρD < ∞. J ≤ D K×K

K

But if the expected number of points of the difference process on R+ is finite, there must be (with probability 1) a finite last occurrence time, say L, of points in N∆,g . This L acts as a coupling time for the two processes N and N † , and strong asymptotic stationarity then follows from the basic coupling inequality of Lemma 11.1.I. The arguments are considerably simplified if the process is unmarked, and are outlined in Exercises 14.7.5–7. Example 14.7(a) Nonlinear Hawkes and ETAS models [see Example 7.3(b)]. Recall that the simple Hawkes model has conditional intensity of the form t λ∗ (t) = λ + 0 µ(t − s) N (ds), with nonlinear version t λ∗ (t) = Φ λ + µ(t − s) N (ds) .

(14.7.22)

0

This falls within the ambit of Proposition 14.7.III provided Φ(·) satisfies a standard Lipschitz condition of the form |Φ(x) − Φ(y)| ≤ α |x − y| (x, y ≥ 0). In this case conditions (14.7.6a–b) reduce, respectively, to µ(t) < ∞ and ∞ ρ = α 0 µ(u) du < 1. When Φ(x) ≡ x this reduces to the usual stability requirement for the Hawkes process. The standard initial condition is to suppose the process empty for t< 0, which certainly satisfies condition (14.7.10), whereas (14.7.9) reduces to Φ(λ) < ∞. Thus a version of the nonlinear process, starting from the empty ∞ initial condition, exists provided that both Φ(λ) < ∞ and α 0 µ(u) du < 1. From Theorem 14.7.IV, the same conditions also imply the existence of a stationary version of the process, with complete conditional intensity λ† (·)

438


as in (14.7.21) but with the integral taken from −∞. The same conditions imply weak asymptotic stationarity of the process started from the empty initial condition. The extra condition (14.7.15) required for strong asymptotic stationarity here reduces to ∞ tµ(t) dt < ∞. 0

Consider next a nonlinear version of the ETAS model as a simple example of a nonlinear marked Hawkes process. As in (7.3.10) we write the conditional intensity for the nonlinear version in the form µ(t − s)A(η) N (ds × dη) f (κ). λ∗ (t, κ) = Φ λ + (0,t)×K

where f (κ) is the density of the distribution of the ‘unpredictable marks’ and A(η) [alias ψ(η) in (7.3.10)] measures the increase in ‘productivity’ with the increase in η. Assuming a simple Lipschitz condition on Φ as above, we can identify the kernel h of (14.7.5) with h(t, η, κ) = α µ(t)A(η)f (κ). ∞ Then condition (14.7.6a) reduces to 0 µ(t) dt < ∞ and the crucial condition (14.7.6b), if we identify the subinvariant vector r with f , to the requirement that ∞ A(η)f (η) K (dη) µ(t) dt < 1. ρ=α K

0

Note that this requires the convergence of the integral K A(η)f (η) K (dη), and that in this case the subinvariant vector is strictly invariant. This condition, together with the condition Φ(λ) < ∞, imply both the existence of a stationary version, and the existence and weak asymptotic stationarity of a version started from the empty initial condition. The extra condition for strong asymptotic stationarity is the same as in the unmarked case. In the more explicit spatial version of Example 6.4(b), without the nonlinear generalization, we leave the reader to check that the conditions reduce to those quoted in the example.

Exercises and Complements to Section 14.7 14.7.1 Poisson embedding results for general histories. Extend Proposition 14.7.I(a) and (b) to the situation where the history Ft can be represented in the form Ft = Ht ∨Gt , where Ht refers to the internal history of the point process, and Gt to the history of a process evolving contemporaneously in parallel with the point process. [Hint: The crucial point for part (a) is to define the underlying Poisson process on a history sufficiently rich for both the point process and the auxiliary process to be H-adapted and H-predictable. See Massoulié (1998).]

14.7.


439

14.7.2 (Continuation). Specialize Proposition 14.7.I to the case that the conditional intensity of the MPP can be represented in the form (14.7.3). [Hint: The main changes relate to the requirement of H-predictability, and the appropriate introduction of the functional ψ.] 14.7.3 Denote by ZN (t) the process whose value at time t is the realization St N− on R− of an MPP with conditional intensity function of the form (14.7.3). Verify the Markov property for ZN (t) and examine the form of its infinitesimal generator. 14.7.4 Verify that the operators H and H ∗ of (14.7.7) do indeed define bounded linear operators on the spaces of functions K1 and K∞ defined in the text, and that their norms are bounded by ρ. [Hint: Use (14.7.6b) and a Fubini theorem argument applied to Hf =

K

|(Hf )(η)| r(η) K (dη) ≤

1 ∗ H g = ess sup r(κ) κ∈K ∞

assuming for H that ess sup[g(κ)] < ∞.

K

K×K

H(η, κ) |f (κ)| r(η) K (dη) K (dκ),

H(η, κ)g(η) K (dη) , |f (κ)| K (dκ) ≡ f < ∞, and for H ∗ , that g ≡

14.7.5 Check and prove directly the following restriction of Proposition 14.7.III to simple (unmarked) point processes. Let Ψ(N ) be a mapping Ψ: NR#− → R+ , satisfying Ψ(∅) = C < ∞ and the Lipschitz condition |ψ(N ) − ψ(N )| ≤ ∞ h(−s) (N ∆N )(ds), where h: R+ → R+ satisfies 0 < 0 h(t) dt < ρ < 1. R −

Suppose also that the initial condition satisfies (t) = E[ R h(t − s) N− (ds)] − ≤ D < ∞. Then there exists a unique point process N with finite mean ground rate, initial condition N− , and conditional intensity λ∗ (t) = Ψ(St N ) for t ≥ 0. State and prove a corresponding extension of Theorem 14.7.IV. [Brémaud and Massoulié (1996) give a version without the restriction to processes with finite mean rate.]

14.7.6 (Continuation). As a variant, prove that similar uniqueness theorems hold under the conditions that Ψ is bounded overall, but the requirement that ρ < 1 is weakened to ρ < ∞. Investigate an analogous theorem for MPPs under the assumption that Ψ(N, κ) ≤ M r(κ) for some finite M and function r that acts as a ρ-subinvariant function for the kernel h(t, η, κ). [For other variants, see also Kerstan (1964b), Brémaud and Massoulié (1996), and, for the marked case, Massoulié (1998).] 14.7.7 (Continuation). Say that a point process or MPP has bounded memory if the functional Ψ depends on N only on either (a) its past in the finite interval (−a, 0), or (b) a finite number of occurrence times t−k with t−k < 0. Investigate existence and stability theorems for point processes and MPPs with finite memory. [See Lindvall (1988) for a treatment of case (b) based on a regeneration point argument. See also Brémaud and Massoulié (1996).]

440


14.8. Point Process Entropy and a Shannon–MacMillan Theorem In this section we return to the discussion in Section 7.6 of point process entropy and information gain. Our aim is to extend and consolidate the theoretical background to the results presented there concerning entropy rates, likelihoods, and information gain. In particular we complete the discussion of Proposition 7.6.II and establish an ergodic result for point process entropy analogous to the Shannon–MacMillan theorem for independent sequences. In introducing any general concept of entropy it is important to bear in mind that entropy, like likelihood, is best regarded as defined relative to a reference measure which can be a probability measure, but need not be totally finite. The entropy of a discrete (i.e., purely atomic) distribution {pk : k = 0, 1, . . .} can be defined directly as the expectation pk log pk , (14.8.1a) Ha = E(− log pk ) = − but the natural analogue for a continuous distribution, with density f (x) say on X ⊆ R, namely, f (x) log f (x) dx, (14.8.1b) Hc = − X

is scale-dependent, and cannot be reached as the limit of approximating discrete distributions. For example, in approximating the continuous uniform distribution on (0, 1) by a sequence of discrete uniform distributions with mass 1/n at each of the points (k − 12 )/n, the entropy of this discrete approximation equals log n and consequently diverges as n → ∞. Similarly, for the uniform distribution on the unit hypercube in Rd , the discrete approximation to the entropy equals d log n = log(nd ) = log(# points used in discrete approximation), and again diverges as n → ∞. Indeed, it was just the differences in the rates of divergence that was recognized by Rényi (1959) as characterizing the dimension of the set on which the limit measure was carried, thus suggesting the definition of the Rényi dimensions introduced around (13.6.1). Intuitively, the infinite limits obtained from the discrete approximations can be regarded as stemming from the unreasonable requirement that observation of a real-valued random variable pins down its value precisely, that is, the observation specifies all the digits in its decimal representation, thus conveying infinite information. One way of overcoming the apparent difficulties in linking discrete and continuous entropies is to consider each entropy relative to a reference measure on the relevant carrying space. Then (14.8.1a) is considered as an entropy relative to the discrete measure with unit masses at each integer, and the continuous version at (14.8.1b) is considered as an entropy relative to Lebesgue measure. This leads to the concept of the relative or generalized entropy, an approach that also permits the definition of the entropy of a distribution on a general probability space. Suppose that (Ω, E, µ) is a measure space, and

14.8.

Point Process Entropy and a Shannon–MacMillan Theorem

441

P µ is a probability distribution on this space. Then the generalized entropy of P with respect to the reference measure µ is given by Λ(ω) log Λ(ω) µ(dω) = − log Λ(ω)P(dω) = EP (− log Λ), H(P; µ) = − Ω

Ω

(14.8.2) where Λ(ω) = (dP/dµ) is the Radon–Nikodym derivative of P with respect to µ. If P is singular with respect to µ, we set H(P; µ) = ∞. If µ = Q, where Q is again a probability measure, we can rewrite (14.8.2) in the form dP P(dω) (14.8.3) log −H(P; Q) = dQ Ω which, apart from the negative sign, identifies the generalized entropy with the expected value of the likelihood ratio (dP/dQ) under the assumption that P is the true distribution. This link to the likelihood ratio underlies the properties of the entropy scores introduced in Section 7.6. Indeed, the righthand side of (14.8.3) is nothing other than the Kullback–Leibler distance between the two probability measures P and Q. Convexity of the function x log x, when applied to the first form in (14.8.2), guarantees that this quantity is always nonnegative, and equals zero if and only if the two measures coincide. When both distributions are absolutely continuous with respect to a reference measure µ, then in the terminology of Section 7.6, the right-hand side of (14.8.3) represents the (expected) information gain, that is, the expected value of the logarithm of the probability gain,resulting from scoring outcomes by − log dP/dµ rather than − log dQ/dµ , when the true distribution is really P. In this formulation the reference measure µ drops out of the comparison, or can be taken to be Q itself as in (14.8.3). Turning to point process entropies, we start from the entropy of a point process observed over a state space X which we take to be a bounded region A ∈ Rd . The distribution of the point process can then be regarded as a symmetric probability distribution on the countable union X ∪ . Observation of a realization of the process conveys information of two kinds: the actual number of points observed, and the location of these points given their number. Assuming absolute continuity with respect to Lebesgue measure for the distribution of locations, and bearing in mind that the points are indistinguishable, we can write the probability density for the latter term in the form k! πksym (x1 , . . . , xk ; A), where πksym denotes a symmetric probability density over A(k) . This suggests defining the entropy of a realization {x1 , . . . , xN } as H ≡ H(N ; x1 , . . . , xN ) = H(N ) + E H(x1 , . . . , xN | N ) ∞ pk log pk =− k=0 ∞ − pk πksym (x1 , . . . , xk ) log[k! πksym (x1 , . . . , xk )] dx1 . . . dxk . k=1

X (k)

(14.8.4)

442


This follows the notation of Section 5.3, with the factor k! arising, just as in the discussion of likelihoods, from the fact that only unordered point sets can be distinguished, so that any given allocation of particles to points is repeated k! times. Rudemo (1964) and McFadden (1965b) introduced point process entropy effectively in this form. From an entropy viewpoint, the extra factor log k! can also be regarded as the loss of information, for given k and a given set of locations, about which particle is located at which location. Under the assumption of indistinguishable points, there are k! permutations from which to choose, all of them equally likely, corresponding to a distribution with entropy log k! . Notice that (14.8.4) can be written sym (x1 , . . . , xN )] H = −EP log[pN N ! πN = −EP log[jN (x1 , . . . , xN )] = EP (− log L), where L is the likelihood, identified as in Section 7.1 with the Janossy density. As with the entropy of distributions with a continuous density considered earlier, the definition at (14.8.4) is scale-dependent: approximating each of the on n points, results in a discrepancy densities πk by a discrete distribution which increases as pk log k log n = E(N ) log n. For this reason E(N ) is sometimes regarded as the dimension of the distribution of a finite point process on R. The implicit reference measure here has d-dimensional Lebesgue measure on each constituent space X d ⊆ Rd and unit mass at each nonnegative integer. The alternative is to proceed as in Section 14.4 and take the reference measure to be the probability distribution of some standard process, usually the Poisson process with unit rate, so that (dP/dQ) reduces to the likelihood ratio relative to this standard process. When this is the Poisson process, and X is a bounded set A ∈ Rd , the net effect is merely to add an extra term (A) to (14.8.4) (see Exercise 14.8.1). The expected value of such a log likelihood ratio is just the information gain as introduced in Section 7.6, which therefore appears as the negative of the corresponding generalized entropy. It takes the same form whether the points are treated as distinguishable or indistinguishable, and is given by G=

∞ k=0

∞

pk log

pk + pk qk k=1

A(k)

πksym log

πksym dx1 . . . dxk . qksym

(14.8.5)

The two expressions (14.8.4) and (14.8.5) illustrate the difference in intent between an absolute and a generalized entropy, giving the entropy relative to a measure reflecting the structure of the space on which it is defined, or else the information gain, which compares the entropies of two probability measures defined within similar structural constraints. For a simple or marked point process in time, entropy can be represented alternatively in terms of conditional intensities. Suppose that P (which we use

14.8.


443

as a shorthand for PT , the probability measure restricted to the events generated by the point process in [0, T ]) corresponds to a MPP on [0, T ] × K, and that we represent the likelihoods in terms of conditional intensities λ∗ (t, κ) relative to the internal history H as in Definition 14.3.I(b). The generalized entropy and the information gain relative to an alternative probability measure P 0 can be found by taking expectations of the likelihood ratio as set out in (14.4.2). In particular, adopting the notation of (14.4.2), the expected information gain for MPPs over an interval (0, T ) can be written in the form GT (P; P 0 ) = EP [dP/dP 0 ] T 0 =E log µ(t, κ) N (dt × dκ) − [µ(t, κ) − 1]λ (t, κ) dt K (dκ) , 0

(0,T )×K

K

where µ(·) is the ratio of H-conditional intensities λ∗ (·) and λ0 (·) under P and Q ≡ P 0 , respectively. Taking predictable versions of the conditional intensity, the last expectation simplifies to give T T λ∗ (t, κ) log µ(t, κ) dt K (dκ) − [mg (t) − m0g (t)] dt, GT (P; P 0 ) = E 0

K

0

(14.8.6) where we have written mg (·) and m0g (·) for the ground rates under P and P 0 respectively. For simple point processes, a similar expression holds but without any marks κ or integral over K. To obtain the generalized entropy, it is necessary to identify the appropriate reference measure. When the point process is simple the reference measure QT is the nonnormalized measure corresponding to eT Poi(1, T), where Poi(λ, T ) is the probability measure of a Poisson process on (0, T ) with constant rate λ. For an MPP the nonnormalized measure is a similar multiple of a compound Poisson process with unit rate and mark distribution π(dκ). Adopting these conventions, we obtain the corresponding generalized entropy in the form T ∗ ∗

∗ λ (t, κ) log λ (t, κ) − [λ (t, κ) − 1] dt π(dκ) , (14.8.7) HT = −E 0

K

In general, the expressions (14.8.6–7) are not easy to evaluate explicitly, although they are usually straightforward to obtain from simulations. It does simplify, however, when the MPP is stationary. In this case we use the extension of the likelihood (14.4.2) to intrinsic conditional intensities (i.e., con0 , so that for ditioned on some initial σ-algebra G0 ) and we take G0 = H−∞ G † t > 0, λ (t) can be identified with the complete intensity λ (t). In this case, for all t, we have E[λ† (t, κ)] = E[λ† (0, κ)] = mg E[f † (κ | 0)], where mg = E[λ†g (0)] is the overall mean rate, which we assume finite. Then, for example, the expected information gain (14.8.6) takes the form † † λ (0, κ) log µ (0, κ) π(dκ) − [mg − m0g ] ≡ T G, (14.8.8a) GT = T E K

444


where G is the expected information gain per unit time of2 (7.6.14). It is often more usefully written as G being equal to f † (κ | 0) f † (κ | 0) log ¯ π(dκ) . E λ†g (0) log λ†g (0)] − [mg − m0g ] + E log λ†g (0) f (κ) K (14.8.8b) Then the first two terms represent the information gain due to the times of the points, and the third term the conditional information gain due to the marks of the points, given their occurrence times. Similarly we can define an entropy rate by HT = −E λ† (0, κ) log λ† (0, κ) π(dκ) + [mg − 1] H≡ T K f † (κ | 0) log f † (κ | 0) π(dκ) + [mg − 1]. = −E λ†g (0) log λ†g (0) + K

(14.8.9)

For reference, the last results are summarized in the proposition below. Proposition 14.8.I. Let N be a simple or marked stationary point process on R, with complete intensity function (determined by the internal history and relative to a reference distribution π(·) on the mark space) λ† (t, κ) = λ†g (t)f † (κ | t). Suppose that E[λ†g (0) log λ†g (0)] < ∞, so that N has finite ground rate mg = E[λ†g (0)]. Then the expected information gain G, relative to a compound Poisson process with unit ground rate and strictly positive mark density f¯(κ) is given by (14.8.8), and the entropy rate H by (14.8.9). Example 14.8(a). Simple, mixed, and compound Poisson processes. The simplest example is a simple Poisson process of constant rate µ. In this case G = µ log µ − (µ − 1), representing the expected information gain per unit time over the unit rate Poisson process. Note the implicit ratio in the first term: each µ is really the ratio µ/1 of rates between the true and reference processes. As a function of µ, G is 0 when µ = 1, that is, when the true process coincides with the reference process, and is otherwise strictly positive. The mean entropy rate is obtained by changing signs and omitting the −1 in the final bracket: H = µ − µ log µ. Exercise 14.8.2 shows that amongst all point processes with given rate λ, the Poisson process has maximum entropy rate. In the case of a mixed Poisson process, where the rate λ is a random variable with distribution function F (·) say, the conditioning on G0 must be taken into account, and the information gain becomes λ[log λ − 1] dF (λ). G= R+

2 Equation (7.6.14) in Volume I contains two errors: in the second term +m E[· · ·] a factor g † fk|0 is omitted and it is wrongly assumed that the process has unpredictable marks, so K

† λ†g (0)fk|0

† fk|0

. In the ensuing display, G fk equals the right-hand side but the middle expression is also flawed and should be deleted. that the term should instead read + E

k=1

log

14.8.


445

κ = 0.2 (G=1.898)

Poisson (G=0)

κ = 5 (G=0.456)

κ = 25 (G=1.204)

Figure 14.2 First 40 points of realizations of four different gamma-distributed renewal processes, same mean but shape parameters κ = 0.2, 1, 5, 25, and (expected) information gains G as shown.

Consider finally a compound Poisson process with ground rate µ and mark distribution {pk : k ∈ Z+ }. The mark distribution is here discrete, and the reference distribution on the mark space can be any distribution f¯k on the nonnegative integers for which all terms are strictly positive. Because the marks are chosen independently of the previous history of the process, (14.8.6) simplifies to ∞ pk log(pk /f¯k ). G = µ log µ − (µ − 1) + µ k=0

As does (14.8.6) more generally, this equation represents a decomposition of the expected information gain into two components: the gain due to modelling the counts, and the gain due to modelling the marks. Example 14.8(b) Renewal processes with gamma interevent distributions. It is shown in Example 7.6(b) that the information gain for a renewal process with lifetime density f (·), relative to a Poisson process with the same mean rate m, takes the form ∞ f (y) dy . (14.8.10) f (y) log G=m 1+ m 0 Figure 14.2, adapted from Daley and Vere-Jones (2004), illustrates the way G can vary with the model as the character of the interevent distribution is changed. Here the mean rate is fixed but the shape parameter is allowed to vary. Details of the computations are sketched in Exercise 14.8.3. We turn next to approximations of point process entropies by entropies of systems of discrete trials, with the aim of consolidating the ideas introduced in Section 7.6; we follow broadly the treatment in Daley and Vere-Jones (2004). Suppose first that the process is unmarked, and that the observation interval (0, T ] is partitioned into subintervals Ai = (ui−1 , ui ], i = 1, . . . , kn , with u0 = 0, ukn = T for which forecasts are required. We observe either the values Xi = N (Ai ) or the indicator variables Yi = IN (Ai )>0 , but not the

446


individual points. Our aim is to show how the information gain and associated quantities for the point process can be approximated by the analogous quantities for the Xi or the Yi , both of which may be thought of as sequences of dependent trials. In order to embed this situation into a limit process, we suppose that the given partition is a member of a dissecting family of partitions, say Tn , n = 1, 2, . . . , as defined in Section A1.6. The crucial partition, however, is not the partition of the state space but the partition this induces on the probability space Ω. For example, a partition of (0, T ] into r subintervals induces a partition of the probability space into 2r events, each corresponding to a particular sequence of values of the indicator variables Yi . We denote by An the algebra of events generated in this way by the partition Tn . To consider the effect of refining the partition, take a particular partition Tn0 , say, and define for each set A ∈ Tn0 the sequence of associated processes η (n) (A), where for n < n0 , η (n) (A) = 0, and for n ≥ n0 , (n) η (n) (A) = Yi = IN (A(n) )>0 . (14.8.11) (n)

i:Ai

⊆A

(n)

i:Ai

⊆A

i

For n > n0 , η (n) (A) counts the numbers of subintervals of A which belong to Tn and contain a point of the process. It is clear that, for increasing n, the η (n) are nondecreasing. Indeed, because the partitions form a dissecting system, and the point process is assumed to be simple, each point of the process will ultimately be the sole contributor to one of the nonzero terms η (n) (A), so that η (n) (A) ↑ N (A). Thus, any event {N (A) = k} can be approximated by the corresponding events {η (n) (A) = k}. More generally, any event defined by the simple point process in (0, T ] can be approximated /∞ by events determined by the processes η (n) (·), or equivalently H(0,T ] = n=1 An . A similar argument holds also for marked point processes. Here we consider a dissecting family Tn of partitions of the product space (0, T ] × K, each of which is of product form Vn × Wn , so that each element of Tn is a rectangle V × W , where V ∈ Vn , W ∈ Wn respectively. A family of processes ζ (n) (·) (n) can be defined much as in (14.8.11), but with the Ai in (14.8.11) interpreted as rectangles from (0, T ] × K. In this case, if (0, T ] is partitioned into r subintervals, and K into s components, the σ-algebra An will be generated by (s + 1)r distinct events, each corresponding to a sequence of length r, each term in which can be any one of the s marks, or a special mark φ to allow for the possibility that no events occur in the subinterval in question. Once again we find that, for any rectangle set A × K from one /∞of the partitions Tn , ζ (n) (A × K) ↑ N (A × K), and consequently H(0,T ] = n=1 An . The following lemma summarizes the conclusions. Lemma 14.8.II. Let N be a marked point process with simple ground process and let ζ (n) be defined as in (14.8.11) with respect to a dissecting family of partitions of (0, T ] × K. Then the processes ζ (n) (A × K) generate the internal history H(0,T ] .

14.8.


447

To proceed further, we need to clarify the role of the underlying σ-algebra in defining the generalized entropy. Here we follow the treatment of Csisz´ ar (1969) and Fritz (1969). Suppose given a measure space (Ω, E), a σ-algebra A ⊆ E, and two measures on (Ω, E), a probability measure P, and a reference measure (not necessarily a probability measure) Q; suppose also that P Q on A, with Radon–Nikodym derivative (dP/dQ)A . As before, but explicitly recording the σ-algebra, we define the generalized entropy on (Ω, A) as dP dP log Q(dω) dQ A Ω dQ A dP dP . log P(dω) = EP − log =− dQ A dP 0 A Ω

H(P, Q; A) = −

(14.8.12)

When Q(·) is itself a probability measure, we prefer to change the sign and refer instead to the information gain dP dP dP 0 log P (dω) = log P(dω) , 0 A dP 0 A dP 0 A Ω dP Ω (14.8.13) with G(P, P 0 ; A) ≥ 0, and it is set equal to +∞ if in fact P P 0 ; note that the integral can diverge even when the absolute continuity condition holds. The following two properties of the generalized entropy, taken from Csisz´ ar (1969) as in Fritz (1969) but restated here in terms of information gains, are of crucial importance (for a sketch of the proof and some related material, see Exercises 14.8.4–6). G(P, P 0 ; A) =

Lemma 14.8.III. (a) The information gain G(P, P 0 ; A) is nondecreasing under refinement of the σ-algebra A. (b) Let {Aα } be a family of σ-algebras generating A, and such that, to every Aα1 , Aα2 , there exists Aα3 ∈ {Aα } for which Aα1 ∪ Aα2 ⊆ Aα3 . Then

G(P, P 0 ; A) = supα G(P, P 0 ; Aα ) . Suppose, in particular, that A(n) is the σ-algebra derived from the ζ (n) (A) associated with a dissecting family of partitions as in Lemma 14.8.II. Then the condition of part (b) of Lemma 14.8.III holds as a consequence of the nested property of the partitions in a dissecting family, and because the family of σ-algebras An is also monotonic increasing with limit H(0,T ] , the two lemmas together imply the first part of the following proposition, which provides an extension and minor strengthening of Proposition 7.6.II. Proposition 14.8.IV. Consider an MPP N defined on (0, T ] × K, with distribution P on (Ω, E), and internal history H, and let P 0 be an alternative probability measure on (Ω, E). Also let Tn = Vn × Wn be a dissecting family

448


of partitions of (0, T ] × K, and An the σ-algebra of events induced by Tn through the approximations ζ (n) of Lemma 14.8.II. (a) If G = G(PT , QT ; H) and Gn = G(PT , QT ; An ) denote the expected information gains associated with N and ζ (n) , respectively; then Gn ≤ G and Gn ↑ G as n → ∞. (b) The same conclusions hold for the information gains relative to the intrinsic histories G = H ∨ G0 and {Gn = An ∨ G0 }. Proof. Part (a) is a direct consequence of Lemmas 14.8.II–III. Part (b) follows by first conditioning on G0 and applying part (a), then taking expectations over G0 . An important extension to these results is to situations where the point process evolves alongside another, stochastically related, process which can provide predictive information about the point process. It is in situations of this kind that more general conditional intensities λF arise, supposing that the joint history is available for observation, but only the point process needs predicting. For such situations it is also possible to consider information gains of the type (14.8.6–10), even though they lose their strict interpretation as expected values of likelihood ratios. We attempt only an informal sketch of this development. We consider just the case where a simple point process N (t) evolves alongside a second cumulative process W (t), both processes being observed over (0, T ], or in terms of their increments over a family of partitions of (0, T ). (n) (n) Specifically, let Yi , Wi denote the observed values of the increments of (n) the processes N (t), W (t) over the set Vi ∈ Vn defined as in the previous (n) (n) (n) discussion, let pi (Y | Fi−1 ) denote the conditional distribution of Yi given (n)

observations (Yj (n)

bution of Yi gain

(n)

(n)

(n)

, Wj ) for 0 ≤ j ≤ i − 1, and pi (Y | Hi−1 ) the distri(n)

given Yj

(0 ≤ j ≤ i − 1) only. This gives an information

(n) (n) (n) pi (Yi | Fi−1 ) (n) (n) (n) Gn = E , pi (Yi | Fi−1 ) log (n) (n) (n) pi (Yi | Hi−1 ) i

(14.8.14)

n ≥ 0 holds essentially as a result of the well-known and the inequality G inequality for entropies, H(X | Y ) ≤ H(X). Lemma 14.8.III can now be applied to show that, in this situation also, refinement of the partitions can only increase the information gain. Introducing a further partition point θ inside a subinterval (ui−1 , ui ] does not affect the information available to the two histories at the beginning of the first new subinterval, (ui−1 , θ], but in the second new subinterval, (θ, ui ], the full process obtains new information about both the point process and the explanatory variables, whereas the reference process obtains only the additional information about the point process. Hence the information gain can only increase with the introduction of the new partition.

14.8.


449

If now we let n → ∞ in (14.8.14), and suppose that conditional intensities λF (t, κ), λH (t, κ) exist, we obtain the limit, which again is also an upper bound, T λF (t, κ) λF (t, κ) dt K (dκ) , log H G(P, P 0 ; AF ) = E λ (t, κ) 0 K the second term in (14.8.6) disappearing because the processes have the same expected rates. In particular, if the two processes are stationary, the information gain per unit time G equals λF f F (κ | 0) g (0) F F + log λ (0) log (0) f (κ | 0) log (dκ) . (14.8.15) E λF K g g λH f H (κ | 0) K g (0) Notice that these equations have the same form as if we had taken expectations of a likelihood ratio, but in fact no true likelihood ratio exists here because the F -intensity by itself does not define a process but only the way the point process component is determined by the full process. See Exercise 14.8.7 for a hidden Markov model example. The last question we take up in this section is that of approximating the expected entropy rate from finite samples, as in MacMillan’s theorem for a discrete ergodic source with finite alphabet [see, e.g., Billingsley (1965)]. Here, the basic statements assert the convergence of the log likelihoods T −1 log LT , either a.s. or in L1 norm. We consider simple point processes only, and follow Papangelou (1978) in deriving the L1 version of this result. Extensions to the MPP context are sketched in Exercise 14.8.11. The main problem is that, although we have derived expressions for the entropy rate from a form of likelihood involving the complete intensity function, in reality the best that is likely to be available is the conditional intensity based on the internal history, starting from an empty history at time 0. Thus there are two main steps in the proof: first, establish the convergence of the pseudolikelihoods in which the complete intensity plays the role of the intrinsic intensity over a finite interval, and second, show that the difference between the true and pseudolikelihoods is asymptotically negligible. These two steps are set out in the next two lemmas. Lemma 14.8.V. Suppose E[λ† (0) log λ† (0)] < ∞ for a simple stationary point process N . Then as T → ∞, 1 T log λ† (t) dN (t) → E[λ† (0) log λ† (0) | I] (14.8.16) T 0 both a.s. and in L1 norm, where I is the σ-algebra of invariant events. Proof. Because N is stationary, the process λ† (t) is stationary. Also, λ† (t) is H† -predictable so the set function log λ† (u) N (du) (bounded A ∈ B) ξ(A) = A

450


may be regarded as a stationary random signed measure with mean density m = E[λ† (0) log λ† (0)]. The result (14.8.16) then follows from the a.s. and L1 ergodic results of Proposition 12.2.IV. [This involves noting that the two processes ξ+ (A) =

[log λ† (u)]+ N (du)

and

A

ξ− (A) =

A

[log λ† (u)]− N (du)

are both nonnegative measures " to which the theorems apply directly, with the " finiteness of E"λ† (0) log λ† (0)" following from x log x ≥ −e−1 (x ≥ 0) and the finiteness assumption in the lemma.] In the next result the monotone family of σ-algebras {H(−T,0) } generated by {N (t): −T < t < 0}, which increase as T → ∞ to H0− , plays a key role. First, the nonnegativity of λ ≡ λ† (0) and assumed finiteness of E(λ log λ) ensure that E(λ) < ∞. Next, the family {λT : 0 < T < ∞}, where λT = E[λ† (0) | H(−T,0) ),

(14.8.17)

constitutes a martingale, which by definition and the finiteness of E(λ) is uniformly integrable, so λT → λ both a.s. and in L1 norm by the martingale convergence theorem. But the function x log x is convex in x, so by Jensen’s inequality, ∞ > E(λ log λ) = E[E(λ log λ | H(−T,0) )] ≥ E(λT log λT ), and because x log x ≥ −e−1 (all x ≥ 0), it follows that E(λT log λT ) is welldefined and finite. Lemma 14.8.VI. Under the conditions of Lemma 14.8.V, with λT as at (14.8.17), " " E"λ† (0) log λ† (0) − λ† (0) log λT " → 0

(T → ∞).

(14.8.18)

Proof. Because λT → λ in distribution and ∞ > E(λ log λ) ≥ E(λT log λT ), and also 0 ≤ λ log(λT /λ) I{λT > λ} ≤ λ(λT /λ − 1) = λT − λ for which λT → λ in L1 norm, it is enough to show that E(λT log λT ) → E(λ log λ). Let x ≥ 1 be a continuity point of the distribution of λ; then for sufficiently large x we can certainly make E(λI{λ>x} log λ) < for arbitrary > 0. Now E[max(x log x, λT log λT )] = E max x log x, E(λ | H(−∞,0) ) log[E(λ | H(−∞,0) )] ≤ E[max(x log x, λ log λ)]

14.8.


451

by Jensen’s inequality because max(x log x, y log y) is convex in y > 0 for x ≥ 0. Because x is a continuity point for λ with x ≥ 1, 0 ≤ E(λT I{λT >x} log λT ) = E[max(x log x, λT log λT )] − x log x Pr{λT ≤ x} ≤ E[max(x log x, λ log λ)] − x log x Pr{λT ≤ x} → E[max(x log x, λ log λ)] − x log x Pr{λ ≤ x} = E(λI{λ>x} log λ) < , with the convergence holding uniformly for T sufficiently large. Theorem 14.8.VII. Let the simple stationary point process N admit Hpredictable complete intensity λ† (t) and H-predictable conditional intensity λ∗ (t) on t ≥ 0 and be such that H ≡ −E λ† (0)[log λ† (0) − 1] is finite, so that m = E[λ† (0)] is finite also. Then as T → ∞, H(0,T ] →H T and

log L(0,T ) → E(Z | I) T

a.s.

(14.8.19)

in L1 norm,

(14.8.20)

where Z = λ† (0)[log λ† (0) − 1] and I denotes the σ-algebra of invariant events for N . Proof. Convergence as in (14.8.19) follows from the definition of H(0,T ] , H, and the conditions and results of the last two lemmas. To prove (14.8.20), consider the difference " " " " log L(0,T ) " − E(Z | I)"", (14.8.21) E" T which by virtue of the triangle inequality is dominated by T1 + T2 + T3 + T4 , where " " T " 1 "" T ∗ † log λ (t) dN (t) − log λ (t) dN (t)"", T1 = E" T 0 0 " " T " 1 "" T ∗ † λ (t) dt − λ (t) dt"", T2 = E" T 0 0 " T " "1 " log λ† (t) dN (t) − E λ† (0) log λ† (0) | I "", T3 = E"" T 0 " T " "1 † " † " λ (t) dt − E λ (0) | I "". T4 = E" T 0

452


Here, T3 → 0 by Lemma 14.8.V, and applying the ergodic theorem to the stationary process λ† (t) implies that T4 → 0. By assumption λ∗ (t) and λ† (t) are predictable on R+ and R, respectively, so both are H-predictable on R+ , and thus T1 is dominated by 1 E T

T

" " " log λ∗ (t) − log λ† (t)"λ† (t) dt.

0

Recall from the projection Theorem 14.2.II that λ∗ (t) can be replaced by a suitably chosen version of E[λ∗ (t) | Ht− ] without altering the value of the integrals in T1 and T2 . Using stationarity, replace (0, T ) by (−T, 0), which leads to 0 1 = E T −T 0 1 T2 = E T −T

T1

" " " log E λ† (t) | H(−T,t) − log λ† (t)"λ† (t) dt , " † " "E λ (t) | H(−T,t) − λ† (t)" dt .

For each fixed t, the expectation of the first integrand → 0 as T → ∞ by Lemma 14.8.VI and stationarity, so the (C, 1) mean also converges to zero. In the proof of the same lemma, the expectation of the second integrand also converges to zero, so the (C, 1) mean does also. A different approach is to ask for entropy rates associated with the point process on R in its dual form as a stationary process of intervals. Extensions of McMillan’s theorem from its original context of a discrete time finitealphabet source (i.e., from a finite state space stochastic process) to stationary sequences of random variables with arbitrary distributions were developed by Perez (1959) and can be applied directly to the process of intervals. In particular, Perez showed that if {Xn : n = 0, ±1, . . .} is a stationary sequence of r.v.s taking their values on the c.s.m.s. X , if Π is a fixed totally finite or σ-finite measure on X , and if the fidi distributions Fk of order k of the sequence are absolutely continuous with respect to the k-fold product measure Π(k) = Π × · · · × Π on (X (k) , B(X (k) )), then −

1 log n

dFk (X1 , . . . , Xn ) dΠ(k) (X1 , . . . , Xn )

→ E(Z | I),

where the invariant r.v. Z has expectation (finite or infinite)

∞

E(Z) = E

∗ dF (x | H(−1) )

0

=E

∞

log 0

log

∗ dF (x | H(−1) )

Π(dx) ∗ ) dF (x | H(−1) Π(dx)

Π(dx)

Π(dx)

∗ dF (x | H(−1) ),

14.8.


453

∗ where F (· | H(−1) ) is a regular version of the conditional distribution of X0 ∗ given the sequence of past values {X−1 , X−2 , . . .} which generate H(−1) . This −µx result can be applied directly to our context if we take Π(dx) = µe dx (x ≥ 0), that is, the stationary interval distribution on R+ associated with the Poisson process with constant mean rate µ. This leads to the result that ∞ µ f (x | τ) log f (x | τ) dx + log µ − , HI (P; Pµ ) = −E0 m 0

where f (· | τ) is the conditional intensity introduced in Corollary 14.3.VI, E0 is used to denote expectations over the vector of past intervals τ, and HI denotes the ‘interval entropy rate.’ If, in particular, we take µ = 1 and use Q to denote the measure corresponding to Π(dx) = e−x+1/m dx, we have similarly that ∞ f (x | τ) log f (x | τ) dx. (14.8.22) HI ≡ HI (P; Q) = −E0 0

This interval entropy rate is easily related to the entropy rate H by appealing to Corollary 14.3.VI. As in the proof of Proposition 14.3.V, for any function h(T, τ) of the backward recurrence time T and the past sequence τ, τ0 ∞ y h(τ0 − x, τ) dx = m E0 f (y | τ) dy h(x, τ) dx E h(T, τ) = m E0 0 0 0 ∞ = m E0 h(x, τ)[1 − F (x | τ)] dx. 0

Now by taking for h(T, τ) the function f (T | τ) f (T | τ) log , 1 − F (T | τ) 1 − F (T | τ) it follows from Corollary 14.8.V that E λ† (0) log λ† (0) ∞ ∞ f (x | τ) log f (x | τ) dx − log[1 − F (x | τ)] f (x | τ) dx = m E0 0 ∞ 0 f (x | τ) log f (x | τ) dx + 1 , = m E0 0

and hence

H = −E λ∗ (0) log λ∗ (0) − λ∗ (0) ∞ f (x | τ) log f (x | τ) dx = mHI . = −m E0 0

Thus, H(P; Pµ ) = mHI (P; Pµ ), which leads to the following statement. Proposition 14.8.VIII. For a simple stationary point process with mean rate m, the entropy rate per unit time equals m times the entropy rate per interval.

454


Exercises and Complements to Section 14.8 14.8.1 Show that, in comparison with (14.8.4), the relative entropy relative to the Poisson process contains the additional term (A). [Hint: Substitute for the Poisson probabilities in the expression at (14.8.5) for EP ( log(dP/dQ)), so the term log qksym reduces to log ([(A)]n ).]

∞ ∞ 14.8.2 (a) Show that, subject to the conditions k=0 pk = 1, k=1 kpk = µ and pk ≥ 0, the Poisson distribution maximizes the sum − ∞ k=0 pk log(k! pk ). (b) Deduce that for a regular point process on a bounded interval D ⊂ Rd with E[N (D)] = µ = const., the point process entropy (14.8.4) is maximized when the process is Poisson with uniform mean rate over D. [Hint: Start by writing (14.8.4) in the form

−H =

pk log(k! pk ) +

pk

D (k)

πk (y) log πk (y) dy.

Now use (a) together with the fact that, conditional on k, the integral is maximized, subject to πk (y) dy = 1, when πk (y) reduces to a uniform distribution over D(k) .] 14.8.3 Entropy of renewal process with gamma lifetimes. For the gamma density function fκ (x; a) = e−ax (ax)κ−1 a/Γ(κ), with shape parameter κ and mean κ/a, so mean rate m = a/κ, the information gain G at (14.8.10) equals ∞ m(1 − log m + 0 fκ (x; a) log fκ (x; a) dx). Show that the integral equals −κ − log Γ(κ) + log a + (κ − 1)ψ(κ), where ψ(z) = Γ (z)/Γ(z) is the digamma function. Known expansions for ψ yield

G = m( 12 +

1 2

log(κ/2π)+ 13 κ−1 +

1 −2 κ 12

+

1 −3 κ 90

−

1 κ−4 120

+ · · · ).

[Hint: Daley and Vere-Jones (2004) give more detail.] 14.8.4 Let (Ω, E, µ) be a measure space and P a probability measure on (Ω, E) with P µ. Let {Aα } be a family of finite or countable subalgebras of E. Define the generalized entropies H(P; µ), H(Pα ; µα ) by −H(P; µ) = −H(Pα ; µα ) =

dP dP dP log µ(dω) = log P(dω), dµ dµ dµ P(Uαj ) P(Uαj ) log , j µ(Uαj )

where {Uαj } is an irreducible (and countable) partition generating Aα . (i) If Aα ⊆ Aβ then H(Pα ; µα ) ≥ H(Pβ ; µβ ). (ii) If {Aα } generates E, then −H(P; µ) = inf α {−H(Pα ; µα )}. (iii) Let PT , QT be defined as in Proposition 14.8.IV, and suppose that {Tα } is a family of finite or countable partitions of the interval (0, T ], so that Tα = {Aαi : i = 1, . . . , Kα } such that {Tα } generates B((0, T ]). Let Aα be the algebra of events generated by Zαi = N (Aαi ). Show that −H(PT ; QT ) = inf α {−H(PαT ; QαT )},

14.8.

Point Process Entropy and a Shannon–MacMillan Theorem where −H(PαT ; QαT ) =

n

pαn log

455

pαn , qαn

n = (n1 , . . . , nKα ), pαn = P{Zαi = ni , i = 1, . . . , Kα }, and qαn = Kα ni i=1 [(Aαi )] . [This result provides a discrete approximation to generalized entropy. See Csiszar (1969) and Fritz (1973).] 14.8.5 (a) Show that H(PT ; QT ) can also be characterized as −H(PT ; QT ) = inf α {

p

log pαn − T log δα },

αn

where δα = maxi µ(Aαi ). (b) Show also that −H(PT ; QT ) = inf

α

παn log

παn + qαn

P{Z

Kα

αi

≥ 1} log µ(Aαi ) ,

i=1

where παn = P{Yαi = ni , i = 1, . . . , Kα } for ni = 0 or 1 and Yαi = I{N (Aαi )>0} . [Hint: Show first that if Tα ⊆ Tβ and each set Aαi of Tα is a union of no more than r sets of Tβ , then

p

αn

log pαn ≤

p

βn

log pβn ≤

p

αn

log pαn + T log r.

See Fritz (1973) for further details. This result shows that in some sense E(N (0, T ]) plays the role of a dimension for the process, as in Rényi’s (1959) discussion of ‘dimensional entropy.’] 14.8.6 Generalize the results of Exercises 14.8.4–5 to the case of a general state space X with Poisson distribution having nonatomic parameter measure µ(·). 14.8.7 Information gain in a Markov modulated point process (MMPP). (a) Let X(t) be a stationary continuous-time Markov process on finite state space X, with stationary distribution πi = Pr{X(t) = i} (all t, i ∈ X), so the rate of jumps between states equals i∈X πi qi = λ say, where the qi are as in Section 10.3 below (10.3.2). Write down an expression for the information gain of the point process consisting of jumps of the Markov process relative to a Poisson process at rate λ. (b) Suppose that, when X(t) = i, points occur in a Poisson process at rate λi [i.e., the point process is a Cox process directed by the states of the hidden Markov process X(·)]. Find the information gain of this point process compared with a Poisson process at rate i∈X πi λi . (c) Compare the results of (a) and (b) and interpret. (d) Consider extensions to (b) when it is a MPP that is directed by X(·), where both the mark distribution and the frequency of observed points msy depend on X(·).

14.8.8 Calculate the entropy rate for a stationary renewal process whose lifetime p.d.f. has unit mean and (a) is exponential on (0, ∞); or else, for some a in (0, 1), is (b) uniform on (1 − a, 1 + a), or (c) triangular on (1 − a, 1 + a). (d) Interpret the limits from (b) and (c) when a → 0.

456


14.8.9 The entropy rate of a stationary renewalÊ process with absolutely continu∞ lifetime d.f. with density f (·) equals 0 f (x) log f (x) dx, where m−1 = Êous ∞ xf (x) dx. One technique used in approximating the behaviour of a sta0 tionary point process N on R (e.g., for simulation purposes) is to replace N by a renewal process whose lifetime d.f. coincides with the stationary interval distribution. Use Proposition 14.8.VIII to show that the entropy rate of such an approximating stationary renewal process is larger than that of N . 14.8.10 Investigate extensions to MPPs of the arguments used to prove Lemmas 14.8.V–VI and Theorem 14.8.VII. Show also that the entropy rate for a stationary MPP can be given in a form analogous to that of Proposition 14.8.VIII by considering the bivariate sequence {(τi , κi−1 )}, where κi is the mark associated with the point initiating the interval τi , and applying the corresponding version of Perez’s result noted before Proposition 14.8.VIII.

CHAPTER 15

Spatial Point Processes

15.1 15.2 15.3 15.4 15.5 15.6 15.7

Descriptive Aspects: Distance Properties Directional Properties and Isotropy Stationary Line Processes in the Plane Space–Time Processes The Papangelou Intensity and Finite Point Patterns Modified Campbell Measures and Papangelou Kernels The Papangelou Intensity Measure and Exvisibility

458 466 471 485 506 518 526

This last chapter provides an introduction to spatial point processes, meaning for the most part results for point processes in R2 and R3 where the order properties of the real line, which governed the development in the preceding chapter, are no longer available. During the last few decades, the rapid growth of interest in image processing has brought about substantial treatments of spatial models, including both engineering and statistical aspects; see in particular Ripley (1981), Baddeley (1998), and van Lieshout (1995). At the same time, the collection of improved quality spatial data in ecology, geography, forestry, geophysics, and astronomy, has maintained a steady demand for spatial statistical models, for which Stoyan, Kendall and Mecke’s text SKM (1995) is an extensive general reference. The material we present falls into two main components. In the first four sections we review mainly descriptive properties, distinguishing between distance and directional properties of spatial point patterns, starting from finite models, moving on to the moment properties of line processes, and then revisiting space–time models, where time reappears so that many of the modelling concepts in Chapter 14 are again available, but spatial patterns also play an important role. The three final sections of the chapter provide an introduction to modelling centred around the concept of the Papangelou intensity; we provide some background and motivation from the statistical and physical settings, then attempt an introduction to the more mathematical theory. 457

458

15. Spatial Point Processes

This chapter also includes an introduction, mainly through the treatment of line processes in Section 15.3, to the rich and diverse territory of stochastic geometry, pioneered by Rollo Davidson, David Kendall, and others in the 1970s [see especially Kendall’s (1974) introduction to Harding and Kendall (1974)]. For present purposes, we may take stochastic geometry to mean the study of families of geometric objects randomly located in one-, two- or threedimensional Euclidean space, where to qualify as an ‘object’ all we demand is that the entity can be specified by a finite (or perhaps countably infinite) set of real parameters that describe aspects such as location, size, and shape. To each such object there corresponds a point in a Euclidean parameter space of suitably high dimension, and random families of such objects can be defined as point processes on this parameter space as state space. Because a characteristic feature of geometric objects is their invariance under rigid motions such as translation, rotation, and reflection, a key question is the implication of such invariance properties on the first and second moment properties of the process. The results for isotropic point processes considered in Section 15.2, and those for line processes in Section 15.3, both illustrate this general theme. The results are still applications of the factorization Lemma A2.7.II or equivalent disintegration results, but as the objects become more complex, so also do the disintegrations become more varied and more intricate. The final three sections of the chapter introduce a rather different aspect of the theory of spatial point processes, where the underlying endeavour is to use the concepts of interior/exterior to provide some kind of weak counterpart to the ideas of past/future on the time axis. Central to this endeavour is the concept of the Papangelou intensity, which underlies recent developments in inference for spatial point processes, such as pseudolikelihood methods and point process residuals. For finite point processes, the properties of the Papangelou intensity can be developed in a relatively elementary manner from the theory of Janossy densities outlined in Chapter 5, and this is undertaken in Section 15.5. The extension to general processes is altogether more demanding, requiring a combination of deep concepts from statistical mechanics and general point process theory. An introduction to this material is contained in Sections 15.5 and 15.6, and centred round the concept of exterior conditioning, meaning a conditioning of the point process on its behaviour outside a bounded set. In this sense the theory can be thought of as a kind of dual to the Palm theory, which is concerned with conditioning on the behaviour within a bounded set, as the dimensions of that set shrink to zero.

15.1. Descriptive Aspects: Distance Properties Faced with a realization of a point process within a bounded region of R2 , or a spatial point pattern as we generally describe it in this chapter, a statistician’s first reaction is likely to be to seek some numerical characteristics with

15.1.

Descriptive Aspects: Distance Properties

459

which to describe its salient features. Spatial point patterns being, in general, objects of some complexity, a variety of different statistics has been developed for this purpose. In this section it is our aim to give a brief overview of some of these quantities, without getting too deeply involved with technical issues such as consistency, unbiasedness, or numerical stability. It is our concern rather to identify and place in context the model characteristics to which the statistics refer. More comprehensive introductions to spatial statistics, including point process models in particular, can be found in Ripley (1981), Diggle (1983, 2003), SKM (1987) and its second edition SKM (1995), Cressie (1991, 1993), van Lieshout (2000), Baddeley et al. (2005), and a broad collection of case studies in Baddeley et al. (2006). Crudely speaking, the models that are available to describe point processes in Rd for d = 2, 3, . . . are derived from Poisson processes, Gibbs processes, or (deterministic) lattice processes that may have undergone modification by translation, clustering, or inhibition. To the extent that features of these modelling mechanisms have been discussed earlier in the book we should have little more to say. Nevertheless it is worth noting briefly some properties that have been developed to describe how particular models and/or datasets may deviate from the simplest underlying structure meaning, most commonly, the ‘complete randomness’ as for a Poisson process. Some of the earliest characteristics to be studied relate to nearest-neighbour distances, which have long been used to assist both in estimating areal densities and in classifying cluster properties. Indeed, they relate to some of the earliest applications of point process ideas in forestry, ecology, and elsewhere [see, e.g., Matérn (1960) and Warren (1962, 1971)]. The functions most frequently used relate to stationary point processes (Definition 12.1.II), often called homogeneous point processes in purely spatial contexts, in which case descriptions in terms of the Palm probability measure P0 can also be used. Homogeneity in space is a major simplifying factor in conceptual models, but is a rare phenomenon in the real world, so that from a practical point of view, a rudimentary understanding of the behaviour of characteristics to be expected when the true model departs in different ways from homogeneity is also important. The first such quantity we consider is a particular example of the avoidance function or avoidance probability [equations (2.3.1), (9.2.11) and Example 5.4(a)], namely, 1 − F (r) = P{N (Sr ) = 0}, where in this section we mostly write Sr = Sr (0) for the circle (or in Rd , the sphere) of radius r with centre at the origin. When N is stationary, the function F (r) is also known as the spherical contact distribution [denoted Hs (r) in SKM (1995, pp. 72, 80)], or empty space function, because F (r) = P{N (Sr ) > 0} is the probability that a sphere of radius r makes contact with a point of N . It is also the distribution function of the distance from the origin to the point x∗ (N ) of (13.3.7).

460


The other function which plays a central role in this context that N is stationary is the nearest-neighbour function itself,

G(r) = P0 N (Sr \ {0}) > 0 , denoted D(r) in SKM (1995). It is the distribution function of the distance from an arbitrary point of the process, selected as origin, to the nearest other point of the process, or equivalently the Palm probability version of the spherical contact distribution. Its form for the Neyman–Scott process is given in Exercise 6.3.10 [see also Example 15.1(a) and Exercise 15.1.3]. The ratio of the two survivor functions, ⎧ ⎨ 1 − G(r) if F (r) < 1, (15.1.1) J(r) = 1 − F (r) ⎩ 1 if F (r) = 1, is an indicator, relative to a Poisson process for which J(r) = 1 (all r), of clustering (when < 1) or ‘regularity’ or inhibition (when > 1) at varying distances r. The notation J(·) follows van Lieshout and Baddeley (1996); Diggle (1983) uses q ∗ (·) for the same function in the setting of a Poisson cluster process as in Example 15.1(a), but the concept dates at least to Warren (1971). Note too that for F (r) < 1, (15.1.1) has the alternative expression

P0 N (Sr \ {0}) = 0 . (15.1.1 ) J(r) = P{N (Sr ) = 0} The similar ratios

P0 N (Sr \ {0}) = k P{N (Sr ) = k}

and

P0 N (Sr \ {0}) ≤ k P{N (Sr ) ≤ k}

(15.1.2)

are both identically 1 in r > 0 for every k in any space Rd when N is Poisson; deviations from 1 for different k may indicate more detail than J(·) concerning clustering or inhibition. More generally, as in SKM (1995, Section 4.1), take a convex compact set B 0 and define the contact distribution function HB by HB (r) = P{N (rB) > 0}, irrespective of P being stationary or not. We consider these contact distributions only for the spherical case underlying J(r). In assessing the behaviour of empirical estimates of the J-function, it is important to bear in mind the possibility of non-stationarity, where the value may depend on the spatial origin. As a general rule, explicit expressions for the J-function are rather difficult to obtain. The first example below summarizes the more tractable results for Poisson cluster processes. Mase (1986, 1990) reviews difficulties in approximating Gibbs processes. Baddeley et al. (2005) and van Lieshout (2006a) examine a range of further results and examples.

15.1.


461

Example 15.1(a) Two-dimensional stationary Poisson cluster process; centresatellite process [see Example 6.3(a) and Exercise 6.3.10]. Equation (6.3.14) of Proposition 6.3.III gives the empty space function for a general Poisson cluster process in a form which here reduces to pSr (y) µc dy , (15.1.3) 1 − F (r) = exp − R2

where µc is the intensity for the Poisson process of cluster centres (here assumed stationary) and pSr (y) is the probability that a cluster with centre at y has no members within Sr . To find the G-function, suppose given a point of the process at the origin, and consider separately the distance to the nearest point from the same cluster, and to the nearest point from a different cluster. For any given cluster structure, there will be a well-defined distribution function tail, Qcl (r) say, for the probability that within a distance r of some given point of a cluster there is no other point of the same cluster. The distance to the nearest point in a different cluster, however, has the same distribution F (r) as in (15.1.3). This implies [Diggle (1983, equation (4.6.5))] that 1 − G(r) = Qcl (r) [1 − F (r)],

(15.1.4)

and hence that J(r) = Qcl (r). Thus, for a stationary Poisson cluster process, J(r) is equal to the probability that no two points from the same cluster lie within a distance r of each other, and therefore satisfies 1 ≥ J(r) ↓ (0 ≤ r ↑ ). Under more specific assumptions, the functions pSr (y) and Qcl (r) can be evaluated explicitly. For example, for a Neyman–Scott process with isotropic normal distributions about the cluster centre, the function pSr (y) is evaluated in Exercise 6.3.10. If qj denotes the probability that a cluster contains j points, ∞ and mcl ≡ j=1 jqj < ∞, then Qcl (·) is evaluated over a cluster with j points with probability jqj /mcl ; see Exercise 15.1.3 and Warren (1971). As a further instructive example, albeit special, consider a stationary Poisson cluster process whose clusters consist of exactly two points, one at the cluster centre and the other uniformly distributed on a circle of radius R around the cluster centre. Then it is impossible for two points from the same cluster to lie within a circle of radius less than 12 R, and certain that they will do so for some circle of any larger radius. Because the clusters consist of exactly two points, it follows immediately that Qcl (r) = 0 or 1 according as r < or ≥ R. For circles of radius < 12 R, the process looks like a Poisson process. Some further details are given in Exercise 15.1.4. This example can be adapted to furnish counterexamples in the case of anisotropy (cf. the next section). When the point process is stationary, we can find expressions for the F and G-functions in terms of the local Janossy measures of Definition 5.4.IV. Thus, the empty space function F (x) is exactly the function J0 ∅ | Sx (0) ,

462


and this allows F (x) to be expressed in terms of the factorial moment densities (when these exist) as in (5.4.14): ∞ (−1)k ··· m[k] (y1 , . . . , yk ) dy1 . . . dyk F (x) = J0 ∅ | Sx (0) = k! Sx Sx =

k=0 ∞ k=0

(−1)k M[k] [(Sx )(k) ]. k!

(15.1.5)

A similar expansion holds for the nearest-neighbour function G(x) in terms of the empty space function and moment densities of the Palm distribution; note that such densities exclude the point at the origin. For a stationary process, this reduces to an expansion in terms of the reduced moment densities of the original process (cf. Proposition 13.2.VI). Thus for G(x), (15.1.5) continues ˘ [k+1] [(Sx )(k) ], where m is the mean rate, ˚[k] [(Sx )(k) ] = m−1 M to hold with M in place of M[k] [(Sx )(k) ]. A careful discussion of these and related expansions is given in van Lieshout (2006b), relating the moment measures for the Palm process to the Papangelou intensities through the Georgii–Nguyen–Zessin formula; see also the discussion of these topics in Section 15.5. The F -, G-, and J-functions can be extended to MPPs provided due care is taken to specify the marks of the points appearing in the definitions. The situation here is analogous to that encountered in defining the Palm distributions P(0,κ) of an MPP, where it is necessary to specify the mark κ of the point at the origin. Thus, in specifying the empty space function for an MPP we need to distinguish between the empty space function for the ground process, Fg (x) say, which determines the distance from an arbitrary origin to any point of the process, regardless of its mark, and the more general family of functions FB (x) determining the distance from such an origin to the first point with mark in the subset B ∈ BK . For nearest-neighbour distances there are in principle four different options to consider: the distance from a point of the process with arbitrary mark to the nearest point with arbitrary mark (giving the nearest-neighbour distribution function Gg (x) for the ground process); the distance from a point with arbitrary mark at the origin to the nearest neighbour with mark in a specified set B [giving the distribution G(g,B) (x), say]; the distance from a point at the origin with specified mark κ to the nearest point of the process regardless of its mark [giving G(κ,g) (x) say]; and the distance from a point with mark κ at the origin to the nearest point with mark in the subset B ∈ BK [giving G(κ,B) (x)]. The next example examines these options for the simplest case of an MPP with independent marks. Example 15.1(b) Processes with independent marks [see Definition 6.4.III]. Since the mark on the point at the origin is independent of the marks and locations of all further points, it has no effect on the nearest-neighbour distances. Thus, for independent marks we find G(κ,g) (x) = Gg (x) ;

J(κ,g) (x) = Jg (x),

15.1.


463

and similarly G(κ,B) (x) = G(g,B) (x) ;

J(κ,B) = J(g,B) (x).

The nearest distance from the origin to a point within the mark set B here corresponds to the same distance for a point process obtained from the original by independent thinnings (Section 11.2) with thinning probability p = B F (dx). The effect is to multiply the kth factorial density in (15.1.5) by the factor pk , so that we obtain FB (x) = =

∞ (−p)k k=0 ∞ k=0

k!

Sx

··· Sx

mg[k] (y1 , . . . , yk ) dy1 . . . dyk

(−p)k g M[k] [(Sx )(k) ], k!

where the moment densities refer to those of the ground process. A similar modification occurs for the corresponding G-function, irrespective of the mark at the origin. Practical estimation of the F - and G-functions raises the usual problems of allowing for edge effects and possible biases arising from nonhomogeneity. Ripley (1988) and Stoyan and Stoyan (1994) are among the several texts which examine such problems in depth. Here we mention only the edge correction for estimates of the nearest-neighbour distribution proposed in Hanisch (1984). This has the advantage of preserving the monotonicity of the estimate as a function of r. It replaces the na¨ıve estimate + G(r) =

N (W ) 1 IN [(Sr (xk ))=0] N (W ) k=1

with the form N (W ) IN [(Sr (xk ))∩W =0] + H (r) = (W ) G N (W ) (W −d(xk ,∂W ) ) k=1

(15.1.6)

where d(x, ∂W ) is the distance from the point x to the boundary c of the observation region W , and A− = {x ∈ A: ρ(x, Ac ) > } = (Ac ) denotes the -interior of A [cf. the -halo set at (A2.2.2); A− is defined for convex A below (12.2.12)]. The interpretation is that when a point x is too close to the boundary of W for the ball Sr (x) to be wholly contained in W , the count from Sr (x) ∩ W is inflated by the weight factor (W )/(W −d(xk ,∂W ) ). Monotonicity and unbiasedness properties are outlined in Exercise 15.1.8. The other quantities we briefly mention in this section are the distributions of point-to-point distances whose properties are summarized in the moment measures, especially the second-order or two-point moment measure. Again we assume stationarity, so that the quantities of principal importance are the

464


reduced moment measures of Section 12.6, or their equivalent representations as moment measures of one order lower for the Palm distributions, as in Proposition 13.2.VI. For spatial processes these reduced moment measures are functions of vector differences u = x1 − x2 , and so can be represented in terms of polar or spherical polar coordinates. Thus, for example, if u = (r, θ), ˘ 2 (dr) and ˘ 2 (u) can be factorized (disintegrated) into a marginal measure K M a family of conditional distributions Γ(dθ | r) describing the distribution of the angle θ for given r. Decompositions of this kind are examined in more detail in the next section. Here we note that Ripley’s K-function at (8.1.21), namely, ˘ 2 (r) = (1/m2 )M ˘ 2 (Sr \ {0}) = (1/m)M ˚1 (Sr \ {0}), (15.1.7) K(r) = (1/m2 )K is widely used alongside the F -, G-, and J-functions as a useful descriptive characteristic of spatial point patterns. It measures the rate of growth of the reduced second moment measure with distance r from the origin, and can be defined for both isotropic and anisotropic processes, although the former ˘ 2 (r) for the isotropic Neyman–Scott is often assumed. The behaviour of K process referred to in Example 15.1(a) is described in Example 8.1(b); see also Exercise 15.1.2.

Exercises and Complements to Section 15.1 15.1.1 The Slivnyak–Mecke Theorem 13.1.VII implies that for a Poisson process, J(r) = 1 for all r. Investigate whether there are analogues of the constructions in Exercises 2.3.1 and 4.5.12 showing that there exist non-Poisson processes with this property. 15.1.2 Divide R2 into unit squares. Independently to each square allocate N points 1 1 , 89 , and 90 for j = 0, 1, 10, with the common distribution Pr{N = j} = 10 respectively, and for each square distribute its N points uniformly over the square. Check that for any Borel set A, the first two moment measures M (A) and M2 (A) for the process are the same as for a Poisson process at unit rate, and hence conclude that Ripley’s K-function is the same for both processes. [Hint: Baddeley and Silverman (1984) give this example with a plot of a realization, remarking that the plot is visually quite different from that of a Poisson process. They reference several other similar counterexamples.] 15.1.3 Consider a stationary Poisson cluster process in R2 as in Example 15.1(a) when a typical cluster member is as in a Neyman–Scott process consisting of j points with probability qj such that mcl ≡ ∞ j=0 jqj < ∞, each point being i.i.d. about the cluster centre with a radially symmetric distribution for r which Pr{point lies within distance r of centre} = 0 uf (u) du. Show that ∞ kqk ∞ Qcl (r) = [1 − P (r | y)]k−1 yf (y) dy, mcl 0

k=1

where P (r | y) =

r−y 0

f (x) dx +

r+y r−y

y+r y−r

f (x)lr (x, y) dx

f (x)lr (x, y) dx

(r > y), (r < y),

15.1.


465

denotes the probability that a particular point of the cluster lies within a circle of radius r and centre at distance y from the cluster centre and lr (x, y) is the length of the segment of a circle of radius x and centre 0 intersected by a circle of radius r and centre at distance y from 0. For the spatial Neyman–Scott process of Exercise 6.3.10, conclude that the function Qcl (·) of (15.1.4) can be represented in the form ∞

Qcl (r) = k=1

kqk mcl

∞

e−x

2

k−1

/2

xr (x, y) dx

.

0

15.1.4 Show that for the Neyman–Scott process of Example 6.3(a), the function r2 (x) in (8.1.19), a standardized conditional intensity for a point at x given a point at the origin, is given by m[2] f (x + u)f (u) du r2 (x) = 1 + m X when the distribution of cluster points from the cluster centre has density function f (·) and m, m[2] are the first two factorial moments of the cluster size distribution [cf. also (6.3.19)]. 15.1.5 Consider the particular Poisson cluster process N with two-point clusters described at the end of Example 15.1(a), setting R = 1. Regard N = Nc +Ns as the superposition of two stationary dependent Poisson processes, each at rate µc , Nc consisting of the cluster centres and Ns of the points of the clusters at unit distance and random orientation relative to the centres. (a) Show that the empty space function F (r) is given by 1 − F (r) = Pr{Nc (Sr ) = 0, Ns (Sr ) = 0} = Pr{Nc (Sr ) = 0} Pr{Ns (Sr ) = 0 | Nc (Sr ) = 0}. 2

The first term in the product equals e−µv2 (1)r . The other term is the same for r ≤ 12 but for larger r we must evaluate Pr{no cluster centre outside Sr has a component point inside Sr }. Use standard Poisson process arguments to exclude points inside Sr from centres at distance y from the centre of Sr , with r < y < r + 1, to conclude that

−2µc πr 2 e

1−F (r) =

(0 < r ≤ 12 ),

2 e−µc πr exp − µc

r+1

r

2 arcos

1+y 2 −r 2 2y

2π

2πy dy

( 12 < r).

As a check, evaluate the latter case in the limit r ↓ 12 . (b) Show that the first and second moment measures for this process are given by Pr{N (dx1 ) = 1} = 2µc (dx1 ),

2 4µc (dx1 ) (dx2 )

if |x1 − x2 | = 1, 2µc (dx1 ) I|x1 −x2 |=1 dθ otherwise, 2π and in the latter case, x2 is expressed in polar coordinates relative to x1 . (c) Evaluate the ratios at (15.1.2) for k = 1, 2, . . . and compare with J(r). [Remark: Since the clusters have two points, this is a Gauss–Poisson process.] Pr{N (dx1 ) = 1, N (dx2 ) = 1} =

466


15.1.6 For a stationary point process in Rd , express the function G(·) in (15.1.1) as G(r) = lim δ↓0

F δ (r) − F (r) , F (δ)

where F (r) = Pr{N (Sr ) > 0}, Fδ (r) = Pr{0 < N (Sδ ) = N (Sr \ Sδ )}, and F (r) = 1 − F (r) [cf. Chapter 3, Ambartzumian (1972), Paloheimo (1971)]. 15.1.7 Consider Matérn’s Model I in R2 [see Example 8.1(c) and Exercise 8.1.8]. Show that for 0 < r < R, G(r) = 0 and hence J(r) > 1 for such r. [The same holds true for any hard-core model as described in Example 5.3(c).] 15.1.8 Hanisch-type edge corrections [Hanisch (1984)] are of the form M 0 (A) =

xk ∈W

N [(xk + A) ∪ W ] , (W −d(xk ,∂W ) )

where d(x, ∂W ) is the distance from the point x to the boundary of the observation region W , A is a test set, and the function being estimated is the first-order moment of the Palm distribution. (a) Show that for the edge-corrected estimate (15.1.6) of the nearest-neighbour distribution, and assuming a simple, stationary, homogeneous process with finite intensity λ (= the mean rate for a simple process), (i) for r1 < r2 , GH (r1 ) ≤ GH (r2 ); and

N (W )

I{N [(Sr (xk ))∩W ]=0} = λG(r). (W −d(xk ,∂W ) ) k=1 [Hint: For (ii), write the left-hand side as the expected value of an integral against N and use the basic formula (13.3.2) for the Palm distribution.] (b) Define analogous edge-corrected estimates of the marked versions Gg (r), G(g,B) (r) defined above Example 15.1(b), and show that they have similar monotonicity and unbiasedness properties. [Hint: van Lieshout (2006b).] (c) Investigate estimates of the same type for other functionals of the Palm process. (ii) E

15.2. Directional Properties and Isotropy Consider first a point process in R2 whose probability structure is invariant under rotations about a fixed point in the plane. Such a process may represent, for example, the distribution of seedlings about a parent plant or of animals or other organisms about a nest or point of release (Byth, 1981). It is natural in such a case to take the fixed point as origin and to represent the points on the plane in terms of polar coordinates (r, θ), with 0 ≤ r < ∞ and 0 < θ ≤ 2π. By omitting the origin, which plays a special role, it can be represented as a + product R+ 0 × S, where R0 = (0, ∞) and S denotes both the circle group and its representation as (0, 2π]. Assuming that a.s. there are no points at the origin (and, we hasten to add, there is little difficulty in incorporating the contribution of an atom at

15.2.

Directional Properties and Isotropy

467

the origin if so desired), we have a process with the same kind of structure as a stationary MPP in time. The distance from the origin constitutes the mark and the angular distance θ from a fixed reference axis corresponds to the time coordinate. The factorization Lemma A2.7.II applies in a similar fashion and leads to the following representation of the first and second moment measures, analogous to Proposition 8.3.II; the proof is left to Exercise 15.2.1. Proposition 15.2.I. Let N (·) be a point process in the plane R2 , invariant under rotations about the origin with N ({0}) = 0 a.s., and having boundedly finite first and second moment measures. Then the first and second factorial moment measures have the respective factorizations M1 (dr × dθ) = K1 (dr) dθ/2π, ˘ [2] (dr1 × dr2 × dφ) dθ1 /2π, M[2] (dr1 × dr2 × dθ1 × dθ2 ) = M where φ ≡ θ2 − θ1 (mod 2π); these factorizations correspond to the integral relations, valid for bounded measurable h(·) with bounded support, namely,

E (R+ ×S)(2)

R+ ×S

2π

dθ 2π

h(r, θ) N (dr × dθ) = 0

∞

h(r, θ) K1 (dr), 0

h(r1 , r2 , θ1 , θ2 ) M[2] (dr1 × dr2 × dθ1 × dθ2 )

2π

= 0

dθ 2π

(2)

R+ ×S

˘ [2] (dr1 × dr2 × dφ). h(r1 , r2 , θ, θ + φ) M

Even without isotropy, for such a ‘centred’ process it is frequently convenient to use polar coordinates and hence to represent the first moment measure (assuming it is boundedly finite) in the form M1 (dr × dθ). Writing r

2π

M1 (ds × dθ)

K1 (r) = E[N (Sr )] = 0

0

for the expected number of points within a distance r of the origin, we can then define a directional rose as the Radon–Nikodym derivative Γ(dθ | r) = M1 (dr × dθ)/K1 (dr).

(15.2.1)

Observe that Γ, in contrast to K1 , is necessarily a probability distribution. In these terms, isotropy embodies two features: the directional rose is uniform over all angles (and equal to 1/2π), and independent of the radius r. When densities exist, we may wish to express M1 (·) in Cartesian coordinates rather than polar coordinates. The densities in these two representations are related by m(x, y) = r−1 k1 (r) γ(θ | r),

468


where r = x2 + y 2 , θ = artan (y/x), k1 (r) = dK1 (r)/dr, and γ(θ | r) = dΓ(θ | r)/dθ. In the isotropic case, γ(θ | r) = 1/2π. For the second-order analysis, we can introduce a factorial moment measure for the counting process on centred spheres N (Sr ) by setting M[2] (dr1 × dr2 × dθ1 × dθ2 ). K[2] (dr1 × dr2 ) = S(2)

In the isotropic case we can then introduce a second-order directional rose Γ2 (dφ | r1 , r2 ) as the Radon–Nikodym derivative ˘ [2] (dr1 × dr2 × dφ)/K[2] (dr1 × dr2 ). Γ2 (dφ | r1 , r2 ) = M

(15.2.2)

K[2] (dr1 × dr2 ) represents the expected numbers of pairs of points located with one point at distance (r1 , r1 + dr1 ) from the origin and the other at distance (r2 , r2 + dr2 ), and Γ2 (dφ | r1 , r2 ) gives the conditional probability distribution of the angular separation φ between the points. The symmetry properties of the second-order moments again lead to Γ2 (dφ | r1 , r2 ) = Γ2 (2π − dφ | r2 , r1 ), but these quantities are in general different from Γ2 (dφ | r2 , r1 ). Example 15.2(a) Isotropic centred Poisson process. Let N (·) be a Poisson process in R2 whose rate parameter has density µ(x, y) = µ(r) for r = x2 + y 2 . Then k1 (r) = 2πrµ(r), k[2] (r1 , r2 ) = 4π 2 r1 r2 µ(r1 ) µ(r2 ), γ(θ | r) = 1/2π = γ2 (φ | r1 , r2 ). Isotropy here implies that the angular separation (at 0) between pairs of points is uniformly distributed. It is also easy to verify that the counting process on centred spheres is a Poisson process on R+ with areal density 2πrµ(r). Example 15.2(b) Isotropic centred Gauss–Poisson process. Define a Gauss– Poisson process, of pairs of points in R2 , by supposing that parent points are located around the origin O according to a Poisson process with density µ(·), as in Example 15.2(a), and that with each such parent point there is associated an offspring or secondary point whose location relative to a parent point on the circle of radius r1 is governed by the probability distribution with density function f (r2 , φ | r1 ), where r2 is the distance of the secondary point from O and φ its angular separation (at O) from the parent point; for isotropy, suppose that this angular separation is independent of the parent’s angular coordinate. The overall intensity k1 (r) at distance r from O is then the sum of two components, an intensity 2πrµ(r) of parent points and an intensity of secondary points obtained by averaging over all locations of parent points: ∞ 2π 2πsµ(s) ds f (r, φ | s) dφ. k1 (r) = 2πrµ(r) + 0

0

15.2.

Directional Properties and Isotropy

469

Similarly, for the second-order radial moment measure density we find k[2] (r1 , r2 ) = k1 (r1 )k1 (r2 ) 2π [2πr1 µ(r1 )f (r2 , φ | r1 ) + 2πr2 µ(r2 )f (r1 , 2π − φ | r2 )] dφ. + 0

The first-order directional rose is of course uniform, but not so the secondorder rose which in general depends on the form of the density function f (r2 , φ | r1 ). If the factorization f (r2 , φ | r1 ) = f (r2 | r1 )g(φ) holds, then k[2] (r1 , r2 ) = k1 (r1 )k1 (r2 ) + 2π[r1 µ(r1 )f (r2 | r1 ) + r2 µ(r2 )f (r1 | r2 )], γ2 (φ | r1 , r2 ) = p(r1 , r2 )/2π + q(r1 , r2 )g(φ), where p(r1 , r2 ) = k1 (r1 )k2 (r2 )/k[2] (r1 , r2 ) and q(r1 , r2 ) = 1 − p(r1 , r2 ). Thus, the second-order directional rose is a mixture of two components: it reflects the relative proportions of pairs of points with radii r1 , r2 coming from independent point pairs and parent–offspring point pairs, respectively. Note that if any given parent point at distance r1 from the origin has an offspring point with probability p(r1 ), then it is enough to change f (r2 , φ | r1 ) into a subprobability density function with R+ ×S

f (r2 , φ | r1 ) dr2 dφ = p(r1 ).

We turn now to consider those planar point processes that are both stationary and isotropic, so they are invariant under the group of rigid body motions in R2 . By Proposition 15.2.I the first moment measure is a multiple of Lebesgue measure in R2 by virtue of stationarity alone, so the effect of isotropy shows up first on the second moment measure, that is, on pairs of points. The only property of a pair of points that is invariant under rigid body motions is the distance between them, so a natural coordinate transformation of R2 × R2 to consider is of the form (x1 , y1 , x2 , y2 ) → (x1 , y1 , x1 + r cos θ, y1 + r sin θ), where 0 ≤ r < ∞ and 0 < θ ≤ 2π, with r the distance between the two points and θ the angle between the directed line joining them and a fixed reference axis. Assuming the point process to be simple and considering just the factorial and cumulant measures, {r = 0} has zero probability and so the second factorial cumulant measure can be represented as a measure on the space R2 × R+ 0 × S, corresponding to the coordinates x, y, r, θ just introduced. Stationarity implies invariance with respect to shifts in the first two coordinates and so yields the usual representation in terms of a reduced ˘ [2] (dr × dθ). factorial measure, which we write in the form M

470


˘ 2 (dr) and a Without yet assuming isotropy, introduce a radial measure K second-order directional rose Γ2 (dθ | r) via 2π ˘ 2 (dr) = ˘ [2] (dr × dθ), K M 0

˘ [2] (dr × dθ)/K ˘ 2 (dr). Γ2 (dθ | r) = M

The function

r

˘ 2 (ds) K

˘ 2 (r) = K 0

can now be interpreted as the expected number of pairs of points separated by a distance r or less and for which the first point of the pair lies within a region of unit area. The directional rose Γ2 (dθ | r) then represents the probability that, given the separation is r, the directed line joining the first point to the second makes an angle with the fixed reference axis falling in the interval ˘ 2 (·) is in terms of the first (θ, θ + dθ). A more natural interpretation of K moment measure of the Palm measure for the process: it equals the product of the mean density m and the expected number of points in a circle of radius r about the origin given a point at the origin, as noted around (15.1.7) and at (8.1.22) in the discussion of what is there called Ripley’s K-function. Consider now the implication of isotropy. A rotation through α transforms the angle θ to a new angle θ that depends in general on x, y as well as α. Given any θ and θ , we can find (x, y) and α such that θ is transformed into θ . Rotational invariance implies therefore that Γ2 (dθ | r) must be invariant under arbitrary shifts θ → θ and so reduces to the uniform distribution on S. We summarize all this as follows. ˘ [2] (·) denote the reduced second factorial moProposition 15.2.II. Let M ˘ [2] (·) ment measure of a simple stationary point process in the plane. Then M can be expressed as ˘ 2 (dr) Γ2 (dθ | r), ˘ [2] (dr × dθ) = K (15.2.3) M corresponding to the integral representation of the second factorial moment measure M[2] (·) (for bounded measurable h with bounded support) h(x1 , y1 , x2 , y2 ) M[2] (dx1 × dy1 × dx2 × dy2 ) R2 ×R2 ˘ 2 (dr) Γ2 (dθ | r), = dx dy h(x, y, x + r cos θ, y + r sin θ) K R2

R+ ×S

˘ 2 (·) is a boundedly finite measure on R+ and for each r > 0, Γ2 (· | r) where K is a probability measure on S. If the process is also isotropic then Γ2 (dθ | r) = dθ/2π

(all r, θ).

˘ [2] (·) here in the stationary case imply that The symmetry properties of M Γ2 (dθ | r) = Γ2 (π + dθ | r), which is the analogue for the representation being used here of the property noted for Γ2 (· | r1 , r2 ) below (15.2.2).

15.3.

Stationary Line Processes in the Plane

471

Exercises and Complements to Section 15.2 15.2.1 Identify R2 \ {0} with R+ 0 × S and consider mappings that lead to invariance of these component factors. Now apply the factorization Lemma A2.7.II and hence complete the proof of Proposition 15.2.I. [Hint: In part (a) of the proposition identify R2 \{0} with the product R+ 0 ×S and consider invariance under the actions of S. For the second moment in part (b) use a diagonal factorization of the components of S × S.] 15.2.2 Let N be a point process on state space the surface of a cone with semiangle α (the extremes α = 0 and 12 π correspond to a cylinder and a disc, respectively). (a) Use R+ 0 × S to describe the points by the distance from the apex of the cone and the angle relative to a fixed plane through the axis of the cone subtended by a plane through the point and the axis (i.e., the ‘longitude’ of a point on the cone). Describe the first and second moment structure of a process invariant under rotations of the cone. (b) For an alternative parameterization, cut the cone down a straight line from the apex and ‘unwrap’ it onto a plane, so that it fills the plane apart from a sector of angle 2π − θ, where θ = 2π sin α. Rotations of the cone correspond to rotations in the plane modulo θ, where the two edges of the missing sector are identified. Rephrase the results in (a) in terms of this parameterization. [See Byth (1981) who uses the term θ-stationary process.] 15.2.3 Exercise 8.1.7 gives some results for the isotropic case of a bivariate Neyman– Scott cluster process. Using the notation from there but now for the nonisotropic case, the directional rose has a density γ2 (θ | r) proportional to 2πm1 + where g(θ, Σ) =

m[2] exp[−r2 g(θ, Σ)/4(1 − ρ2 )] , 4πσ1 σ2 (1 − ρ2 )1/2 2ρ cos θ sin θ sin2 θ cos2 θ − + , 2 σ1 σ1 σ2 σ22

that is, γ2 (θ | r) = p(r)/2π + q(r) exp{· · ·}, where p(r), q(r) are nonnegative involving a Bessel function arising from the normalizing condition Êfunctions, ∞ γ2 (θ | r) dθ = 1, and p(r) → 1 as r → ∞. 0

15.3. Stationary Line Processes in the Plane Stationary line processes constitute the paradigm for many of the recent developments in stochastic geometry. In particular, Davidson’s conjecture that all stationary isotropic line processes are doubly stochastic and his imaginative early investigations of this question inspired important further studies by Krickeberg, Papangelou, Kallenberg, and others, and these in turn laid the foundation for recent work on the relations between conditional intensities and Gibbs potentials in the models of statistical physics. Here we give a brief introduction to the properties of line processes, based largely on the early

472


sections of Harding and Kendall (1974) and the work of Roger Miles. The same circle of ideas is introduced in Stoyan and Mecke (1983, Chapter 7), and more extensively in SKM (1995, Chapters 8–9). It is convenient to characterize a directed line in the plane by its coordinates (p, θ), where θ satisfying 0 < θ ≤ 2π is the angle between the line and a fixed reference direction, and p is the signed perpendicular distance from the line to a fixed origin, being positive when, in looking in the direction of the line, the origin is to its left. We define a random process of directed lines to be a point process on the cylinder R×S, each (random) point on the cylinder being identified with the (random) line in R2 via its specification (p, θ). Thus, by the distribution of a stochastic line process, we mean a probability measure on the point process in R × S. In this text we assume this point process (and hence the line process) to be simple. For example, by a Poisson process of lines in the plane at rate λ, we mean a Poisson process on R × S at rate λ/2π, {(pi , θi )} say, representing directed lines whose directions are i.i.d. uniformly over (0, 2π] and whose signed perpendicular distances from a fixed origin form a Poisson process on R at rate λ [see also Exercise 15.3.1(a)]. Note that, given a planar Poisson process at rate λ and locating through each point a line with direction uniformly distributed over (0, 2π), we obtain infinitely many lines intersecting any unit interval with probability one (cf. Exercise 15.3.2). A process of undirected lines can be treated as a point process on either R+ × S or R × (0, π]: the latter fits more easily into our discussion and follows, for example, Stoyan and Mecke (1983) [see also Exercise 15.3.1(b)]. Another representation of a line process is as a point process in R2 in which the first coordinate x say, denotes the intercept by the line on a fixed reference line, and the second equals x cot θ, its intercept on a line orthogonal to the reference line, where θ ∈ (0, π] is the angle that the line makes with the reference line [in terms of (p, θ) these two intercepts are (p sec θ, p cosec θ)]. Clearly, when p = 0 the direction θ is not determined by these two intercepts. However, either representation can be used to describe stationary processes (see below) for which the event p = 0 has probability zero. It will be evident that these two representations of a given line process lead to different distributions on different spaces. As in Section 15.2 the principal questions we study relate to the effects of stationarity or isotropy on the moment structure of the process. By these conditions we mean of course invariance of the process of lines under translations and rotations in the plane, so our first task is to examine the effect of these motions on the coordinates (p, θ) of a line. Rotation through an angle α about the fixed origin corresponds to rotation of the cylinder through the same angle, (p, θ) → (p, θ + α). Translation of the plane a distance d in a direction making an angle φ with the fixed reference axis induces the transformation (p, θ) → p − d sin(θ − φ), θ (−∞ < d < ∞, 0 < φ ≤ π) (15.3.1) on the cylinder, corresponding to a shear whereby points on the cylinder are

15.3.


473

displaced parallel to its axis through a distance varying (when d = 1) from −1 at θ = φ + 12 π through 0 at θ − φ = 0 mod π to +1 at θ = φ − 12 π (here, addition of angles is taken modulo 2π). We start by showing that any Borel measure on the cylinder that is invariant under the action of the shears (but not necessarily under rotations) has the product form (dp) G(dθ), where (·) denotes Lebesgue measure on R. This is the first occasion where we encounter an invariance result that cannot be handled via the factorization Lemma A2.7.II: this is so because shears do not generate translations of the cylinder along its axis. Nevertheless, the result we require is still a simple corollary of the more general theorems about the decomposition of invariant measures, which can be established via the general theory of the disintegration of measures, as for example in Krickeberg (1974b, Theorem 2). For the sake of completeness we sketch a simplified version of the theorem as it applies here. Lemma 15.3.I. Let M (dp × dθ) be a boundedly finite Borel measure on the cylinder R × S, and let M (·) be invariant with respect to the action of the shears at (15.3.1). With (·) denoting Lebesgue measure on R, there then exists a totally finite Borel measure G on S such that M (dp × dθ) = (dp) G(dθ).

(15.3.2)

Proof. In outline, we find a factorization of M of the form K(dp | θ) G1 (dθ), and then show that K(dp | θ) factorizes as λ(θ) (dp). To start with, there exists a function f (p), as, for example, (n 0 for all p and f (p) M (dp × dθ) < ∞. Introduce the measure R×S f (p) M (dp × dθ) (θ ∈ S). G1 (dθ) = p∈R

For all Borel sets A, M (A × dθ) G1 (dθ), so by appealing to the usual arguments leading to the existence of regular conditional probabilities, we deduce the existence of a kernel K(dp | θ) such that K(A | θ) is a measurable function of θ for each bounded Borel set A, K(· | θ) is a Borel measure on R for G1 -almost all θ, and M (dp × dθ) = K(dp | θ) G1 (dθ). Invariance under the action of a given shear (15.3.1), with parameters (d, φ) say, implies that for any bounded measurable h(·) of bounded support, h(p, θ) M (dp × dθ) = h(p, θ) K(dp | θ) G1 (dθ) R×S R×S = h p + d cos(θ − φ), θ K(dp | θ) G1 (dθ) R×S = h(p, θ) K dp − d cos(θ − φ) | θ G1 (dθ). R×S

474


Because this is true for all such h, the measure K(· | θ) must coincide with its shifted version K(· − d cos(θ − φ) | θ) for G1 -almost all θ. By choosing two appropriate values of d and φ, we infer that for such θ, the measure K(· | θ) is invariant under the action of two incommensurate shifts. This in turn implies that K(· | θ) is a multiple of Lebesgue measure, λ(θ) (·) say (see Exercise 15.3.3 for details). Thus, K(A | θ) = (A)λ(θ), where the left-hand side is a measurable function of θ, hence there is a measurable version λ∗ (θ) of λ(θ) such that λ∗ (θ) = λ(θ) for G1 -almost all θ. Setting G(dθ) = λ∗ (θ) G1 (dθ) proves the lemma. Corollary 15.3.II. Let a stationary line process have first moment measure M on R × S. Then M factorizes in the form (15.3.2). Because the measure G is totally finite, it can be normalized to give a first-order directional rose Π(·) on S Π(dθ) =

G(dθ) G(dθ) = . G(S) G(dθ) S

Π(dθ) may be interpreted as the probability that an arbitrary line has orientation θ, and the total measure m ≡ G(S) has an interpretation as the mean density of the line measure induced by the process. To explain this idea, observe that for any line W and any closed bounded convex set A ⊂ R2 , there exists a well-defined length (W ∩ A). Given any configuration {Wi } of lines on the plane, we can introduce a corresponding line measure (Wi ∩ A). Z(A) = i

This set function Z(·) is clearly countably additive and extends to a measure on arbitrary Borel sets in the plane. Furthermore, if Wi has coordinates (pi , θi ) in the cylinder R × S, then the mapping (pi , θi ) → (Wi ∩ A) is measurable, so that if the {Wi } constitute a realization of a stochastic line process, each Z(A) is a random variable. From Proposition 9.1.VIII it follows that Z(·) is a random measure, which we call the line measure associated with the original line process. Proposition 15.3.III. Let Z be the line measure associated with a stationary line process W in R2 . Then Z is a stationary random measure on R2 , and if W has finite first moment measure M (·), Z has mean density M (dp × dθ). (15.3.3) m = G(S) = (0,1]×S

Proof. Writing A (p, θ) = (W (p, θ) ∩ A) for a line with coordinates (p, θ) and any bounded Borel set A ⊂ R2 , we can express Z(A) = A (p, θ) N (dp × dθ) R×S

15.3.


475

in terms of the point process N on the cylinder. Because A ≥ 0, we have A (p, θ) M (dp × dθ) = G(dθ) A (p, θ) (dp). E[Z(A)] = R×S

S

R

For fixed θ, the integral over R is simply the area of A evaluated as the integral of its cross-sections perpendicular to the direction θ. Writing 2 for Lebesgue measure in R2 , we have E[Z(A)] = G(dθ) 2 (A) = m 2 (A), S

establishing (15.3.3). Stationarity of Z(·) follows from the stationarity of the line process defining Z(·). Given a configuration {Wi }, an alternative to the line measure Z(A) is the number of lines hitting A. This set function is subadditive but not in general additive over sets, and thus not a measure, although for each convex set A it defines a random variable whose distribution and moments can be investigated. If, however, A is itself a line, we obtain the point process on the line formed by its intersections with the lines Wi , which is a random measure. Proposition 15.3.IV. Let a stationary line process in R2 be given with mean density m and directional rose Π(·). (i) Let V be a fixed line in R2 with coordinates (pV , α), and let NV (·) be the point process on V generated by its intersections with the line process. Then NV is a stationary point process on V with mean density mV given by (15.3.4) mV = m | cos(θ − α)| Π(dθ). S

If the line process is isotropic, mV is independent of V with mV = 2m/π

(all V ).

(15.3.4 )

(ii) Let A be a closed bounded convex set in R2 , and let Y (A) be the number of distinct lines of the line process intersecting A. If the line process is isotropic then E[Y (A)] = mL(A)/π, (15.3.5) where L(A) is the length of the perimeter of A. Proof. For any bounded measurable function h of bounded support in R, we have h(x) NV (dx) = h x(p, θ) N (dp × dθ), R

R×S

where x = x(p, θ) denotes the distance from a fixed origin on V to a point of intersection of a line with coordinates (p, θ), and N (·) refers to the point

476


process on the cylinder representing the given line process. Because of stationarity, there is no loss of generality in taking the fixed origin on V as the origin for the cylindrical (p, θ) coordinates. Then x(p, θ) = p sec(θ − α), and on taking expectations, we obtain

E R

h(x) NV (dx) =

R×S

h p sec(θ − α) (dp) m Π(dθ).

(15.3.6)

Substituting u = p sec(θ − α), (dp) = | cos(θ − α)|(du), from which (15.3.4) follows because NV , being stationary, has ENV (dx) = mV (dx). In the isotropic case, Π(dθ) = dθ/2π and integration at (15.3.4) leads to 2m/π as asserted. To prove (ii), suppose first that A is a convex polygon. Apply the result of (i) to each side of the polygon in turn, so that adding over all sides shows that the expected number of intersections of lines from the line process with the perimeter of A equals 2mL(A)/π. Convexity implies that each line intersecting the polygon does so exactly twice (except possibly for a set of lines of zero probability), so the factor 2 cancels and (15.3.5) is established for convex polygons. A limiting argument extends the result to any closed bounded convex set A. Propositions 15.3.III and 15.3.IV can be extended to processes of random hyperplanes and more generally random ‘flats’ in Rd : see Exercises 15.3.4–5 for some preliminary results and the extensive series of papers by Miles (1969, 1971, 1974). Krickeberg (1974b) sets out a general form of the required theory of moment measures. When further distributional aspects are specified, the results can be sharpened as in the basic example below. For extensions to higher dimensions see Exercise 15.3.6 and the cited papers by Miles. Example 15.3(a). We define a stationary Poisson process of lines in terms of the associated point process N (·) on the cylinder R×S. For the line process to be stationary in R2 , the point process N (·) must be invariant under shears of the cylinder, and its first moment measure must decompose as at (15.3.2). But the first moment measure of a Poisson process coincides with its parameter measure µ(·), so µ(dp × dθ) = µ (dp) Π(dθ) (15.3.7) must hold for some density µ and probability distribution (here, the directional rose of the line process) Π(·). We can therefore write for the p.g.fl. of the point process N (·), with suitable functions h(·), log h(p, θ) N (dp × dθ) G[h] = E exp R×S h(p, θ) − 1 Π(dθ) . = exp µ (dp) R

S

15.3.


477

Thus, for example, the second factorial moment measure is given by M[2] (dp × dp × dθ × dθ ) = µ2 (dp) (dp ) Π(dθ) Π(dθ ) = M (dp × dθ) M (dp × dθ ), which is of course of the product form expected for a Poisson process. The p.g.fl. for the point process NV of intersections of the line process on a fixed line V follows from an extension of the reasoning leading to (15.3.6). In the notation used there, log h(x) NV (dx) GNV [h] = E exp R log h p sec(θ − α) N (dp × dθ) = E exp R×S h p sec(θ − α) − 1 (dp) Π(dθ) = exp µ R×S h(u) − 1 du , = exp µ | cos(θ − α)| Π(dθ) S

R

which we recognize as the p.g.fl. of a stationary Poisson process on V with density mV as at (15.3.4). In particular, for a stationary isotropic Poisson line process, the density mV = 2µ/π is independent of the orientation α of V , and the number of lines crossing a closed convex set A has a Poisson distribution with parameter µL(A)/π with µ equal to the mean line density. We now discuss second-order properties of line processes, confining our attention to stationary isotropic processes. Now it is clear that one invariant of a pair of lines, under both translations and rotations on the plane, is the angle φ between them where 0 < φ ≤ 2π (this allows for directed lines), so we take coordinates in the form (p, θ, p , θ ) → (p, θ, p , θ + φ). Invariance under rotations then implies that the second factorial moment measure M[2] of the point process N representing the stationary isotropic line process in R2 factorizes into a product ˘ [2] (dp × dp × dφ) dθ/2π. M[2] (dp × dp × dθ × dθ ) = M ˘ [2] we proceed much as in Proposition 15.2.II to deduce To handle the term M that ˘ [2] (dp × dp × dφ) = K[2] (dp × dp | φ) G[2] (dφ). M Invariance of N under shears implies that for almost all φ and at least for (say) rational r and ψ = θ − α, K[2] (dp × dp | φ) = K[2] (dp − r cos ψ) × (dp − r cos(ψ + φ)) | φ .

478


Provided φ = 0 or π, the equations r cos ψ = u and r cos(ψ + φ) = v have unique solutions (r, ψ) for all real u and v, and therefore K[2] (dp × dp | φ) is invariant under at least a countable dense family of translations (p, p ) → (p + u, p + v), including incommensurate pairs (u, v), (u , v ). For such values of φ then, K[2] (dp × dp | φ) reduces to a multiple of Lebesgue measure 2 in R2 , λ(φ) 2 (dp×dp ) say, where as earlier we can take λ(φ) to be a measurable function of φ. The exceptional cases φ = 0 and π correspond to the occurrence of pairs of parallel and antiparallel lines, respectively. In both cases, the signed distance y between the lines is a further invariant of the motion. In the first case, invariance under translation implies that K[2] (dp × dp | 0) = K[2] (dp + r cos(θ − α) × dp + r cos(θ − α) | 0), so that setting p = p + y, the measure factorizes into the form + (dy) (dp) K[2] (dp × dp | 0) = K + on R. Similarly, for φ = π, the measure for some boundedly finite measure K K[2] (dp × dp | π) is invariant under the transformations (p, p ) → p + r cos(θ − α), p − r cos(θ − α) so that setting now p = y − p, where y is the distance between parallel lines oriented in opposite senses, (dy) (dp) K[2] (dp × dp | π) = K − − . Finally, because M[2] is symmetric under the transfor boundedly finite K formation (p, p , θ, θ ) → (p , p, θ , θ), + and K − are symmetric under reflection in their respective all three of G[2] , K origins. We have proved the following result. Proposition 15.3.V [Davidson (1974a), Krickeberg (1974a, b)]. Let M[2] be the second factorial moment measure of a stationary isotropic line process in R2 . Then M[2] admits a representation in terms of the factors: (i) a totally finite symmetric measure G[2] (dφ) on (0, π) ∪ (π, 2π) governing the intensity of pairs of lines intersecting at an angle φ; + (dy) on R governing the inten(ii) a boundedly finite symmetric measure K sity of pairs of parallel lines distance y apart; and − (dy) governing the intensity of pairs of antiparallel (iii) a similar measure K lines a distance y apart.

15.3.


479

The representation is realized by the integral relation, valid for bounded measurable nonnegative functions h of bounded support on R × R × S × S = R2 × S2 , R2 ×S2

h(p, p , θ, θ ) M[2] (dp × dp × dθ × dθ ) = G[2] (dφ) h(p, p , θ, θ + φ) dp dp dθ (0,π)∪(π,2π) R2 ×S + (dy) + h(p, p + y, θ, θ) dp dθ K R R×S − (dy) + K h(p, y − p, θ, θ + π) dp dθ. (15.3.8) R

R×S

A similar representation holds for the factorial covariance measure. As Rollo Davidson (1974a) showed, many remarkable corollaries follow from the representation (15.3.8). Corollary 15.3.VI. With probability 1, a stationary isotropic line process in R2 either has no pairs of parallel or antiparallel lines, or has infinitely many pairs of parallel lines, or has infinitely many pairs of antiparallel lines, − (R) = 0, or K + (R) > 0, or K − (R) > 0, respectively. + (R) = K according as K Proof. Let A be a bounded Borel set in R and V a fixed line in the plane. + (A)/π as Then by the preceding discussion and (15.3.4 ) we can interpret 2K the mean density of the stationary point process on V generated by its intersections with those lines of the process which have other lines of the process + (R) = 0, any such process of parallel to them and at separation y ∈ A. If K interaction has zero mean density and is therefore a.s. empty. Letting A ↑ R, we deduce that with probability 1 no line of the process has another line of + (R) > 0, we can find a bounded the process parallel to it. Conversely, if K Borel set A with K+ (A) > 0 and a line V such that the process of associated points on V is stationary with positive mean density and therefore has an infinite number of points (see Proposition 12.1.VI). The argument concerning antiparallel lines is similar. Corollary 15.3.VII. M[2] is invariant under reflections if and only if the process has a.s. no pairs of antiparallel lines, in which case it is also invariant under translations p → p + y of the cylinder parallel to its axis. − (R) = 0, it follows from (15.3.8) that M[2] is invariant under Proof. If K the transformation (p, p , θ, θ ) → (p, p , −θ, −θ ) on account of the symmetry properties of G[2] . This mapping corresponds to reflection in the reference axis. Hence, by isotropy and stationarity, it

480


is invariant under any other reflection. A similar conclusion holds for the transformation (p, p , θ, θ ) → (p + y, p + y, θ, θ ). − (R) > 0, we can choose h(·) in (15.3.8) so that a conConversely, if K tradiction arises if we assume that M[2] is invariant under either of these transformations. + ((0, T ]) and T −1 K − ((0, T ]) both vanish in the limit Provided that T −1 K as T → ∞, we can show that the measure G[2] of Proposition 15.3.V is positive + (R) = definite: we proceed under the more restrictive assumption that K K− (R) = 0, that is, that there are a.s. no parallel or antiparallel pairs of lines. Let a(θ) be a bounded measurable function on (0, 2π) and in (15.3.8) set h(p, p , θ, θ ) = I(0,T ] (p) I(0,T ] (p ) a(θ) a(θ ) = h(p, θ) h(p , θ ). In place of M[2] , consider the ordinary second moment measure M2 for which M2 (A × B) = M[2] (A × B) + M1 (A ∩ B). We have 2 h(p, θ) N (dp, dθ) 0≤E R×S = h(p, θ)h(p , θ ) M[2] (dp × dp × dθ × dθ ) (R×S)2 + h2 (p, θ) M1 (dp × dθ) R×S

= T2 S

S

a(θ)a(θ + φ) G[2] (dφ) dθ +

S

a2 (θ)mT dθ . 2π

Division by T 2 and rearrangement yield S

S

a(θ)a(θ + φ) G[2] (dφ) dθ ≥ −

S

a2 (θ)m dθ →0 2πT

(T → ∞);

that is, the measure G[2] (·) is positive definite (equivalently, in the terminology of Definition 8.6.I, it is a p.p.d. measure). This result immediately suggests asking whether G[2] (·) can be interpreted as a covariance measure; accordingly, we look for some random measure Y on S with which G[2] (·) may be associated. The appropriate candidate (as we now show) is the ergodic limit of the original point process on R × S with respect to translations of the cylinder parallel to its axis, or, equivalently, the conditional expectation E(N (·) | IS ) of the original process with respect to the invariant σ-algebra IS generated by these translations. That such ergodic

15.3.


481

limits exist follows directly from Theorem 12.2.IV, and we then have, for A, B ∈ B(S), E Y (A)Y (B) = lim

1 T →∞ T 2

= S2

(R×S)2

M[2] (dp × dp × dθ × dθ )

× I(0,T ] (p)I(0,T ] (p )IA (θ)IB (θ ) dp dp dθ dθ IA (θ)IB (θ + φ) G[2] (dφ) dθ.

An even more surprising corollary is the following. Consider the Cox process N ∗ on the cylinder R × S directed by the random measure × Y . It is readily checked that N ∗ is invariant under rotations and translations of the cylinder and that for A, B ∈ B(S) and U the unit interval, E[N ∗ (U × A)] = m(A)/2π = EY (A), ∗ (U × U × A × B) = E Y (A)Y (B) . M[2] Thus, N and N ∗ have the same first and second moment measure. We summarize this as follows. Proposition 15.3.VIII. If a stationary isotropic line process has boundedly finite second moment measure and has a.s. no parallel or antiparallel lines, then the reduced moment measure G[2] (·) of Proposition 15.3.V is positive definite and can be identified with the second moment measure of the random measure on S " 1 " N (dp × A) a.s., Y (A) = E N (U × A) I = lim T →∞ T p∈(0,T ] where I is the invariant σ-algebra associated with shifts of the cylinder parallel to its axis. Furthermore, N has the same second moment measure as the Cox process N ∗ directed by × Y . This proposition, coupled with his failure to find counterexamples, led Davidson to formulate his celebrated conjecture [‘the big problem’ of Davidson (1974b, p. 70)] that any stationary isotropic line process with a.s. no parallel or antiparallel lines and boundedly finite second moment measure must be a Cox process. Davidson showed that no counterexample could be constructed by taking a point process on a line and putting lines through its points [cf. Proposition 15.3.IV(i)], nor by taking a stationary point process in R2 and putting lines through its points, nor seemingly ‘by tinkering with a Poisson line process.’ The structure of stationary isotropic line processes, as well as of the more general stationary hyperplane processes in spaces R2d , differs radically from those of stationary point processes in R1 . That Davidson’s conjecture is false was shown by Kallenberg (1977b) in which the main idea is

482


the construction of a process from a lattice configuration in a parametrization with respect to a fixed line. To describe this example, it is necessary to adopt the alternative representation of a line as a pair (x, y) ∈ R2 . Here x is the x-coordinate of the intercept of the line with the x axis, and y = x cot θ as described earlier in this section. (Our attempt in the first edition to apply similar arguments to a lattice in the cylinder representation contained a fundamental flaw: see Exercise 15.3.10.) Example 15.3(b) Kallenberg’s randomized lattice. We start from the line system specified in the alternative representation above by the square lattice of points in the plane with integer coordinates. We randomize the location of the lattice by translating it by a vector X uniformly distributed over the unit square, and rotating it by an angle Φ uniformly distributed over the interval (0, 2π]. The resulting point process has unit mean density, and the average density of points in a realization (‘sample density’) is also a.s. unity. It is still of lattice type, and is invariant under arbitrary rigid motions of the plane. The crucial requirement, however, is to produce from this randomized lattice a point process that is invariant under shears, Σα : (x, y) → (x, x+αy) say, for these correspond to translations in the space of lines. Because the process is already invariant under rotations, it is enough to consider just the shears Σα parallel to the y-axis. To this end we consider a sequence of further randomizations: first select α uniformly over the interval (−n, n), then let n → ∞. This yields a sequence of point processes in the plane, which become more and more nearly invariant under shifts as n → ∞. Moreover, each such process is still of lattice type (although no longer a square lattice), is invariant under translations and rotations, has mean density 1, and has mean sample density a.s. equal to 1. Boundedness of the mean densities implies that the sequence of point processes is tight in the topology of weak convergence (see Exercise 11.1.2). Thus, there exists at least one weakly convergent subsequence. However this argument does not preclude the possibility that the resulting limit measure might be null. To eliminate this last possibility, Kallenberg considers the corresponding Palm measures, and shows that these are tight, implying, because a Palm measure necessarily has a point at the origin, that the resultant limit is nonzero. The resultant line process is not locally bounded, but the line process obtained from considering just the points in a vertical strip will be so and will still be invariant under vertical shears (corresponding to translations in the space of lines) and to vertical shifts (corresponding to rotations). The limit is not a Poisson process because it still has an a.s. lattice character, but from the construction it is invariant under both rotations and shears. Finally, its second factorial moment measure exists, and a further analysis shows that − (R) = 0, and G[2] (·) is uniformly distributed on S. This is + (R) = K K the same second moment structure as for the simple Poisson process itself. Because it is not the Poisson process, it cannot be a Cox process either.

15.3.


483

The corresponding line process is thus invariant under translations and rotations of the plane, has finite first and second moment measures, has a.s. no parallel or antiparallel lines, and is not a Cox process. It therefore refutes Davidson’s conjecture. Details of the proof and further remarkable properties of the process in the plane constructed in this way are given in Kallenberg (1977b) (which includes a further comment by Kingman), and in Mecke’s (1979) subsequent paper. In particular, Mecke gives both an explicit construction for the process and an algebraic characterization of it as the unique process having lattice character and invariant under all affine translations of the plane. The singular character is clearly evident from SKM (1995, Figure 8.3) illustrating the process.

Exercises and Complements to Section 15.3 15.3.1 (a) The line process in R2 represented as a Poisson process on R×S is defined initially with respect to a specified origin and reference direction in R2 . Show that the line process in fact is homogeneous and isotropic in R2 . [Hint: Consider the transformation on (p, θ) effected by moving the origin as underlying (15.3.1). The Poisson process on the cylinder is preserved under both this transformation and change of the reference direction.] (b) For each of the three point process representations below of a process of undirected lines, describe the point process that represents a line process that is (a) homogeneous; or (b) isotropic; or (c) both. (i) Take the distance p > 0 from the origin to the line as one parameter and the angle made by the line and a fixed reference axis as the other. The line process is represented as a point process on R+ × S. (ii) As in (a) except that the distance is signed and the angle is restricted to the range (0, π]. Then the line process is a point process on R × (0, π]. (iii) Describe a line by its intercepts on the x and y axes as parameters. Then a line process is represented as a point process in R2 . 15.3.2 Suppose given a Poisson process in R2 at unit rate; let Wδ denote those of its points (x, y) lying in the wedge 0 < y/x < tan δ with δ < 12 π. Independently through each point construct a line with orientation θ uniformly distributed on (0, 2π). Let S denote the circle with centre at the origin and radius > 0. Show that Pr{no line through a point of Wδ intersects S } = 0. Conclude that with probability one, infinitely many lines intersect S . 15.3.3 Let µ be a measure on R invariant under shifts Ta and Tb for incommensurate a and b. Set F (x) = µ((0, x]) for x > 0, = −µ((x, 0]) for x ≤ 0, and let U be the set of points u ∈ R such that µ is invariant under shifts Tu . Show that u ∈ U implies −u ∈ U , and that for u, v ∈ U , F (u + v) = F (u) + F (v), so that U is an additive group and thus contains all points of the form ja + kb for positive or negative integers j, k. Deduce that F (x) = αx for all x and some α ≥ 0. [This result, like Exercise 12.1.8, is a variant on the Hamel equation at (3.6.3).] 15.3.4 Random hyperplanes. A hyperplane is a (d − d )-dimensional linear subspace of Rd shifted through some vector x ∈ Rd for some positive integer d < d.

484

15. Spatial Point Processes (a) Show that a directed hyperplane is uniquely specified by a pair (p, θ), where θ lies on the d-dimensional unit ball S d , p ∈ R, and the sense of the hyperplane (whether the normal to the origin is directed toward or away from the hyperplane) is determined by the sign of p. (b) A process of random d − d hyperplanes can be represented as a point process in S d × R. Rotation of the original plane corresponds to rotation by an element of S d ; translation of the original plane corresponds to the transformation (p, θ) → (p + x, θ, θ), where x, θ is the inner product in Rd . (c) Such a process is homogeneous and isotropic if and only if the point process is invariant under both rotations and generalized shears as defined above, and its first moment measure, if it exists, is then a multiple of Lebesgue measure on S d × R, md (·) say. (d) Define a random measure ξ(A) for bounded Borel A in Rd as the sum of the hypervolumes A ∩ Si , where the particular hyperplanes of the process are denoted by {Si }. Then md (·) is the mean density of this random measure. (e) If L is an arbitrary fixed line in Rd , the points of intersection of L with {Si } form a stationary point process with mean density mΓ( 12 d)/2π 1/2 . [Hint. See references preceding Example 15.3(a).]

15.3.5 The special case of random lines in R3 uses the representation of such lines as points in S2 × R2 , where the component in S2 determines the direction of the line and the point in R2 its point of intersection with the orthogonal plane passing through the origin. Find analogues to (d) and (e) of Exercise 15.3.4 for the ‘line density’ of the process (a random measure in R3 ) and the point process generated by the points of intersection of the lines with an arbitrary plane in R3 . 15.3.6 Extend the result of Example 15.3(a) to the context of Exercises 15.3.4(e) and 15.3.5 (i.e., show that if the original process is Poisson so are the induced processes on the line and plane, respectively). 15.3.7 Given a homogeneous isotropic Poisson directed line process in R2 , form a ‘clustered’ line process with pairs of lines in each cluster in one of the following three ways. (a) Railway line process (i): To each line (pi , θi ) of the process, add the line (pi +d, θi ) for some fixed positive d. In the notation of Proposition 15.3.V, G[2] and K− are null, and K+ has an atom at d. The process is invariant under translation, rotation, and reflection. (b) Davidson’s railway line process (ii): To each line (pi , θi ) add the antiparallel line (−pi − d, π + θi ). Then G[2] and K+ are null, K− has an atom at d, and because of the built-in handedness, this process is not invariant under reflections of the plane. (c) To each line (pi , θi ) add the line (pi , θi + α) for some fixed α in 0 < α < π. The resulting line process is no longer translation invariant. [Each process here is a possible analogue of the Poisson process of deterministic cluster pairs as in Bartlett (1963, p. 266) or Daley (1971, Example 5).]

15.4.

Space–Time Processes

485

15.3.8 (a) Show that two distinct lines (p, θ) and (p , θ ) intersect in a point inside the unit circle if and only if |p| < 1, |p | < 1 and p2 + p2 − 2pp cos(θ − θ ) < sin2 (θ − θ ). The expected number of line pairs intersecting within the circle is thus found by integrating the second factorial moment measure over the region defined by these inequalities. (b) More generally, the first moment measure M of the process of intersections is found from integrals of the form

R2

h(x, y) M (dx × dy) =

R2 ×S2

h(x(p), y(p)) M[2] (dp × dp × dθ × dθ ),

where h(·) is a bounded measurable function of bounded support, and p denotes the vector of coordinates p, p , θ, θ and the two lines (p, θ) and (p , θ ) intersect in the point (x(p), y(p)). (c) When the line process is homogeneous and isotropic, M[2] reduces to the form described in Proposition 15.3.V. Assuming there are a.s. no parallel or antiparallel lines, use the representation of (b) to show that the intersection process is stationary and has mean density given by sin φ G[2] (dφ).

4π (0,π)

15.3.9 Show that if a stationary isotropic line process has a.s. no parallel or antiparallel lines, then it cannot be a Poisson cluster process. [Hint: Consider the form of the second factorial moment measure; a Poisson cluster process with nontrivial cluster distribution cannot factorize in the form of Proposition 15.3.V.] 15.3.10 (a) Consider a line process represented by a lattice on the cylinder R × S. Show that its properties are quite different from those of a line process represented by a lattice in R2 using the alternative representation using the intercepts (x, y) = (p sec θ, p cosec θ). (b) Let {(pi , θi )} denote a stationary line process in R2 . Investigate whether the point process in R with realizations {pi } is stationary [i.e., invariant under all shears at (15.3.1)]. [The question is due to Dietrich Stoyan.]

15.4. Space–Time Processes Space–time models combine elements from the evolutionary processes studied in Chapter 14 and from the descriptive properties of spatial patterns covered earlier in this chapter. Because the spatial location can always be considered as one component of a multi-dimensional mark, some aspects, such as the likelihood theory based on conditional intensity functions, are essentially special cases of the more general discussion of Chapter 14. For applications, on the other hand, the evolution of spatial features with time is often of special

486


interest. In this section we review some basic features of space–time point processes, trying to select those that most warrant more careful examination. The earliest statistical models for space–time processes of which we are aware were prompted by an agricultural setting. As agricultural trials continued on experimental stations, fluctuations in soil fertility were studied, and it was observed that the spatial correlations decayed remarkably slowly. Pioneer studies by Whittle (1954, 1962), using diffusion methods, showed that such long-term correlations could be caused by a sequence of perturbations (applications of fertilizer or other treatments) followed by gradual diffusion. Whittle also observed that space–time models are likely to be more insightful, by penetrating farther into the physical processes generating a spatial point pattern, than a purely static model for that pattern. Despite such considerations, studies of space–time models have lagged well behind those of simple temporal models, and even those of purely spatial models. No doubt the reasons have been largely practical, notably the difficulty of compiling good space–time datasets and the heavy computations needed to analyze them. Their importance, however, can only grow as time goes on and these difficulties are overcome. Another point to bear in mind about space–time models is their diversity. The models on which we focus here—models for earthquakes form a paradigm example—are for events which can be regarded as points in both time and space dimensions. With earthquakes and forest fires, a point pattern can be obtained only by accumulating events over time (e.g., the fires which have occurred over the last year). Models for particles moving through space constitute a different class. Although they can be viewed as spatial point patterns evolving in time, in space–time they form families of trajectories rather than families of points. A similar situation arises for models for storm centres or rainfall cells within a storm; in the discussion in Wheater et al. (2000), these phenomena are treated as points that persist a while until they disappear. A great deal of flexibility is added by moving from simple space–time point processes to space–time point processes with an associated mark. The spatial location itself may be viewed as a mark for a simple point process in time, thereby providing one route to likelihood analyses of space–time models. Further characteristics, such as magnitude, spatial extent, or even duration, can be added as additional marks. Deft use of this procedure, such as has been employed for some decades in applications to queueing systems and networks, can be very helpful in making complex models more tractable. Finally it should be observed that behind many observed space–time point processes lie evolving but unobserved spatial fields: earthquakes may be regarded as a response to some evolving stress field, forest fires as a response to some underlying spatial field determining the ignition potential, and so on. Thus, the study of space–time point processes leads almost inevitably to the more general study of evolving spatial fields, although practical modelling in this direction is still limited and very subject-specific. We turn to a more systematic study of space–time point processes and focus

15.4.


487

on the most commonly occurring situation, namely, the process is stationary in time but not necessarily homogeneous1 in space or in the mark distribution. The two main aspects we discuss are the first- and second-order moment properties, thereby revisiting and elaborating the discussion in Sections 8.3 and 12.3, and the extension to space–time processes of the conditional intensity and likelihood arguments of Chapter 14. For simplicity of exposition we suppose throughout that space here refers to Euclidean space R2 ; other options, such as point processes on the circle or the sphere, are indicated briefly in Exercises 15.4.1–2. For further examples, discussion and references, see Vere-Jones (2007). We start with first moments. Lemma 15.4.I. If a stationary marked space–time point process has finite overall ground rate mg , then there exist a distribution Φ(dx) in space, and a family of conditional distributions Ψ(dν | x) for the residual mark ν, such that the first moment measure can be decomposed as M (dt × dx × dν) = mg dt Φ(dx) Ψ(dν | x).

(15.4.1)

Proof. We know from Proposition 8.3.II that, if an MPP is stationary and has finite ground intensity mg , then its first moment measure can be written in the general form M (dt × dκ) = mg dt Π(dκ). The mark κ here has two components, the location x and a residual mark, ν say, so the measure Π here is a bivariate distribution on the product of the location space and the space of the residual mark. The decomposition (15.4.1) is then just the standard disintegration of Π(·) into the marginal distribution in space and a family of conditional distributions for the residual mark, given the spatial location. In this lemma, we allow the distribution of the residual mark ν to depend on the spatial location x, but not (from stationarity) on the time t. The lemma implies in particular that a stationary space–time Poisson process must have an intensity measure Λ which can be disintegrated as in (15.4.1). Assuming densities exist, its intensity in (space–time, mark) space will be of the form λ(t, x, ν) = λg φ(x) ψ(ν | x), where λg is the overall intensity (ground rate). [We follow convention by calling the process ‘space–time’, and write the components of a typical ‘space– time’ point (t, x) in reverse order.] The process can be otherwise interpreted as a space–time compound Poisson process with spatially varying space–time intensity λ(t, x) = λg φ(x) and spatially dependent mark distribution with density ψ(ν | x). 1

For a space–time process on, e.g., R+ × R2 we adopt the convention of using stationarity to refer to invariance with respect to time-shifts, and homogeneity to refer to invariance with respect to shifts in space. Thus, a process that is invariant under shifts in both time and space is an homogeneous stationary space–time process.

488


Example 15.4(a) Models for persistent points; spatial M/G/∞ queue. This example illustrates in simple form some of the issues which have been mentioned. Although in essence it is a model for particles that persist in time, it can be reduced to a marked space–time point process, in the narrow sense in which we have defined it, by treating the duration, as well as the location, as part of the mark. It can be regarded as a spatial version of an M/G/∞ queue, but many models for population and other processes have a similar general structure: particles which arrive at times ti and locations xi , persist for some time τi and then die or otherwise disappear. The spatial birth-and-death process of Example 10.4(e) is a more complex example, where the duration τi may depend on the evolving history of the whole set of particles. It in turn is a special case of the wider class of branching diffusion models, which has a considerable literature of its own (see Section 13.5). One basic approach to the process is to consider it as an MPP in time, say N (dt × dκ), with time points ti and marks κi = (xi , τi ) embracing both the locations xi and the durations τi . Equally, it may be regarded as a space– time point process, with locations (ti , xi ) say, and associated marks τi . Once this underlying process has been specified, all other characteristics should be derivable from it. Observations, however, may be restricted to snapshots of the time-varying spatial point pattern Nt (dx) representing the locations of the particles extant at time t. Here Nt may be regarded as a stochastic process taking values in X ∪ ; the locations of the points at time t can also be represented as a vector xt , anticipating the notation to be used in Section 15.5. A slightly different approach is to fix sets Ai in X and look at the joint evolution of the processes Xi (t) = Nt (Ai ) as a multivariate time series. In any case, one initial question is to find a representation of the spatial processes Nt in terms of the underlying process N, and to examine how far the structure of N can be reconstructed from observations on the Nt . The basic relation is of simple linear form: ∞ t N (ds × dx × dτ ) . Nt (A) = s=−∞

A

τ =t−s

This representation is immediately useful in obtaining the first moment measure for Nt (·). Suppose that the MPP N is stationary in time, and, adopting notation similar to Lemma 15.4.I, that its first moment measure has a density which can be written in the form m(t, x, τ ) = mg φ(x) ψ(τ | x), where φ(x) is a time-invariant probability density over the spatial region of interest, and ψ(τ | x) is the time-invariant probability density function for the life of a particle started at location x. Taking expectations in the expression for Nt we obtain E[Nt (dx)] = mt (x) dx, where t ∞ ∞ ψ(τ | x) dτ ds = mg φ(x) τ ψ(τ | x) dτ mt (x) = mg φ(x) −∞

t−s

0

15.4.


489

∞ and 0 τ ψ(τ | x) dτ ≡ L(x) is the mean lifetime of a particle started at location x. A similar (albeit more involved) representation for the second moment measure of Nt is outlined in Exercise 15.4.2. Only the mean of the lifetime distribution can be obtained from the above expression, even in the case that the spatial mean L(x) is independent of x and the ground rate mg could be independently estimated. If the aim were to obtain further information about the lifetime distribution, it would be necessary to combine observations on Nt over a sequence of values of t. Let us then consider, as a second step, the first moment structure for observations on two snapshots Nt1 and Nt2 . The combined observations can be treated as a single realization of a multivariate point process on the location space X , with points of three different types: Type 1 is observed at t1 but not t2 , Type 2 at both times, and Type 3 at t2 but not t1 . (We assume, here and above, that the particles themselves cannot be distinguished by their ages: they are either present or not present.) Arguing much as above, and writing ∆ = t2 − t1 , we obtain for the first moment measures of the three components t1 t2 −s m1 (x) = mg φ(x) ψ(τ | x) dτ ds = mg φ(x) A1 (∆ | x), −∞ t1

m2 (x) = mg φ(x)

t1 −s ∞

−∞ t2 −s t2 ∞

ψ(τ | x) dτ ds = mg φ(x) A2 (∆ | x),

ψ(τ | x) dτ ds = mg φ(x) A3 (∆ | x),

m3 (x) = mg φ(x) t1

t2

where

A1 (∆ | x) = A3 (∆ | σ) = A2 (∆ | x) =

∆

σψ(σ | x) dσ + ∆ 0 ∞ ∆

σψ(σ | x) dσ − ∆

∞

∆∞

ψ(σ | x) dσ, ψ(σ | x) dσ.

∆

Similar decompositions can be obtained for the first moments of larger numbers of snapshots, and begin to piece together information about ψ. Because of the simple linear relation between Nt and N , it is possible to extend the moment results into results for p.g.fls. For h ∈ V we obtain I(t − ti < τi ).1 + I(t − ti ≥ τi )h(xi ) Gt [h] = E ti R. It is not our intention, however, to pursue the Ising problem as such, but rather to consider the question of how quantities analogous to the Papangelou intensities of Section 15.5 can be introduced and related to other characteristics of the point process. One special feature of (15.6.2) is worth noting at this stage. For given location y and {xi : i = 1, 2, . . .}, the function at (15.6.1) is dependent on the set B only through the normalization constant CB and the requirements that y ∈ B, xi ∈ B c (i = 1, 2, . . .). We are therefore led to let B ↓ {y} to recover, in the finite case, the functions ρ(y | x) that we could there interpret as the conditional intensity for the occurrence of a particle at y given the realization of the process throughout the remainder of the state space, that is, in X \{y}. From the interaction potential viewpoint, the term in the exponent at (15.6.2) can be related to the work required to introduce a new particle into the position y keeping the locations of the existing particles fixed. But although these ideas lead to a straightforward definition in the finite case, the situation is more complicated in the general case. It is necessary to distinguish three different random measures, each of which embodies some aspect of the conditional intensity ρ(· | ·) of Section 15.5. These quantities are as follows. (i) The Papangelou kernel R(· | ·) defined by integral relations extending (15.5.10). (ii) A random measure π(·) describing, loosely speaking, the atomic part of these kernels. (iii) The Papangelou intensity measures ζ(·) as originally introduced by Papangelou (1974b) in terms of the limit ζ(B) = lim

n→∞

kn c E N (Ini ) | Ini N , i=1

(15.6.3)

15.6.

Modified Campbell Measures and Papangelou Kernels

521

where T = {Ini : i = 1, . . . , kn }: n = 1, 2, . . . is a fixed dissecting system of partitions of B as in Definition A1.6.I. In all approaches to this topic, Condition Σ below plays a fundamental role, and we work under it unless explicitly stated otherwise. Definition 15.6.I. The simple point process on the c.s.m.s. X satisfies Condition Σ if for all bounded Borel sets B, P{N (B) = 0 | B cN } > 0 a.s.

(15.6.4)

This requirement generalizes the assumption of Section 15.5 that the Janossy densities be positive everywhere. Its essential role is to preclude situations where the behaviour inside B is deterministically controlled by the behaviour outside B, as can occur in (1◦ ) and (2◦ ) of Example 15.6(a). Example 15.6(a) On Condition Σ. (1◦ ) Let N be a point process on X for which P{N (X ) = r} = 1 for some fixed integer r. Then for any nonempty set A ∈ BX , P{N (A) = 0 | N (Ac ) < r} = 0, and thus Condition Σ is violated. (2◦ ) Let N1 be a point process with exactly one point uniformly distributed over the bounded state space X ∈ B(Rd ), for example, a circle of unit area. Let N2 be a Poisson process at unit rate on X with N2 independent of N1 . Then the point process N = N1 + N2 violates Condition Σ because for any Borel set A ⊆ X of positive Lebesgue measure, P{N (A) = 0 | N (Ac ) = 0} = 0. (3◦ ) With N1 and N2 as in (2◦ ), the process N equal to N1 with probability p and to N2 with probability q = 1 − p, with pq > 0, satisfies Condition Σ (details are left to the reader). In order to set down a general form of the integral equations (15.5.10) for the Papangelou kernel, denoted by R(A | N ) for A ∈ BX and N ∈ NX#∗ , we start from (15.5.12) and (15.5.14), treating only the first-order case. As in (15.5.14), we can rewrite the left-hand side of this equation in terms of modified Campbell measure CP! of Definition 13.1.I(b) to yield h(u, N ) CP! (du × dN ) ≡ h(u, N − δu ) N (du) P(dN ) #∗ X ×NX

#∗ X ×NX

=

#∗ X ×NX

h(u, N ) ρ(u | N ) du P(dN )

=

#∗ X ×NX

h(u, N ) R(du | N ) P(dN ). (15.6.5)

For the finite case considered in Proposition 15.5.III, this exhibits R(· | N ) du as being derived from a disintegration of the second component of CP! with respect to P, namely, for all bounded A ∈ BX and U ∈ B(NX#∗ ), R(A | N ) P(dN ) = CP! (A × U ), (15.6.6) U

522


and leads us to seek such a disintegration in general. To justify such a disintegration, we use the absolute continuity condition that (15.6.7) CP! (A × ·) P(·) for each fixed A ∈ BX . The proof of this condition in Lemma 15.6.III below gives us an immediate illustration of the role that Condition Σ plays. The proof also makes use of the following results, in which σ{N }, σ{B cN } denote the σ-algebras generated by random variables {N (A), A ∈ BX }, {N (A), A ∈ BB c }, and so on. Lemma 15.6.II. (a) For any U ∈ σ{N } ⊆ B(NX#∗ ), and any B ∈ BX , there exists U ∗ ∈ σ{B cN } such that U ∩ {N (B) = 0} = U ∗ ∩ {N (B) = 0}.

(15.6.8)

(b) For any σ{N }-measurable function g(N ), and any B ∈ BX , there exists a σ{B cN }-measurable function g0 (N ) such that g(N ) I{N (B)=0} (N ) = g0 (N ) I{N (B)=0} (N ).

(15.6.9)

Furthermore, if E|g(N )| < ∞, then for any bounded σ{B cN }-measurable Y , E Y (N ) g(N ) I{N (B)=0} (N ) = E Y (N ) g0 (N ) P{N (B) = 0 | B cN } . (15.6.10) Proof. Any U ∗ ∈ σ{B cN } is generated by sets of the form {N (Ai ) = ji : Ai ∈ B(B c ), nonnegative integers ji }, and for any U ∈ σ{N }, U ∩ {N (B) = 0} is generated by sets of the form {N (B) = 0, N (Ai ) = ji : A ∈ B(B c ), ji = 0, 1, . . .}. This proves (a). Part (b) follows from (a) by a standard extension argument: start from indicator functions IU (N ) for U ∈ σ{N }, for which IU ∩{N (B)=0} = IU ∗ I{N (B)=0} and IU ∗ is σ{B cN }-measurable. Lemma 15.6.III. Let N be a simple point process on the c.s.m.s. X satisfying Condition Σ. Then for all bounded A ∈ BX , the absolute continuity condition (15.6.7) holds. Proof. We have to show that for any U ∈ B(NX# ) such that P(U ) = 0, CP! (A × U ) = 0 (all bounded A ∈ BX ). Suppose first that U ⊆ {N (A) = 0}, so that U = U ∩{N (A) = 0}, and thus by Lemma 15.6.II(a), IU (N ) = IU ∗ (N )I{N (A)=0} (N ) for some U ∗ ∈ σ{AcN }. Noting that for y ∈ A, N − δy ∈ U ∗ if and only if N ∈ U ∗ (i.e., the behaviour of N inside A is irrelevant to whether N ∈ U ∗ ), IU (N − δy ) N (dy) P(dN ) CP∗ (A × U ) = #∗ NX

=

#∗ NX

=

#∗ NX

A

I{N (A)=0} (N − δy ) N (dy)IU ∗ (N ) P(dN ) A

I{N (A)=1} (N ) IU ∗ (N ) P(dN ) ≤ P(U ∗ ).

15.6.


523

Equally, using (15.6.10), 0 = P(U ) = E IU ∗ (N ) P{N (A) = 0 | AcN } . By Condition Σ, the coefficient of the bounded function IU ∗ (N ) is positive a.s., so we have a contradiction unless IU ∗ (N ) = 0 a.s.; that is, P(U ∗ ) = 0, and hence CP! (A × U ) = 0 for such U . Suppose next that P(U ) = 0 for some U ⊆ {N : N (A) ≤ k}

for some given integer k ≥ 1, and let T = {Ani : i = 1, . . . , kn }: n = 1, 2, . . . be a dissecting family of partitions for A. Write also Uni = U ∩ {N (Ani ) = 0}

and

Uni = U \ Uni .

Then for n = 1, 2, . . . we have CP! (A × U ) =

kn

CP! (Ani × U ) =

i=1

=

kn

kn ! CP (Ani × Uni ) + CP! (Ani × Uni ) i=1

CP! (Ani × Uni )

i=1

because by the earlier argument, CP! (Ani × Uni ) = 0. For the last sum write kn i=1

CP! (Ani × Uni )=

kn i=1

≤k

kn

# NX

Ani

IU \Uni (N − δy ) N (dy) P(dN )

P{N (Ani ) ≥ 2, N (A) ≤ k}

i=1

(for this last step, N (Ani ) ≤ N (A) ≤ k for N ∈ U , and any y ∈ Ani that is an atom of N can contribute to the integral only if also (N − δy )(Ani ) ≥ 1, and thus N (Ani ) ≥ 2 for such y). The assumption that N is simple implies that the last sum → 0 as n → ∞, and hence that CP! (A × U ) = 0. To complete the proof, use monotone convergence to deduce that, whenever P(U ) = 0, CP! (A × U ) = lim CP! A × (U ∩ {N : N (A) ≤ k}) = 0. k→∞

Standard arguments based on this result can now be used to establish the existence and uniqueness properties of the disintegration of the modified Campbell measure, leading to the following theorem whose proof is left to Exercise 15.6.1. Kallenberg (1983a, Chapter 13) gives a more extended treatment; part (iv) of the theorem follows Gl¨ otzl (1980).

524


Theorem 15.6.IV. Suppose given a simple point process N defined on the c.s.m.s. X , with probability measure P, and satisfying Condition Σ. Then there exists a unique kernel R(A | N ) satisfying (i) for each bounded A ∈ BX , R(A | ·) is a Borel-measurable function on NX#∗ ; (ii) for each N ∈ NX#∗ , R(· | N ) is a bounded finite Borel measure on BX ; and (iii) for all nonnegative, measurable functions h(u, N ): X × NX#∗ → R+ , vanishing for u outside a bounded Borel set of X , h(u, N ) CP! (du × dN ) = h(u, N ) R(du | N ) P(dN ). #∗ X ×NX

#∗ X ×NX

(15.6.11)

(iv) If also CP! (du × dN ) × P

on BX × BN #∗ , X

(15.6.12)

then there exists a BX × BN #∗ -measurable function ρ(u | N ) such that X (15.6.5) holds and for all A ∈ BX , ρ(x | N ) (dx) (P-a.s. in N ). (15.6.13) R(A | N ) = A

Definition 15.6.V. When they exist, the kernel R(· | ·) defined by (15.6.11) of Theorem 15.6.IV is the Papangelou kernel associated with the point process N of the Theorem, and the density ρ(x | N ) of (15.6.13) is the Papangelou intensity of N . Strictly speaking, the kernel R(· | ·) is the first-order Papangelou kernel associated with N , because higher-order kernels can be defined via higher! . These we now introduce and discuss order modified Campbell measures CP,k briefly, setting ! (A1 × · · · × Ak × U ) CP,k k IU N − δyi N [k] (dy1 × · · · × dyk ) P(dN ) = # NX

A1 ×···×Ak

(15.6.14)

i=1

for bounded A1 , . . . , Ak ∈ BX and U ∈ B(NX#∗ ), where N [k] (·) denotes the k-fold factorial product measure defined by N (·) as above Proposition 9.5.VI. Much as in Lemma 15.6.III, it can be shown that under Condition Σ, ! (A1 × · · · × Ak × ·) P(·), and hence a kth-order Papangelou kernel CP,k Rk (A1 × · · · × Ak | N ) is well-defined P-a.s. for bounded A1 , . . . , Ak ∈ BX by Rk (A1 × · · · × Ak | N ) P(dN ) U ! = CP,k (A1 × · · · × Ak × U ) all U ∈ B(NX#∗ ) . (15.6.15)

15.6.


525

! Furthermore, regarding CP,k+ as a measure on X () × (X (k) × NX#∗ ), it ! ! can be shown that CP,k+ CP,k with respect to subsets of X (k) × NX#∗ . The corresponding kernel can be identified, up to the usual equivalence, with the kernel function k " " δxi , R A1 × · · · × A " N + i=1

thus justifying the extension of the multiplicative relation (15.5.7) to the form (using the simplicity of N in an essential way) k " " δxi Rk (dx1 × · · · × dxk | N ) R dy1 × · · · × dy " N + i=1

= Rk+ (dy1 × · · · × dy × dx1 × · · · × dxk | N ).

(15.6.16)

Finally, defining R0 (· | N ) = 1 for N (X ) = 0 and zero otherwise, the Papangelou kernels of all orders can be combined into a portmanteau kernel on the space X ∪∗ when N (X ) < ∞ via the equation G(V | N ) =

∞

! Rk (V ∩ X (k) | N ) k!

V ∈ B(X ∪∗ ) .

(15.6.17)

k=0

Under Condition Σ, this Gibbs kernel G(· | ·) is a density for the portmanteau Campbell measure defined much as in (15.6.14) but allowing k to vary, so the resultant set A is any Borel subset of X ∪∗ . Many properties of the Papangelou kernels can be assumed under a general treatment of the Gibbs kernel: see Kallenberg (1983a, Chapter 13) for details. Moreover, we can take the disintegrations the other way, assuming for any given k that the kth-order factorial moment measure exists, and disintegrating the kth-order Campbell measure relative to this moment measure in just the same way as for the ordinary Palm measures in Chapter 13. Then all these disintegrations can be combined to give a decomposition of the portmanteau Campbell measure into a family of Palm measures and an associated portmanteau factorial moment measure. Again we refer to Kallenberg (1983a, 1984) for further details. We conclude this section with another property of the first-order Papangelou kernel; it is an extension of the conditional probability relation (15.5.8). Proposition 15.6.VI. Let N be a simple point process defined on the c.s. m.s. X satisfying Condition Σ. Then for any bounded A, B ∈ BX with A ⊆ B, R(A | N ) =

P{N (A) = N (B) = 1 | B cN } P{N (B) = 0 | B cN }

a.s. on {N : N (B) = 0}. (15.6.18)

Proof. In (15.6.5) substitute h(u, N ) = IA (u) I{N (B)=0} (N ) IU (N + δu ),

526


where U ∈ σ{B cN }. When u ∈ A ⊆ B, N + δu ∈ U if and only if N ∈ U , so (15.6.5) yields R(A | N ) I{N (B)=0} (N ) P(dN ) = N (A) I{N (B)=1} (N ) P(dN ) U U = E N (A) I{N (B)=1} | B cN P(dN ) U = P{N (A) = N (B) = 1 | B cN }P(dN ). U

On the other hand, using Lemma 15.6.II(b), noting that U ∈ σ{B cN }, we can write E R(A | N ) I{N (B)=0} (N ) = E R0 (A | N ) P{N (B) = 0 | B cN } , where R0 (A | N ) is σ{B cN }-measurable. Because we have equality for all U on {N (B) = 0}, R0 (A | N ) P{N (B) = 0 | B cN } = P{N (A) = N (B) = 1 | B cN }. Moreover, by Condition Σ, the coefficient of R0 (A | N ) > 0 a.s., and on N (B) = 0, R0 (A | N ) = R(A | N ), so (15.6.18) follows.

Exercises and Complements to Section 15.6 15.6.1 Prove Theorem 15.6.IV. [Hint: Cf. the proof of Lemma 15.6.III.]

15.7. The Papangelou Intensity Measure and Exvisibility In this final section we turn to an investigation of the two other measures, π and ζ, referred to in the discussion around (15.6.3), and their relation to the first-order Papangelou kernel R(· | ·). Theorem 15.6.IV(iv) exhibits the Papangelou intensity as the Radon–Nikodym derivative w.r.t. Lebesgue measure of the kernel R(· | ·), under the condition that CP! is absolutely continuous with respect to both × P and P (the latter property holds by definition). Atoms of π are precisely what hinders the existence of a Papangelou intensity in general. If the atomic component is absent, then the Papangelou intensity of the previous section coincides with the intensity defined via exvisibility, that is, as a density of ζ. Thus far, we have considered R(· | N ) as a kernel on the canonical space NX#∗ . It is more convenient for this section to regard R(A | N ) as defined on the space (Ω, E, P) itself. By an abuse of notation we sometimes write R(A) = R(A, ω) = R A | N (ω) , (15.7.1) and treat the quantity on the left-hand side as a random measure. That it is

15.7.

The Papangelou Intensity Measure and Exvisibility

527

a random measure follows from the measurability of R(A | N ) as a function of N , which implies that R(A) is a random variable for each A ∈ BX and hence that the requirements of Proposition 9.1.VIII are satisfied. As we show shortly, this random measure has some properties in common with the compensator for temporal processes, but difficulties arise with the atoms in particular, and we start on a different tack, following broadly the approach adopted by Papangelou (1974b) and Kallenberg (1983a). Suppose given a fixed dissecting system

of partitions for X [recall (15.6.3)] T = {Inj : j = 1, . . . , kn }, n = 1, 2, . . . . We need the following lemma. Lemma 15.7.I. For bounded Borel sets A, B with A ⊆ B,

P · ∩ {N (B \ A) = 0} | B cN a.s. on {N (B) = 0}, (15.7.2) P{· | Ac N } = P{N (B \ A) = 0 | B cN } the denominator being a.s. positive on {N (B) = 0}. Proof. E[I{N (B)=0} (N ) P{N (B) = 0 | B cN }] = E[P{N (B) = 0 | B cN }] a.s. If N ∈ {N (B) = 0}, then either N is in a set of measure zero, or else P{N (B) = 0 | B cN } > 0, so that in either case the last probability is a.s. positive on {N (B) = 0}, and therefore the denominator in (15.7.2) is a.s. positive on {N (B) = 0}. The relation itself is just a version of P (U | V ∩ W ) = P (U | W )/P (V | W ) for V ⊆ W , where because A ⊆ B, we can take P to be P(· | B cN ) and W to be {N (B \ A) = 0}. In what follows, we take A, B to be elements of T and note that, because T is countable, we can assume that (15.7.2) holds simultaneously a.s. for all such choices of A, B ∈ T . First we examine any atomic component of R. Proposition 15.7.II. Let N be a simple point process on the c.s.m.s. X . Let T be a dissecting system of partitions of X , x a general point of X , and {In (x)} a sequence of elements of T with In (x) ↓ {x} (n → ∞). Then the limit (15.7.3) π{x} = lim P N In (x) ≥ 1 | Inc (x)N } n→∞

exists a.s., is independent of T , and can be identified a.s. on the set {N (In (x) \ {x}) = 0} with the ratio

P N {x} = N In (x) = 1 | Inc (x)N

, π{x} = (15.7.4) P N In (x) \ {x} = 0 | Inc (x)N the ratio being interpreted as 1 when both numerator and denominator vanish. The equation π{xi } = π{xi }δxi (A) (15.7.5) π(A) = π(A, N ) ≡ xi ∈supp(N )∩A

defines a random measure on X .

528


When Condition Σ holds, π{x} < 1 for all x, the atoms of π include the atoms of R, and ! R{x} = π{x} (1 − π{x}) (15.7.6) unless N {x} = 1, in which case R({x}) = 0. Proof. In Lemma 15.7.I take A = In (x), B = Im (x) with n > m. Omitting the dependence on x for notational brevity, (15.7.2) implies that a.s. on {N (Im ) = 0}, P{N (In ) > 0 | Inc N } = 1 − P{N (In ) = 0 | Inc N } P{ N (Im ) = 0 | Inc N } . =1− c N} P{N (Im \ In ) = 0 | Im

(15.7.7)

For increasing n, the numerator here remains fixed, and the denominator c N }. We deduce decreases monotonically to the limit P{N (Im \ {x}) = 0 | Im that the limit exists a.s. on the set {N (Im \ In ) = 0} for every n > m, hence a.s. on {N (Im \ {x}) = 0}, and hence a.s. because N (Im \ {x}) → 0 a.s. as m → ∞. Also the ratio equals (15.7.4) except possibly for realizations where the denominator vanishes. If the latter holds, the ratio for finite n will tend to ∞ unless in the limit the numerator also vanishes. In this case, we have P{N (In ) = 0 | Inc N } → 0; that is, P{N (In ) ≥ 1 | Inc N } → 1, implying that π{x} = 1, in accordance with the interpretation here that ‘0/0 = 1’. Next, let T1 and T2 be dissecting systems with T1 ⊆ T2 . Then the limits π1 {x}, π2 {x} say exist for each system, and taking {Inj (x)} ⊆ Tj with Inj (x) ↓ {x} (nj → ∞) for j = 1, 2, so {In1 (x)} ⊆ T1 ⊆ T2 , it follows that π1 {x} = π2 {x} because we can always find In (x) ↓ {x} with successive terms taken alternately from {In1 } and {In2 }. In general, any two systems T1 , T2 generate by their intersection a third system T3 with T3 ⊇ T1 and T3 ⊇ T2 , so π{x} is independent of T . For bounded A ∈ BX , N (A) < ∞, so the defining sum at (15.7.5) can be expressed as a finite sum of limits as at (15.7.3) over disjoint sets Inj for sufficiently large n, and thus it is a random variable. From Proposition 9.1.VIII, we deduce that π(A, N (ω)) is a random measure. When Condition Σ holds, if π{x} = 1 for some x, then for this x and all n P{N In (x) ≥ 1 | Inc (x)N } = 1

a.s.

on account of the monotonicity of the denominator in (15.7.7). Consequently, P{N (In (x)) = 0 | Inc (x)N } = 0 a.s., contradicting Condition Σ. Thus, π{x} < 1. To demonstrate (15.7.6), refer to (15.6.18). By putting A = Im (x) and B = In (x), deduce that on {N (In ) = 0}, P{N (Im ) = 1 = N (In ) | Inc N } . R Im (x) | N = P{N (In ) = 0 | Inc N }

15.7.


529

Here, R(· | N ) is a measure and Im (x) ↓ {x} as m → ∞, so the left-hand side → R({x}). On the right-hand side, {N : N (Im ) = 1 = N (In )} ↓ {N : N {x} = 1 = N (In )}, so on {N (In ) = 0},

P N {x} = 1 = N (In ) | Inc N . R{x} ≡ R({x}) = P{N (In ) = 0 | Inc N } The numerator here coincides with that of (15.7.4), and the denominator equals

P N (In \ {x}) = 0 | Inc N − P N (In ) = N {x} ≥ 1 | Inc N . Because N is simple, the last term equals P{N (In ) = N {x} = 1 | Inc N }. Then (15.7.6) follows from (15.7.4) whenever {N (In ) = 0} for sufficiently large n. On the complementary event, x is an atom of N because (

{N (In ) ≥ 1} = N {x} ≥ 1 . n

In this case, we choose some positive integer k and substitute h(y, N ) = IA (y) I{N (B)≤k} (N ) I{N (A)≥1} (N ) in the relation (15.6.5) with B = In (x), A = Im (x) with m ≥ n. The left-hand side yields E R(Im ) I{N (In )≤k} (N ) I{N (Im )≥1} (N ) , and the right-hand side is bounded above by (k + 1)P{N (Im ) ≥ 2, N (In ) ≤ k + 1}. Now repeat this with Im replaced by a dissecting partition {Inj } for Im , and sum over j. Proceeding to the limit, the sum → 0 by simplicity, and the sum of the left-hand side converges to R{x}N {x}, the sum being taken over all atoms lying in Im . Consequently, R{x} = 0 whenever N {x} > 0. The following simple examples may help illustrate the nature of atoms of R and N , and especially the role played by π(·). Example 15.7(a) On Condition Σ [continued from Example 15.6(a)]. Let N be a point process on X for which P{N (X ) = r} = 1 for some fixed integer r ≥ 1 as earlier in (1◦ ). For x ∈ supp(N ) and a given dissecting system T let {In } ≡ {In (x)} be a sequence of elements of T contracting to {x}. Then P{N (In ) ≥ 1 | N (Inc ) ≤ r − 1} = 1 = lim P{N (In ) ≥ 1 | N (Inc ) ≤ r − 1} n→∞

= π{x}. The set {N : N (In − {x}) = 0} ∩ {N (Inc ) ≤ r − 1} consists precisely of those realizations N for which N (Inc ) = r −1, N (In ) = N {x} = 1. If we assume that

530


the points of N are r points i.i.d. over X , which is a bounded Borel subset of Rd —for example, the interior of a circle or a sphere—then assuming the sets In have positive Lebesgue measure, we should have P{N (In ) = 1 | N (Inc ) = r − 1} = 1

= P N (In ) = 1 = N {x} | N (Inc ) = r − 1 = 0

= P N (In − {x}) = 0 | N (Inc ) = r − 1 . Thus, this example justifies the interpretation of π{x} = π({x}, N ) as unity when the expression at (15.7.4) equals ‘0/0’. Example 15.7(b) A particular Gauss–Poisson process. Let there be given a Gauss–Poisson process in the plane, which, in its Poisson cluster process representation, consists of a cluster centre process that is Poisson at unit rate, and for which any point x in the centre process produces clusters of either zero or one additional points, with probability p for there being one point, which is then located at x + a for some fixed position a relative to the cluster centre. Consider such a process on a bounded subset X ∈ B(R2 ). It can be checked that this process satisfies Condition Σ. Suppose the state space X contains all of x, x − a and x + a for some x; consider realizations N for which N {x} = 1, and let {In } ≡ {In (x + a)} be a sequence of sets of positive Lebesgue measure belonging to some dissecting system for X and ↓ {x+a}. On realizations N for which N {x} = 1 = N {x−a} and N (X \ ({x} ∪ {x − a} ∪ In )) = 0, P{N (In ) ≥ 1 | Inc N } = (In )(1 + o(1)), so the ratio at (15.7.4) → 0. On realizations for which N {x} = 1 = N (Inc ), P{N (In ) ≥ 1 | Inc N } = p + (In ) 1 + o(1) and

P N (In \ {x + a}) = 0 | Inc N = 1 − (In ) 1 + o(1) ,

so the ratio at (15.7.4) → p = π{x} ≡ π({x}, N ). Example 15.7(c) Discrete Bernoulli process [cf. Example 7.2(d)]. Consider a simple point process N on the finite set X = {x1 , . . . , xn } satisfying, independently for each point, N {xi } = 0 or 1 with probabilities q and p = 1 − q, respectively. The Janossy measures are purely atomic, with J0 = q n and Jk (xr1 . . . , xrk ) ≡ Jk ({xr1 }, . . . , {xrk }) = pk q n−k for any subset Sk of k distinct points in X . Observe that n k n−k p q Jk (Sk ) = . all Sk k The integral equation defining R(· | ·) reduces to a definition as a ratio of Janossy measures ! R(y | x1 , . . . , xk ) = Jk+1 (y, x1 , . . . , xk ) Jk (x1 , . . . , xk ) = p/q.

15.7.


531

Also, if B = {y, x1 , . . . , xk },

P N {y} = N ({y, x1 , . . . , xk }) = 1 | B cN p pq k

= , = q k+1 q P N ({y, x1 , . . . , xk }) = 0 | B cN consistent with (5.6.18). On the other hand,

P N {y} = N ({y, x1 , . . . , xk }) = 1 | B cN pq k

π{y} = = k = p, c q P N ({x1 , . . . , xk }) = 0 | B N and π{y}/(1 − π{y}) = p/(1 − p) = p/q. Finally, if N {y} = 1, π{y} is unchanged, but for R({y} | ·) we should have a ratio of Janossy measures with the argument y repeated in the numerator, which is zero on account of the process being simple. The last of the three quantities mentioned around (15.6.3), the Papangelou measure ζ(·), is arguably the one that has the most important applications. Papangelou (1974b) devised it primarily as a means of tackling certain problems in stochastic geometry quite distinct from the present context (see Kallenberg (1983b) and related discussion for an informal account and references). As we show shortly, its importance in statistical applications is that under weak conditions it has a density which can be identified with the Papangelou intensity. We start by establishing its existence under Condition Σ, relating it to the random measures R(· | ·) and π(·) discussed already. Proposition 15.7.III. Let N, X , T , π, and R be as in Proposition 15.7.II, and suppose that Condition Σ holds. Then as n → ∞, the limit ζ(B) = lim

n→∞

kn

c P{N (Bnj ) = 1 | Bnj N },

Bnj ≡ B ∩ Inj ,

(15.7.8)

j=1

exists a.s. for all bounded B ∈ BX and defines a random measure given a.s. by (15.7.9) ζ(·) = π(·) + Rd (· | ·), where Rd (· | ·) is the diffuse (i.e., nonatomic) component of the random measure R(· | ·). Proof. Without loss of generality assume that B ∈ T ; then each Bnj = Inj ∈ T (although we continue to write Bnj ). Write kn j=1

c P{N (Bnj ) = 1 | Bnj N} =

kn

c P{N (Bnj ) = 1 | Bnj N } I{N (Bnj }=0}

j=1 c + P{N (Bnj ) ≥ 1 | Bnj N } I{N (Bnj }≥1} c − P{N (Bnj ) ≥ 2 | Bnj N } I{N (Bnj }≥1}

≡ Σ0 (n) + Σ1 (n) − Σ2 (n),

say.

532


For Σ0 (n), observe from (15.6.18) that c c c N } = R(Bnj | Bnj N ) P{N (Bnj ) = 0 | Bnj N }, P{N (Bnj ) = 1 | Bnj

so we can rewrite Σ0 (n) in the form c IBn (x) hn (x) R(dx | Bnj N ),

Σ0 (n) = B

where, for fixed N , Bn is the union of those Bnj where N (Bnj ) = 0, and c N } on Bnj . {Bn } is a monotonic increasing hn (x) = P{N (Bnj ) = 0 | Bnj sequence of sets, with limit B \ {supp N }, and hn (x) ↑ 1 − π{x} a.s. by Proposition 15.7.II. By monotone convergence, therefore, Σ0 (n) →

B\{supp N }

(1 − π{x}) Ra (dx) + Rd (dx)

with Ra denoting the atomic component of R. By using (15.7.3), Ra πa a.s., so the first term here equals π(B \ {supp N }), and because π{x} = 0 Rd -a.s., and R{x} = 0 for x ∈ {supp N }, the second term equals Rd (B). For Σ1 (n), for given N , the sum reduces for n sufficiently large to a sum over sets Bnj containing exactly one of the atoms of N (·) in B. By Proposition 15.7.II again, the limit as n → ∞ reduces to π(B ∩ {supp N }). Thus, Σ0 (n) + Σ1 (n) → Rd (B) + π(B) = ζ(B)

a.s.,

a.s. Just as for Σ1 (n), the sum and it remains to prove that Σ2 (n) → 0 say, over precisely N (B) sets reduces for n sufficiently large to a sum, Bnj , where N (Bnj ) = 1; that is, Σ2 (n) = =

c P{N (Bnj ) ≥ 2 | Bnj N}

P{N (Bnj ) ≥ 2, N (B \ Bnj ) = 0 | B cN } P{N (B \ Bnj ) = 0 | B cN }

on using Lemma 15.7.I. As n increases, each of the N (B) terms in the numerator → 0 because N is simple, and for the denominator, which is decreasing to P{N (B \ {x}) = 0 | B c N } for some x ∈ {supp N }, Condition Σ implies that 0 < P{N (B) = 0 | B cN } ≤ P{N (B \ {x}) = 0 | B cN }. The Papangelou intensity measure is related to a first moment in much the same way as the first moment and intensity measures coincide (Propositions 9.3.IX–X).

15.7.


533

Corollary 15.7.IV. Suppose the first moment measure EN (·) exists. Then for all bounded B ∈ BX , ζ(B) at (15.7.8) also has the representation lim

n→∞

kn c E N (Bnj ) | Bnj N = ζ(B)

a.s. and in L1 mean.

(15.7.10)

j=1

Proof. For the a.s. convergence, write c N E N (Bnj ) | Bnj

c c N } + E N (Bnj ) I{N (Bnj )≥2} (N ) | Bnj N , = P{N (Bnj ) = 1 | Bnj

and use an extension of Lemma 15.7.II to write kn c E N (Bnj ) I{N (Bnj )≥2} (N ) | Bnj N j=1

kn E N (Bnj ) I{N (Bnj )≥2} I{N (B\Bnj )=0} | B cN ) P{N (B \ Bnj ) = 0 | B cN } j=1 kn c E j=1 N (Bnj ) I{N (Bnj )≥2} (N ) | B N . ≤ P{N (B) = 0 | B cN }

=

This conditional expectation is bounded above by E(N (B) | B cN ) < ∞ a.s., and N being simple implies that each term in the sum → 0 a.s. (cf. also Exercise 9.3.10), so by dominated convergence we have the required result. To establish L1 convergence, observe in the proof of the proposition that Σ0 (n) increases monotonically to its limit, which has expectation bounded by E(N (B)), so its a.s. convergence implies its L1 convergence. Also, Σ1 (n) − Σ2 (n) =

kn

c P{N (Bnj ) = 1 | Bnj N } I{N (Bnj )≥1}

j=1

≤

kn

I{N (Bnj )≥1} ≤ N (B),

j=1

so here the a.s. convergence of Σ1 (n) − Σ2 (n) implies its L1 convergence by the dominated convergence theorem. Finally, kn kn c E N (Bnj ) I{N (Bnj )≥2} (N ) | Bnj N =E N (Bnj ) I{N (Bnj )≥2} E j=1

j=1

→0

(n → ∞)

from simplicity and the assumption that E N (B) < ∞.

534


So far the development in this chapter has been based mainly on disintegrations and limits, having much in common with the material of Chapter 13. It is possible to base the derivation of the Papangelou intensity measure on arguments much closer to those used in the discussion of the compensator and its density, the conditional intensity λ∗ (·, ω). With a state space X more general than R or R+ as in Chapter 14, the concept of predictability used there is replaced by that of exvisibility due to Van der Hoeven (1982) [we follow the terminology of Kallenberg (1983a)]. Write σ{B cN } for the completion of the B-external σ-algebra σ{B cN } with respect to the null sets of σ{N } of the process N . Then on the product space X × Ω (or, more specifically, X × NX# in the canonical set-up), define the exvisible σ-algebra Z to be the σ-algebra generated by sets of the form B × U , where B ∈ BX , U ∈ σ{B cN }. A stochastic process on (Ω, E, P ) is then exvisible if it is measurable with respect to Z on X × Ω. Given a random measure ξ, a ‘dual exvisible projection’ of ξ can be introduced as the unique random measure satisfying conditions (i)–(iii) of Proposition 15.7.V below. A direct proof of this assertion requires arguments from the general theory of processes analogous to those needed to give a direct proof of the properties of the compensator A(·) in Section 14.1 [see Van der Hoeven (1982, 1983)]. However, when ξ is a point process satisfying the special conditions assumed in this chapter (namely, it is simple and satisfies Condition Σ), it is not too difficult to see that the dual exvisible projection is nothing other than the Papangelou intensity measure ζ itself: we conclude with a formal statement and proof of this result. Proposition 15.7.V. Under the conditions of Proposition 15.7.III, and assuming EN (·) exists, the Papangelou intensity measure ζ is the unique (up to equivalences) random measure satisfying the conditions (i) ζ is determined by the point process N (i.e., ζ(B) is σ{N }-measurable for every B ∈ BX ); (ii) the process Z(x) ≡ ζ{x} is exvisible; and (iii) for every nonnegative exvisible process Y and bounded B ∈ BX , E Y (x) ζ(dx) = E Y (x) N (dx) . (15.7.11) B

B

Proof. The function ζ as defined at (15.7.8) is the limit of a σ{N }-measurable r.v., and therefore (i) holds for ζ. To prove (ii), suppose x is an atom of ζ. Then in the notation used in the proof of Proposition 15.7.III, a.s., ζ{x} = 1 − π{x} = lim 1 − hn (x) n→∞

kn c where hn (x) = j=1 P{N (Bnj ) = 0 | Bnj N } IBnj (x) is clearly exvisible. The limit is thus a.s. equal to an exvisible process, and if the σ-fields are

15.7.


535

complete we can allow modifications on sets of measure zero without upsetting measurability, so all versions are exvisible. For (iii), take Y in (15.7.11) to have the special form Y (x, ω) = IB (x) IU (ζ)

for U ∈ σ{B cN }.

c Then Corollary 15.7.IV implies that because U ∈ σ{Bnj N } for every Bnj ,

kn c E IU N (B) = E IU E N (Bnj ) | Bnj N → E IU ζ(B) . j=1

Because the left-hand side here is fixed, (15.7.11) follows for this particular function Y , and then for processes Y as described by standard extension arguments. Thus, ζ satisfies (i)–(iii): suppose η is some other random measure satisfyc N }, (15.7.11) implies that ing the conditions. Whenever U ∈ σ{Bnj c N = E IU η(Bnj ) = E IU N (Bnj ) E IU E η(Bnj ) | Bnj c N , = E IU E N (Bnj ) | Bnj from which it follows that c c E η(Bnj ) | Bnj N = E N (Bnj ) | Bnj N

a.s.,

and hence that lim

n→∞

kn kn c c E η(Bnj ) | Bnj N = lim E N (Bnj ) | Bnj N = ζ(B) n→∞

j=1

a.s.

j=1

Each of these two sums may be further analysed by the same procedure as used in forming the sums Σ0 (n), Σ1 (n), Σ2 (n) in the proof of Proposition 15.7.III. c N }-measurability of η(·) on {N (Bnj ) = 0}, In particular, using the σ{Bnj kn c E η(Bnj ) I{N (Bnj )=0} (N ) | Bnj N I{N (Bnj )=0} j=1

=

kn

c η(Bnj ) P{N (Bnj ) = 0 | Bnj N } I{N (Bnj )=0}

j=1

→

B\{supp N }

(1 − π{x}) η(dx) = ηd (B) +

(1 − π{xi }) η{xi },

where in the second step we have used the limit behaviour of the function hn (x) as in the proof of the earlier result, ηd (·) denotes the diffuse component of η, and summation is over the atoms in B of η(·). Thus, we have a.s. (1 − π{xi }) η{xi } = ηd (B) + (1 − π{xi }) ζ{xi }. (15.7.12) ηd (B) +

536


Now it follows from conditions (ii) and (iii) that the atomic parts of η and ζ must be equal, because if we let V = {(x, ω) ∈ X × Ω: Z(x, ω) > ζ({x}, ω)}, then V is an exvisible set and from (15.7.11),

Z(x, ω) − ζ({x}, ω) η(dx, ω) − ζ(dx, ω) P(dω) = 0.

V

Only atoms of η and ζ contribute to this integral, and indeed only those for which η{xi } − ζ{xi } > 0. This leads to a contradiction unless η{xi } ≤ ζ{xi } a.s., and by reversing the argument we deduce that η{xi } = ζ{xi } a.s. for all atoms; that is, η and ζ have the same atoms and the atoms are of the same size a.s. Then (15.7.12) implies that the diffuse components agree a.s.; that is, η and ζ coincide except possibly on a set of measure zero. Now suppose that Condition Σ holds and that the point process N admits a Papangelou intensity ρ(x | N ). It means that the random measure R is a.s. absolutely continuous with ρ as its density. Because its atomic component π is null, it then follows from (15.7.9) that ζ also has no atomic component, and that the diffuse components of ζ and R coincide. Hence, ζ also is a.s. absolutely continuous with density ρ. This gives the following corollary of Proposition 15.7.V, in which we repeat (15.7.11) for convenience. Corollary 15.7.VI. If the simple point process N admits a Papangelou intensity ρ and satisfies Condition Σ, then the random measures R and ζ coincide, and both have density ρ. Moreover, in this case, for all nonnegative, exvisible, FN -measurable processes Y E Y (x) N (dx) = E Y (x) ζ(dx) = E Y (x) ρ(x | N ) dx . B

B

B

This equation may be compared with the extended form (15.5.15 ) of the Georgii–Nguyen–Zessin formula. There the role of the exvisible process Y is taken by the function h(x, v), where v denotes (points of) a realization N , so h must embody any condition of FN -measurability. If h does not depend on v, or depends on it only through the function ρ(u | v), which is itself exvisible, then (15.5.15 ) will hold. A further illustration is in Exercise 15.7.1.

Exercises and Complements to Section 15.7 15.7.1 Show that if a process Y (x) is a.s. continuous and FN -measurable, and vanishes outside a bounded set, then it is exvisible.


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Subject Index

[Page references in slanted font, such as 382, are to Volume I, but these references are not intended to be comprehensive.] Adjacency relation ∼, 120, 127 ∪ ∩ binary relations ∼, ∼, 125 x set-dependent adjacency (∼), 128 Area-interaction point process, 124 as equilibrium birth-and-death process, 130 Asymptotic independence mixing, 206 weakly mixing, 206 ψ-mixing, 206 ergodicity, 206 expressed via functionals, 210 Asymptotic stationarity, 223 (C, 1)-asymptotic stationarity, 223 shift-coupling sufficient, 230 weak and strong coincide, 326 conditions for convergence of moment measures, 236 strong asymptotic stationarity, 223 Atomic measure, 382 counting measure correspondence, 4 moment measure characterization, 66 Avoidance function, 2, 33, 459 determines distribution of simple point process, 35 of Cox process, 38 557

use in limit theorems, 166 Avoidance probability, 459 Bartlett spectrum, 303 atom at origin, 205 Batch Markovian arrival process (BMAP), 110 convergence to equilibrium, 228 E–M algorithm for, 114 Q-matrix structure, 110 representations, 111 stationary interval distribution and correlation properties, 118 Bayesian-type formulae likelihood ratio, 406 Bernoulli process, discrete Papangelou kernel properties, 530 Bijective point map, 309 Binomial probability bounds, 166 Birth-and-death process, 99 conditional mark distribution, 100 death process in reliability, 100 spatial, 126, 130 as space–time process, 488 Blackwell renewal theorem, 83 generalized, 331

558 BMAP, 110, see Batch Markovian arrival process Bonferroni inequalities p.g.fl. bounds, 71 Borel measure, 384 Boundedly finite Borel measure, 2 Bounded measurable function space BM(X ), 52 B-selective point map, 311 ( , 1)-asymptotic stationarity, 326 see also Asymptotic stationarity Campbell measure, 268 ——, basic properties characterization, 272, 282 definition via extension, 270 factorization, 269 first moment measure relation, 273 for a.s. atomic random measure, 275 structure is singular, 275 characterization for point process, 275 for KLM measure Q, 272, 282 for random measures, 284 higher-order analogues, 272 ‘modified’ v. ‘reduced’, 270 Radon–Nikodym approach, 270 refinement of first moment, 270 terminology origin, 271 ——, invariant, 285, 293 factorization, 293 Palm measure characterization, 294 stationary random measure invariance characterization, 285 ——, marked from marked cumulative process, 379 of MPP, reduced, 331 product of predictable kernel and Campbell measure for Ng , 380 semi-Markov process example, 381 Campbell theorem, 271 original, 66 Campbell measure precursor, 66 refined, 288 Cantor dust, 20 Central limit properties simple point process martingale, 412 with random scaling, 413 Characteristic functional, 54 Taylor series moment expansion, 68 remainder terms, 71

Subject Index

Characteristic functions continuity condition, 63 convergence condition, 64 Cliques, 121 properties, 121, 129 maximal, 121 Cluster iterates, 334 Poisson case, 334 infinitely divisible, 334 method of backward trees, 336 stable/unstable dichotomy, 336 for critical cluster member process, 337 for Neyman–Scott process, determined by random walk, 337 for stable cluster member processes, weakly singular infinitely divisible limit, 338 Cluster process, stationary cluster components, construction, 193 from nonstationary components, 192 stochastic condition to be well-defined, 192 Compensator, 241, 358 absolute continuity from conditional d.f.s, 364 characterizations Cox process, 419 Poisson process, 420 continuous equals quadratic variation for simple point process, 370 variance of integrated martingale, 376 convergent sequence of condition for Poisson/Cox limit, 384 dependent-thinning example, 387 dual predictable projection, 378 Campbell measure derivation, 377 integral of conditional intensity, 390 on whole of R, 394 complete conditional intensity, 394 one-point process, 358 unbounded, 376 uniquely defined, determines process, 365 Complete conditional intensity, 394 Hawkes process, 395 renewal process with density, 395 stationary process, history H† , 396 hazard function representation, 397

Subject Index

Complete separable metric space (c.s.m.s.), 124, 2 Completely independent MPP when ground process simple, 86 Completely monotone set function, 35 rˆ ole in determining point process, 36 Completely random measure, 77 fidi distribution structure, 85 Laplace functional representation, 86 moments, 67 sample path structure, 79 Laplace transform, 82 stationary Laplace–Stieltjes transform representation, 85 Compound Poisson process extended, 161 from random time change, 422 moment measures, 72 stationary, 180 MPP formulation, 181 Concentration function of distribution, 167 tight convergence rate, 175 Conditional intensity function 211, 231 existence of MPP, 429, 432 stationary solution, 434 weak and strong asymptotic stationarity, 434 law of evolution, 432 Lipschitz conditions, 430 Conditional intensity complete (for stationary process), 443, see also Complete conditional intensity F -mark-predictable Radon–Nikodym derivative of marked Campbell measure, 391 integral as F -predictable intensity of ground process, 391 F -predictable conditional mark distribution, 391 in information gain, 443 on coarser G w.r.t. F -conditional intensity, 393 for mixed Poisson process, 393 Conditional intensity measure (for MPP), 399 product form for kernel, 399 Condition Σ, 521 examples, 521, 529

559

Configurations of points, ergodicity, 202 Contact distribution function, 460 spherical, 459 Continuity condition for fidi distributions, 28 Controlled thinning of point process, 387 Convergence modes of, 131 of Campbell measures, 297 of fidi distributions, 132 of KLM measures, 147 of Laplace functionals weak convergence conditions, 138 of moments, 141 of moment measures, 144 of p.g.fl.s weak convergence conditions, 138 of point processes of probability distributions strong, 132 implies weak convergence, 132 weak, 132 does not imply strong, 134 Convergence to equilibrium, 223 as limit distributions interpretation via inversion theorem , 325 via weak convergence, 326 Palm, from limit of nth point, 323 stationary, from t → ∞, 323 conditional intensity conditions, 427 via Poisson embedding, 428 Foster–Lyapunov conditions, 228 in variation norm, 223 space Z of initial conditions, 226 from Palm distribution, 323 see also Asymptotic stationarity Convex averaging sequence, 196 Copy or version of process same fidi distributions, 11 Counting measure on R one-to-one correspondence with sequence of intervals, 24 Coupling, 132, 229 conditions for, 231 coupling inequality, 133, 229 coupling time, 132 equivalences, 231 with stationary process

560

implies strong asymptotic stationarity, 230 see also Shift-coupling Covariance density, 69 Covariance measure of random measure, 69 condition to be singular, 70 Covariant mapping, 309 Cox process avoidance function, 38 class invariant under rarefaction, 166 contraction of thinned process, 157 convergence to conditions on compensators, 384 via dependent-thinning, 387 directed by Markov chain, 101 likelihood, 102 E–M algorithm, 103 finite state space, 108 MPP extension, 117 directed by Markov diffusion, 409 directed by partially observed Markov process, 410 Neyman–Scott example, 410 directed by stationary G-process, 87 from scale-contraction of Rd , 157 stationary, 181 iff directing measure stationary, 192 long-range dependent, 254 preserved under thinning and translation, 181 C.s.m.s. (= Complete separable metric space), 124, 2 Cumulant measures, 69 Cumulative process, 356 adapted to right-continuous history F has unique F -compensator, 367 family for MPP, 356 weak convergence, 143 with density, 358 Cycle stationarity point-stationarity, R1 analogue, 308 Davidson’s ‘big problem’, 481 Kallenberg counterexample, 482 Determining class of set functions, 372, 27 Deterministic component of random measure, 86 Deterministic H-intensity characterizes Poisson process, 420

Subject Index Deterministic lattice process in Rd , 192 Deterministic map of bounded set into itself, fractal dimension, 351 moment growth conditions, 352 consistent if mixing, 352 Deterministic point process, 76, 137 in Rd , 192 unit rate, 137 Diagonal shifts and stationarity, 182 reduced measure, 183 Dirac measure, 382, 3 Directed lines, random process of, 472 see also Line process Directional rose, 467 Ripley’s K-function, 467 process in R2 , moment factorization, 467 Dirichlet distribution, 24 Dirichlet process, 11 random probability distribution, 11 moment measures, 74 Discrete-time renewal process F -compensator, 373 Dissecting system, 382 use in entropy approximation, 446 monotone under refinement, 454 information gain in limit, 454 tiling, infinite analogue of, 311 tool to study sample paths, 39 Dobrushin’s lemma, 45 Doob–Meyer decomposition, 241, 430, 355 Dust, 20 see also Cantor, Lévy, 20 Edge-effects in nearest-neighbour estimation, 463 Hanisch-type correction, 466 E–M algorithm, 101 numerical procedures, 103 uniformization algorithm, 109, 117 Empty space (F -)function, 2, 459 estimation, edge-effects, 463 Hanisch-type corrections, 466 Poisson cluster process, 461 via local Janossy measure, 461 see also Avoidance function

Subject Index

Entropy, 440 atomic distribution, 440 continuous distribution, 440 generalized or relative, 440 Poisson has maximum entropy, 454 Entropy dimension, 341 Entropy rate (process in R1 ), 444 finite sample approx’n, 449 convergence conditions, 450 L1 and strong convergence, 451 mixed and compound Poisson, 444 renewal process, 455 Entropy rate for intervals, 452 relation to entropy rate, 453 Enumeration, measurable, 14 Ergodic theorem averaging over group, 196 convex averaging sequence, 196 for MPP, 197 for random measure, 197 for weighted averages, 201 general, 199 higher order, 242 individual, 196 statistical versions of, 204 statistical, 197 ETAS model, 203 nonlinear, 437 as space–time model, 499 Event- (= interval-)stationarity, 327 Extended Laplace functional, 57 sequence of, convergence, 58 Extended MPP, 7 MPP counterexamples, 22 purely atomic random measure, 8 Extended p.g.fl., 60 convergence, 64 Exvisibility (spatial predictability), 513 of Papangelou intensity measure, 534 Factorial moment measure, 133, 69 advantages, 70 Family of probability measures uniformly tight, 136 F -adapted process, 236 on Z+ , 372 F -predictable process, 425, 358 on Z+ , 372 F -function, see Empty space function

561

Fidi distributions, 2, 25 consistent family, 26 convergence of random measures, 135 equivalent to weak convergence, 135, 137 Filtering problem, 400 Finite-dimensional (fidi) distributions, 25, see Fidi distributions Finite point process existence, 32 Fixed atom of random measure, 39 at most a countable infinity, 39 component of random measure, 86 Flow, 269 on probability space, 177, 179 stationary Cox process, 182 Fractal dimension, Rényi, 340 controlled Palm growth, 354 deterministic map of set into itself, 351 from small-scale clustering, 349 kth order, controlled diagonal growth, 347 mean-square consistent estimate, 347 multinomial measures, 352 Functionals, linear, 52 Gamma random measure, 167 stationary case, 162, 11, 30 Gaussian measures on Hilbert space, 54 Gauss–Poisson process, 185, 465 as limit of u.a.n. array, 154 isotropic centred, 468 KLM measure structure, 94 marked, 332 reduced moment densities, 332 Papangelou kernel aspects, 530 Generalized Blackwell theorem, 331 Generalized compound Poisson process, 61 Generalized entropy expected likelihood ratio, 441 Kullback–Leibler distance, 441 reference measure in, 441 Generalized functions, random, 53 Georgii–Nguyen–Zessin formula, 462 GNZ equation, 511 extended form, 513, 536

562

Germ–grain models, 503 see also Lilypond model, Particle process, 205 G-function, 460 see also Nearest-neighbour function Gibbs kernel, 525 portmanteau Campbell measure, 525 Gibbs process with pairwise interactions, 507 interaction potential, 507 process on circle or sphere, 509 G-random measure, 83 directing measure of Cox process, 84 Lévy representation, 83 shot-noise, 84, 87 Grassberger–Procaccia estimates, 355 Ground measure, 3 Ground process, 194, 7 condition to be well-defined, 22 Haar measure, 408 in transformation invariance, 409, 188 MPPs as canonical example, 190 Hamel equation, 64, 192 application in Rd , 192 variant of, 482 Hammersley–Clifford theorem, 122 extended, 129 Hawkes process, 183 complete conditional intensity, 395 convergence to equilibrium, 232, 236 strong asymptotic stationarity, 234 exponential decay, 99 moments, 145 nonlinear, 437 Lipschitz condition, 437 without ancestors, 141, 145 Hazard function, conditional rˆ ole in likelihoods, 402 Hereditary class of subsets, 122 Hereditary function, 122, 128 Hidden Markov model (HMM), 101, 400 Cox process, 103, 400 Neyman–Scott example, 410 Poisson observations for, 106 state estimation, 105 Higher order ergodic theorem, 242 Hill estimate, 346

Subject Index

History, 357 minimal H, 357 right-continuous, 357 F(+) , 373 counterexample, 373 HMM, see Hidden Markov model Homogeneous point process, 459 Hougarde process, 83 Hurst index, 250 cluster process, 251 Cox process, 252, 254 of superpositions, 252 stationary renewal process, 254 Hyperplanes, directed, intersections of, 484 IHF, see Integrated hazard function Independence of random measures, 63 Infinitely divisible distributions, nonnegative, 82 Infinitely divisible MPP, 94 cluster process representation, 94 ground process, 94 infinite divisibility, 94 KLM measure, 94 structure, 94 Infinitely divisible point process, 87 a.s. finite characterization, 92 convergence, 147 equivalent KLM Q-weak convergence, 147 KLM measure, 89 regular/singular components, 92 representations cluster process, 89 p.g.fl., 91 Poisson randomization, 89 transformation invariant iff KLM measure invariant, 221 ——, stationary iff KLM measure stationary, 216 KLM measure ergodicity, mixing conditions, 218 Palm factorization, 295 regular/singular dichotomy if regular then ergodic, 217 regular iff Poisson cluster process representation, 216 singular if Poisson randomization representation, 216 weak/strong singular dichotomy, 220

Subject Index

strongly singular iff Poisson randomization superposition exists, 221 weakly singular example, 185 Infinitely divisible random measure, 87 fidi distributions infinitely divisible, 88 representation, 95 Laplace functional representation, 93 Infinite particle systems external configuration, 519 Information gain, expected, 276 generalized entropy, 441 in discrete approx’n to entropy, 447 limit of refinement, 454 mixed and compound Poisson, 444 MPP decomposition (points/marks), 444 renewal process, 445 rˆ ole of conditional intensity, 443 simplifies when stationary, 443 two processes evolving simultaneously, 448 Inhomogeneous Poisson process, as Markov point process, 121 Innovations (‘residual’) process, 513 Integrated hazard function (IHF), 108, 359 for one-point process, 359 Integration by parts for Lebesgue– Stieltjes integral, 107, 376 Intensity measure of point process, 44 Korolyuk equation, 46 Interacting point processes, 244 Interaction function, 122 Interaction potential in Gibbs process, 127, 507 Internal history H, 234, 357 H-predictability of process, 374 Intervals, process of, 302 definition of point process, 13 marked, stationary, 333 Interval-stationarity, 268, 299 Intrinsic history, 234, 357 Invariant σ-field I trivial invariant functions constant, 204 Inversion formulae, Palm measure stationary point process, 291, 300

563

Isotropy, 297, 466 moment measure factorizations, 467 rotationally invariant probability, 466 MPP analogue, 467 Iterated convolutions conditions for Poisson limit, 169 see also Cluster iterates, Random translations Janossy density, 125 in Papangelou intensity, 507 Markov point process density, 119 J-function, 460 clustering/regularity indicator, 460 Poisson cluster process, 461 Poisson process, 464 Jordan–Hahn decomposition, 374, 38, 252 Khinchin existence theorem, 45 Khinchin orderliness, 47 K-function, 464 see Ripley’s K-function KLM measure, 91 extended, convergence of, 147 Gauss–Poisson process, 94 infinitely divisible MPP, 94 Palm factorization for stationary infinitely divisible point process, 295 regular, strongly or weakly singular, 295 Q-weak convergence, 147 see also Infinitely divisible point process Kolmogorov consistency conditions, 27 Kolmogorov existence theorem for point process, 30 for random measure, 28 Korolyuk’s theorem, 45 generalized Korolyuk equation, 46 purely atomic random measure, 50 Kullback–Leibler distance expected information gain, 441 generalized entropy, 441 Laplace functional, 57 characteristic functional analogue, 64 completely random measure, 86 convergence conditions, 64 expansion, first-order, 75 Taylor series, moments, 75 extended, 57

564

Palm kernel characterization, 280 random measure characterization, 57 Lebesgue–Stieltjes integration by parts, 107, 376 Level-crossing, Poisson limit, 159 Lévy dust, 20 Lévy process nonnegative, 82 subordinator, 20 Lifted operator, 178 Likelihood, MPP, point process existence when H-adapted, 401 from ground process intensity and conditional mark density, 403 reference probability measure, 401 Likelihood ratio, point process as Radon–Nikodym derivative, 411 from H-conditional intensity, 402 H-local martingale, 404 in Bayesian-type formulae, 406 Poisson process with mixed atomic and continuous parameter, 411 Lilypond protocol models, 503 algebraic specification, 505 germ–grain model, 503 absence of percolation, 504 volume fraction, 504 Linear functionals, 52 Linear process of random measure, 74 shot-noise example, 74 Line process clustered, 484 coordinate representations, 472 directed/undirected, 472, 483 in R3 , 484 line measure, 474 Poisson, 472 stationary, 476 no parallel/antiparallel lines, 485 railway line process, 484 ——, stationary directional rose, 474 isotropic, 474 second-order properties, 477 parallel/antiparallel lines, 478 reflection invariant, 478 mean density, 474 moment structure, 473 shear-invariant, 473, 482 moment factors, 474

Subject Index

Line segment process from lilypond model, 504 yielding random measure, 23 Lipschitz conditions on conditional intensity functions, 430 Local martingale, 358 integral w.r.t. martingale, 373 Local Palm distribution P(x,κ) , 273 element of Palm kernel, 273 for modified Campbell measure, 298 for MPP, 317 for nonsimple point process, 282 for second-order Campbell measure, 298 for stationary process, 274 higher-order family, 281, 282, 284 random measure with density, 274 Long-range dependence, 250 renewal process, 250, 254 cluster process, 251 covariance measure decomposition, 252 deterministic process, 253 ‘power-law decay’, 254 MAP, see Markovian arrival process Mark distribution, stationary in ergodic theorem for MPP, 197 Marked Campbell measure, 317, 379 Marked cumulative process, 378 marked Campbell measure, 379 Marked point process (MPP), 194, 7 embedded regeneration points, 328 Palm moment inequalities, 333 ergodic theorem, 199 stationary mark distribution, 197 extended MPP, 7 individual ergodic theorem, 318 nonergodic case extension, 321 for marked random measure, 322 information gain, 443 points v. marks decomposition, 444 likelihood existence when H-adapted, 401 Palm distributions for mean Palm distribution P 0 , 323 local Palm distributions P(x,κ) , 278, 317 for ground process, 279, 283 representation on X × K∪ , simple, 23 with simple ground process, 22

Subject Index

random rescaling to compound Poisson, 422 rescaling counterexample, 425 rescaling to two-dimensional Poisson process, 424 ——, identified processes completely random, 84 with simple ground process, 84 structure of, 85 exponentially distributed intervals, 332 marked Gauss–Poisson process, 332 reduced moment densities, 332 on S, stationary, 190 conditions for stationarity, 191 ——, stationary, 197 asymptotic stationarity, 223 (C, 1)-asymptotic stationarity, 223 weak and strong coincide, 231 inversion theorem analogue, 327 averaged (= mean) Palm distribution P 0 , 319, 323 complete conditional intensity, 399 convergence to equilibrium, 323 coupling, shift-coupling equivalences, 230 ergodic, L2 convergence for, 204 higher-order mark distributions, 321 independent unpredictable marks, ergodic limits, 319 inversion theorem for Palm and stationary distributions, 324 kth order stationarity, 237 mark-dependent Palm distributions, 317 on Rd × K a.s. zero–infinity dichotomy, 205 P, P0 invariant σ-algebras equivalent, 333 reduced Campbell measure, 318, 331 reduced moment measures, 238 higher-order ergodic theorem, 247 stationary mark distribution, 197, 317 Markov density function, 120 Markov modulated Poisson process (MMPP), 101 convergence to equilibrium, 228 Markov point process, 118 density function for, 119 Janossy densities, 119

565

Hammersley–Clifford theorem, 122 extended, 129 products of Markov density functions, 125 Papangelou conditional intensity, 120 simple finite point process with density, 120 Markov renewal function, 115 mean Palm, subadditive, 332 Markov renewal process (MRP), 96 as MPP, 98 point process properties, 98 ground process, renewal function, 332 factorial moment densities, 99 Markov renewal function, 115 Markov renewal operator, 98 observed on subset, 115 renewal function analogue, 99, 332 subadditivity, 332 semi-Markov process equivalence, 97 see also Semi-Markov process Markovian arrival process (MAP), 110 Q-matrix structure, 110 Martingale, 427 integral w.r.t., local martingale property, 373 quadratic variation of, 368 increments as conditional variances, 369 see also Point process martingale Maximal clique, 121 Mean Palm distribution P 0 for MPP, 323 Measurable enumeration, 14, 24 Measure, decomposition of, 4 Method of backward trees for cluster iterates, 336 Method of reduced trees for cluster iterates, 336 Metric transitivity implies ergodicity, 204 of transformation, 194 ergodic theorem for, 201 M/G/∞ queue, spatial MPP in time, 488 space–time model, 488 Minimal history H, 357

566

Mixed Poisson process conditional intensities on G ⊂ F , 393 conditions to be ordinary, 51 contraction of thinned process, 156 defined by moments, condition, 73 fidi distributions, 62 limit of points moving with random velocities, 172 limit of random translations of nonergodic process, 171 moment measures, 72 overall Palm probability in ergodic limit, 323 p.g.fl., 61 random translation invariant, 172 ——, stationary mixing iff Poisson, 212 Mixed random measure, moments, 67 Mixing process, 206 kth order mixing, 215 trivial tail σ-algebra, 215 MMPP, see Markov modulated Poisson process Modified Campbell measure, 271 analysis of Papangelou kernel, 521 disintegration of, 524 in GNZ equation, 512 local Palm distributions for, 298 ‘reduced’ terminology, 270 Moment measure, 65 Campbell theorem, 66 expectation measure, 65 higher-order, symmetric, 75 isotropy, factorization, 467 Ripley K-function and directional rose, 467 kth order, 66 of diagonal of power set, 75 bound on multiplicity, 75 stationary diagonal factorization, 237 reduced moment estimation, 244 MPP, see Marked point process MRP, see Markov renewal process Multidimensional random measure, 7 Multivariate characteristic function, 54 characteristic functional, 54 Multivariate infinite divisibility, 94 Multivariate Poisson process independent components, as limit, 152

Subject Index

Multivariate random measure, 7 Mutual nearest-neighbour matching, 311 Natural history, 357 Nearest-neighbour distances, 459 stationary MPP, 462 independent marks, 462 Nearest-neighbour (G-)function, 460 as Palm–Khinchin limit, 466 estimation, edge-effects, 463 Hanisch-type corrections, 466 Matérn’s Model I, 466 stationary MPP, 462 independent marks, 462 via local Janossy measure, 461 Nearest-neighbour matching, 311 Negative binomial process, 10, 23, 73 defined via two mechanisms, 73 local properties, 73 Neighbourhood relation, see Adjacency Neyman–Scott cluster process, 181 analogue on circle S, 189 bivariate, nonisotropic, isotropic, 471 hidden structure, 410 in R2 , 464 conditional intensity, 465 space–time, singular components, 349 different orders of growth, 350, 355 stationary regular representation, 185 Nonergodic process Bartlett spectral mass at origin, 205 Nonnegative increment process, 11 Nonnegative infinitely divisible distributions, 82 Nonnegative Lévy process, 82 Nonnegative stable processes, 83 Laplace–Stieltjes transform, 85 Nonstationary point process counterexamples avoidance function stationary, 191 MMPP, 455 one-dimensional distributions stationary, 191 One-point process convergence, 137 H-compensator, 358 absolutely continous or continuous, 360 randomized hazard function, 362

Subject Index

square integrable quadratic variation, 371 with prior σ-algebra F0 compensator w.r.t. intrinsic history, 360 ——, MPP compensator, 389 for ground process, 389 Orderly point process, 46 equivalent conditions, 51 Khinchin orderly, 47 µ-orderly, 46 terminology: orderly, ordinary, 51 Ordinary point process, 46 simple, 51 Ord’s process, 130 Palm, theory, 268 queueing theory applications, 269 stochastic geometry applications, 269 Palm distributions, measure, 273 for MPP, 277 from Campbell measure, 273 local, 273 Palm kernel, 273 ——, moment measures, 292 as Radon–Nikodym derivatives, 293, 298 relation to stationary moments, 292 Palm kernel, for Poisson process, 280 Laplace functional characterization, 280 see also Local Palm distribution Palm measure, σ-finite Campbell measure characterization, 293 factorization characterization, 294 ——, for MPP P(x,κ) , 317 mean (‘average’) P 0 , 323 Sigman’s alternative version, 332 ——, stationary point process inversion formulae, 291, 300 Palm probabilities as rates, 290 Palm–Khinchin equations generalization, 302 point process in Rd , 308 Papangelou (conditional) intensity, 120, 506 conditional probability interpretation, 507 defined via Janossy densities, 507

567

density of Papangelou kernel, 524 density of Papangelou measure, 536 higher-order, 507 integral relations, 508 Markov density function rˆ ole, 120 multiplicative relations, 510 conditional Papangelou kernels, 525 relation to Palm densities, 507 Papangelou (intensity) measure exvisibility property, 534 relation to first moment, 533 Papangelou kernel, 520 atomic part, 520, 527 discrete Bernoulli example, 530 Gauss–Poisson example, 530 higher-order, 525 Gibbs (portmanteau) kernel, 525 Papangelou (intensity) measure, 520 Papangelou intensity as density, 524 viewed as random measure, 526 Penetrable spheres model, 124 P.g.fl., 59 see Probability generating functional PH-distributions, 111 representation (Q, π), 111 Laplace transforms, 117 Planar point process (= point process in R2 ) isotropic, 466 Poisson, centred, 468 Gauss–Poisson, centred, 468 moments, polar coordinates and factorization, 467 stationary isotropic, 469 counterexample, 205 moment measure factorization, 470 Ripley K-function and directional rose, 470 Point map bijective, 309 B-selective, 311 covariant mapping, 309 for point-stationarity, 308 inverse, 309 properties, 317 Point process (see also individual entries) ——, basic properties distribution, 26 determined by fidi distributions, 26

568

ergodic limit a.s. constant trivial invariant σ-algebra, 206, 208 metrically transitive, 206 formal definition, 7 integer-valued random measure, 7 intensity measure, 44 interval sequence definition, 13, 302 Kolmogorov consistency conditions, 30 nonnull, 188 orderly, ordinary, 46 random variable mapping characterization, 8 simple, simplicity of, 47, 7, 43 unaligned, 312 ——, general properties asymptotic independence expressed via functionals, 210 mixing, 206 ψ-mixing, 206 weakly mixing, 206 conditional intensity local Poisson character, 370 entropy discrete trial approximation, 445 dissecting partitions, 446 filtering, 400, 407 Cox process and HMM, 400 unit rate Poisson as reference, 407 mixed Poisson example, 407 updating formulae, 407 time series analogies, 401 likelihood existence when H-adapted, 401 likelihood ratio from H-conditional intensity, 402 H-local martingale, 404 martingale, for simple process, 247, 368, 412 F -compensator via IHFs of conditional d.f.s, 364 quadratic variation, 370 randomly scaled to normal limit, 415 (mixed) Poisson example, 416 ——, stationary, in R, 178 complete conditional intensity, 396 MPP extension, 399 kth order moments and k-point configurations, 244 kth order stationarity, 237 reduced moment measures, 238

Subject Index

long-range dependence, 250 Hurst index, 250 Palm measure a.s. zero–infinity dichotomy, 299 infinite intensity possible, 302 ——, stationary, in Rd , 304 points enumerated by distance from origin, 305 a.s. no points equidistant from origin, 304 Palm measure bijective point map invariant, 312 ——, identified processes controlled thinning, 387 in R2 , see Planar point process multivariate, 7 quadratic variation, 370 see also MPP on circle S stationary, 289 reduced moment measures, 298 on surface of cone, 471 two parameterizations, 471 on Z+ , 372 F -adapted, -predictable, 372 F -compensator, 372 discrete-time renewal process, 373, 375 quadratic variation, 375 Point-stationarity, 269, 299, 312 Poisson approximants inequality, 153 tool for convergence conditions, 154 Poisson cluster process convergence to equilibrium, 236 strong and geometric asymptotic stationarity, 236 empty space function, 461 stationary, regular representation, 183 nonuniqueness, 184 Khinchin measures, 184 Neyman–Scott process, 185 Poisson distribution maximum entropy, 454 Poisson embedding for general history, 438 history-dependent thinning, 426 Poisson probabilities bounds, 166

Subject Index

Poisson process, convergence to conditions on compensators, 384 dependent-thinning example, 387 Poisson process ——, basic properties existence, 31 infinitely divisible, 91 isotropic centred, 468 moment measures, 72 ordinary iff simple, 51 p.g.fl., 60 stationary, 180 ——, characterization class invariant under rarefaction, 165 deterministic H-intensity, 420 inverse compensator scaling, 421 Palm kernel and Slivnyak–Mecke theorem, 281 Watanabe’s, 418 ——, general properties configurations of points, 202 information gain, 444 maximum entropy rate, 454 Palm kernel, 280 Poisson property lost under random shift, 316 ——, limit properties conditions on compensators, 384 dependent-thinning example, 387 contraction of thinned process, 155 multivariate, 151 of high level-crossing, 159 of dilated superposition, 150 extensions, 154 variation norm limit, 161 Poisson randomization representation for infinitely divisible process, 89 KLM measure totally finite, 93 Power-law growth, 342 Predictability, 355 Probability generating functional (p.g.fl.), 144, 59 characterization for point process, 60 convergence conditions, 64 counterexample, 64 expansions of via factorial moments, 70 factorial cumulant measure, 71 moments, 70

569

extended p.g.fl., 60 condition for convergence, 64 infinitely divisible point process, 91 Prohorov metric, 145 ψ-mixing stationary random measure central limit property, 214 Purely atomic measure, 3 see also Random measure, purely atomic Quadratic random measure, 9 integral as random measure, 42 moments, 67 Quadratic variation process of martingale, 368 atomic and continuous components, 370, 375 equals compensator if continuous for simple point process, 370 for simple and multivariate point processes, 370 square integrable one-point process, 371 Q-weak convergence, 147 Railway line process, 484 Random distribution, 11, 23 absolutely continuous, 50 nonatomic condition, 75 Random linear functionals strict and broad sense, 53 Random Markov shifts, 173 Random measure, 6 ——, basic properties characterization characteristic functional, 56 distribution, 26 determined by fidi distributions, 26 Laplace functional, 57 random variables, 54 sample path components, 86 deterministic, 86 fixed atom, 39, 86 condition to be free of, 39 nonatomic a.s., condition, 41 set-indexed family of r.v.s, 17 tail σ-algebra, 208 ——, general properties superposition, 63

570

——, identified completely random, see Completely random measure integral of nonnegative process, 23 as random sum of discs or lines, 23 linear process of, 74 shot-noise example, 74 multidimensional, 7 multivariate, 7 nonnull, 188 on R+ , history for F -adapted process, 357 internal (natural, minimal), 357 with χ2 density, 9, 42 ——, purely atomic, 7 characterization as point process, 275 via Campbell measure, 275 via construction, 49 extended MPP, 8 finitely many atoms, example, 277 Palm kernel, 277 Korolyuk equation for, 50 moment condition for, 66 on countable state space, 282 Campbell measure for, 282 ——, stationary, 178 Bartlett spectrum atom at origin, 205 nonzero iff process nonergodic, 205 Campbell measure characterizations, 285 kth order stationarity, 237 long-range dependence, 250, 254 Hurst index, 250 moment properties when absolutely continuous, 248 on Rd , zero–infinity dichotomy, 187 Palm measure, distribution, 288 reduced moment measures, 238 Random probability distribution, see Random distribution Random process of directed lines, 472 see also Line process Random Schwartz distributions, 53 Random time change (= Randomly rescaled point process) Poisson, 421 proofs using p.g.fl., 419 proofs via interval distributions, 426 stationary compound Poisson process, 422

Subject Index

see also Poisson embedding Random thinning, see Thinning Random translations, 166 conditions for Poisson limit, 169 iterated convolution, u.a.n. property, 167 Random variable mapping characterization of point process or random measure, 8 Random walk, point process boundedly finite, 24 Random walk renewal process as space–time process, 494 moment measures, 505 Random walk on circle S, 193 Rarefaction, see Thinning Rayleigh–Lévy dust or flight, 21 Reduced measure by diagonal factorization, 183 Campbell measure for MPP, 331 moment measures for stationary random measures, 238 properties, 239 scale factors in, 238 see also Modified Campbell measure Reference measure in generalized entropy, 441 in likelihood, 401 Refined Campbell theorem, 288 Regeneration points in MPP, 328 Regenerative measure stationary, 186 Regular conditional probabilities 380 on product space, 25 Regular infinite divisibility, 92 a.s. finite, 92 KLM measure totally finite, 92 Poisson cluster process representation, 93 Relative entropy, 440 Renewal process, 67 class invariant under rarefaction, 165 compensator and martingale for, 366 convergence to equilibrium, 227 entropy, gamma lifetimes, 454 not a Markov point process, 127 on circle S, 193 phase type (PH-distributions), 111 Poisson limit under rarefaction, 165 recurrence relation for p.g.fl., 65

Subject Index stationary, 186 Palm–Khinchin equations, 302 reduced moment, cumulant measures, 247 with density complete conditional intensity, 395 Renewal theorem, 83 Blackwell, generalized, 331 for random walk on S, 193 Rényi dimensions, 341 correlation integrals, 341 quasi factorial moment estimator, 343 consistent estimators, 343 discrete entropy approximation, 440 multifractal, 341 consistent estimator for, 343 unifractal, 341 Rényi’s dimensional entropy, 455 Rényi–Monch theorem, 35 Poisson limit from one-dimensional distributions, 162 use of variation norm, 162 Repairable system model, 235 Residual or ‘innovations’ process, 513 exvisibility, 513 Right-continuity of histories, 357 F(+) , 373 counterexample, 373 Ripley’s K-function, 297, 464 radial component of moment factorization of process in R2 , 467 Scale-invariance, 255 Schwartz distributions, random, 53 Self-similarity, 83, 255 in point configurations, 249 of random measure, 83 stable random measures, 83 Semi-Markov process, 96 compensator components, 381 conditional intensity and mark distribution, 381 Markov process simplification, 382 equivalent Markov renewal process, 97 likelihood, 116 point process properties, 98 see also Markov renewal process Set function, subadditive under refinement, 50

571 x

Set-dependent adjacency (∼), 128 Shift-coupling, 229 in limit theorems, 269 with stationary process implies strong (C, 1)-asymptotic stationarity, 230 see also Coupling Shift transformations, 177 shift operator, 178 Shot-noise process, 163, 170 as linear process, 74 from G-random measure, 84, 87 Signed random measure, 19 Wiener motion counterexample, 19 Simple birth process maximum likelihood estimator asymptotically mixed normal, 416 Simple counting measure, 3 Simple point processes distribution determined by avoidance function, 35 sample path property, 43 second moment measure sufficient condition, 66 sequence of, sufficient conditions for convergence, 140 Singular infinite divisibility, 92 Skorohod metric, 145 Slivnyak–Mecke Poisson process characterization theorem, 281 Smoothing problem, 400 Space of counting measures NX# , 3 as c.s.m.s., 6 closed subspace of M# X, 6 Space of measures, as c.s.m.s., 6 Space–time process ——, general 485 estimation in, 490 evolving spatial field, 486 residual analysis diagnostics, 502 second moment estimation, 496 Bartlett spectrum, 497 boundary (edge) effects, 497 conditional intensity, 498 stochastic declustering, 491 stochastic reconstruction, 491 variety of processes, 486 with associated mark, 486

572

——, models 505 cluster, Bartlett–Lewis, 505 cluster, Neyman–Scott, 505 ETAS model, 499 M/G/∞ queue, 488 Poisson process, 348 spatial birth-and-death, 488 ——, stationary family of Palm distributions, 494 first moments, 487 Poisson, 487 Poisson cluster process, 495 terminology, 487 reduced second moments, 491 alternative representations, 492 Fourier transforms, 493 simplified when homogeneous, 492 stationary-time, homogeneous-space, 487 Spatial birth-and-death process, 126 Spatial point pattern, 458 models, 459 statistics, 459 diagnostic tests, 514 residual variances, 516, 518 reduced moment measures, 464 Spherical contact distribution, 459 Stable convergence, 419 identifying Poisson/Cox limit of convergent compensators, 384 dependent-thinning example, 387 point process martingale, 412 randomly scaled to normal limit, 415 Stable random measures, 83 self-similarity, 83 Stationarity, 178 on half-line extension to whole line, 223, 235 preserved by random thinning, 181 preserved by random translation, 181 strict v. weak, 178 see also Asymptotic stationarity Stationary cluster process asymptotic independence determined by cluster centre process, 213 Stationary cubic lattice process, 192 Stationary gamma random measure, 162, 11, 30

Subject Index

Stationary independent increment process, 81 Stationary isotropic planar point process, 469 Stationary MPP, 179 Stationary point process on Rd , 178 a.s. zero–infinity dichotomy, 187 MPP extension, 205 functional equivalence, 180 on circle S, 188 extension to MPP, 190 Palm measure inversion formulae, 291, 300 Stationary random measure ergodicity implies nontrivial mixture impossible, 217 on Rd , 178 a.s. zero–infinity dichotomy, 187 functional equivalence, 180 Stein–Chen methods, 163 Stochastic continuity sets, 134 form an algebra, 135 Stochastically continuous, 11 Stochastic declustering, space–time model, 491 stochastic reconstruction, 491 Stochastic integral w.r.t. Poisson process, 428 stochastic d.e. driven by Poisson process, 428 thinning construction, 427 Stopping time sequence properties of limits, 373 Strauss process, 123 characterization, 123 extended, 130 Stress-release model, 239 limit properties of estimators, 418 MPP variant, 235 Subadditive set function, 43, 50 under refinement, 50 Sums of independent random measures, 62 Superposition of point processes, 146 convergence, 65 p.g.fl. condition for convergence, 154 Superposition of random measures, 63 conditions for weak limit, 153 convergence, 65 Sup metric on space of d.f.s, 145

Subject Index

Support counting measure, 4 Tail σ-algebra, 208 tail event, 208 trivial, implies mixing, 209 Takacs–Fiksel estimation procedure, 514 Telegraph signal process, 101 Thinning of point process, 155 condition for Poisson limit, 155 Cox process as limit, 157, 387 dependent-thinning, limit via convergent compensators, 387 random, nonstationary, with stationary output, 192 Tiling of c.s.m.s., 15, 311 ‘infinite’ dissecting system, 311 Triangular array, 146 conditions for Poisson limit, 150 independent array, 146 u.a.n. condition, 146 U.a.n., see Uniform asymptotic negligibility Unaligned point set, 312 Uniform asymptotic negligibility (u.a.n.), 146 sufficient for infinite divisibility, 149 Uniform integrability rˆ ole in convergence, 141 Uniform random measure, 9 Uniform tightness of probability measures, 136 Updating formulae, likelihoods 407 estimation/detection separation, 407 simplified in Markovian case, 408 Vacuity function, 2 see also Avoidance function Variation norm, 144 Poisson limit property, 162

573

Version or copy of process same fidi distributions, 11 Voronoi polygon, 305 about point at origin of N0 , 306 Watanabe’s theorem (Poisson characterization), 365 analogue of proof, 390 basic form, 420 extension to Cox process, 419 Weak convergence, 390 totally finite measures, 2 weak-hash (w# ) convergence, 2 equivalence and non-equivalence, 3 example not weakly convergent, 22 ——, of random measures, 132 convergence of fidi distributions, 134 equivalent convergence modes, 135, 137 Laplace functional conditions, 138 p.g.fl. conditions, 138 Weakly asymptotically uniformly distributed measures, 175 Weighted averages, ergodic theorem, 201 Wiener’s homogeneous chaos, 19 Wold process, 92 complete conditional intensity, 399 convergence to equilibrium, 227 recurrence relation for p.g.fl., 65 stationary Palm–Khinchin equations, 302 Workload-dependent queueing process, 235 Zero–infinity dichotomy, a.s. stationary random measure/point process on Rd , 187 for MPP on Rd × K, 205

An Introduction to the Theory of Point Processes: General theory and structure

An introduction to the theory of point processes. General Theory and Structure

An Introduction to the Theory of Point Processes, Volume II: General Theory and Structure, 2nd Edition