Stein's Method and Applications

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I D RPPLICHTIONS

LECTURE NOTES SERIES Institute for Mathematical Sciences, National University of Singapore Series Editors: Louis H. Y. Chen and Denny Leung Institute for Mathematical Sciences National University of Singapore

Published Vol. 1

Coding Theory and Cryptology edited by Harald Niederreiter

Vol. 2

Representations of Real and p-Adic Groups edited by Eng-Chye Tan & Chen-Bo Zhu

Vol. 3

Selected Topics in Post-Genome Knowledge Discovery edited by Limsoon Wong & Louxin Zhang

Vol. 4

An Introduction to Stein's Method edited by A. D. BarbourA Louis H. Y. Chen

Vol. 5

Stein's Method and Applications edited by A. D. Barbour& Louis H. Y. Chen

Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore

£ £ A ^ ^ 9

HND RPPLICHTIONS A D Barbour University of Zurich, Switzerland

Louis HY Chen

National University of Singapore, Singapore

SINGAPORE UNIVERSITY PRESS N E W

J E R S E Y

• L O N D O N

^p

World • B E I J I N G • Scientific S H A N G H A I • H O N G

NATIONAL UNIVERSITY OF SINGAPORE

• S I N G A P O R E

K O N G

• TAIPEI

•

C H E N N A I

Published by Singapore University Press Yusof Ishak House, National University of Singapore 31 Lower Kent Ridge Road, Singapore 119078 and World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

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STEIN'S METHOD AND APPLICATIONS Copyright © 2005 by Singapore University Press and World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN 981-256-281-8

Printed in Singapore by World Scientific Printers (S) Pte Ltd

CONTENTS

Foreword

vii

Preface

ix

Charles Stein (photographs)

xiii

Memories of the Stein Workshop (photographs)

xv

Zero biasing in one and higher dimensions, and applications Larry Goldstein and Gesine Reinert

1

Poisson limit theorems for the appearances of attributes Ourania Chryssaphinou, Stavros Papastavridis and Eutichia Vaggelatou Normal approximation in geometric probability Mathew D. Penrose and J. E. Yukich

19

37

Stein's method, Edgeworth's expansions and a formula of Barbour 59

Vladimir Rotar

Stein's method for compound Poisson approximation via immigrationdeath processes Aihua Xia 85 The central limit theorem for the independence number for minimal spanning trees in the unit square Sungchul Lee and Zhonggen Su 103 V

vi

Contents

Stein's method, Markov renewal point processes, and strong memoryless times Torkel Erhardsson 119 Multivariate Poisson-binomial approximation using Stein's method A. D. Barbour

131

An explicit Berry-Esseen bound for Student's i-statistic via Stein's method Qi-Man Shao 143 An application of Stein's method to maxima in hypercubes Z. D. Bai, L. Devroye and T. H. Tsai

157

Exact expectations of minimal spanning trees for graphs with random edge weights James Allen Fill and J. Michael Steele 169 Limit theorems for spectra of random matrices with martingale structure F. Gotze and A. N. Tikhomirov 181 Characterization of Brownian motion on manifolds through integration by parts Elton P. Hsu 195 On the asymptotic distribution of some randomized quadrature rules Wei-Liem Loh

209

The permutation distribution of matrix correlation statistics A. D. Barbour and Louis H. Y. Chen

223

Applications of Stein's method in the analysis of random binary search trees Luc Devroye 247

FOREWORD

The Institute for Mathematical Sciences at the National University of Singapore was established on 1 July 2000 with funding from the Ministry of Education and the University. Its mission is to provide an international center of excellence in mathematical research and, in particular, to promote within Singapore and the region active research in the mathematical sciences and their applications. It seeks to serve as a focal point for scientists of diverse backgrounds to interact and collaborate in research through tutorials, workshops, seminars and informal discussions. The Institute organizes thematic programs of duration ranging from one to six months. The theme or themes of each program will be in accordance with the developing trends of the mathematical sciences and the needs and interests of the local scientific community. Generally, for each program there will be tutorial lectures on background material followed by workshops at the research level. As the tutorial lectures form a core component of a program, the lecture notes are usually made available to the participants for their immediate benefit during the period of the tutorial. The main objective of the Institute's Lecture Notes Series is to bring these lectures to a wider audience. Occasionally, the Series may also include the proceedings of workshops and expository lectures organized by the Institute. The World Scientific Publishing Company and the Singapore University Press have kindly agreed to publish jointly the Lecture Notes Series. This volume on "Stein's method and applications" is the 5th of this Series. We hope that through regular publication of lecture notes the Institute will achieve, in part, its objective of promoting research in the mathematical sciences and their applications. December 2004

Louis H. Y. Chen Denny Leung Series Editors vii

PREFACE

Probability theory in the first half of the twentieth century was substantially dominated by the formulation and proof of the classical limit theorems — laws of large numbers, central limit theorem, law of the iterated logarithm — for sums of independent random variables. The central limit theorem in particular has found regular application in statistics, and forms the basis of the distribution theory of many test statistics. However, the classical approach to the CLT relied heavily on Fourier methods, which are not naturally suited to providing estimates of the accuracy of limits such as the CLT as approximations in pre-limiting circumstances, and it was only in 1940 that Berry and Esseen, by means of the smoothing inequality, first obtained the correct rate of approximation in the form of an explicit, universal bound. Curiously enough, the comparable theorem for the conceptually simpler Poisson law of small numbers was not proved until 26 years later, by Le Cam. These theorems all concerned sums of independent random variables. However, dependence is the rule rather than the exception in applications, and had been increasingly studied since 1950. Without independence, Fourier methods are much more difficult to apply, and bounds for the accuracy of approximations become correspondingly more difficult to find; even for such frequently occurring settings as sums of stationary, mixing random variables or the combinatorial CLT, establishing good rates seemed to be intractable. It was into this situation that Charles Stein introduced his startling technique for normal approximation. Now known simply as Stein's method, the technique relies on an indirect approach, involving a differential operator and a cleverly chosen exchangeable pair of random variables, which are combined in almost magical fashion to deliver explicit estimates of approximation error, with or without independence. This latter feature, in particular, has led to the wide range of application of the method.

ix

x

Preface

Stein originally developed his method to provide a new proof of the combinatorial CLT for use in a lecture course, and its first published application, in the Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability in 1972, was to give bounds for the accuracy of the CLT for sums of stationary, mixing random variables. Since then, the scope of his discovery has expanded rapidly. Poisson approximation was studied in 1975; the correct Lyapounov error bound in the combinatorial CLT was obtained in 1984; the method was extended to the approximation of the distributions of whole random processes in 1988; its importance in the theoretical underpinning of molecular sequence comparison algorithms was recognized in 1989; rates of convergence in the multivariate CLT were derived in 1991; good general bounds in the multivariate CLT, when dependence is expressed in terms of neighborhoods of possibly very general structure, were given in 1996; and Stein's idea of arguing by way of a concentration inequality was developed in 2001 to a point where it can be put to very effective use. Despite the progress made over the last 30 years, the reasons for the effectiveness of Stein's method still remain something of a mystery. There are still many open problems, even at a rather basic level. Controlling the behavior of the solutions of the Stein equation, fundamental to the success of the method, is at present a difficult task, if the probabilistic approach cannot be used. The field of multivariate discrete distributions is almost untouched. There is a relative of the concentration inequality approach, involving the comparison of a distribution with its translations, which promises much, but is at present in its early stages. Point process approximation, other than in the Poisson context, is largely unexplored: the list goes on. Due to its broad range of application, Stein's method has become particularly important, not only in the future development of probability theory, but also in a wide range of other fields, some theoretical, some extremely practical. These include spatial statistics, computer science, the theory of random graphs, computational molecular biology, interacting particle systems, the bootstrap, the mathematical theory of epidemics, algebraic analogues of probabilistic number theory, insurance and financial mathematics, population ecology and the combinatorics of logarithmic structures. Many, in their turn, because of their particular structure, have led to the development of variants of Stein's original approach, with their own theoretical importance, one such being the coupling method.

xi

Preface

This volume contains the proceedings of a workshop which took place during the meeting STEIN'S METHOD AND APPLICATIONS: A PROGRAM IN HONOR OF CHARLES STEIN, held in Singapore at the Institute for Mathematical Sciences, from 28 July to 31 August 2003. The 16 papers provide a broad cross-section of the research work currently being undertaken in this area. The accompanying volume, consisting of the four tutorial lectures on Stein's method given in the course of the same meeting, provides a detailed introduction to the theory and application of Stein's method, in a form suitable for graduate students who want to acquaint themselves with the method. We hope that the content of the two volumes will encourage the reader to become as enthusiastic about Stein's method as we are. December 2004

A. D. Barbour Louis H. Y. Chen Program Co-Chairs

CHARLES STEIN

Around 1980 (courtesy Susan Holmes)

xiv

Charles Stein

At a department picnic in 1981 (courtesy Susan Holmes)

In 1986 (courtesy Susan Holmes)

MEMORIES OF THE STEIN WORKSHOP

To be with Master (seventh from XV left in first row): Charles Stein

xvi

Memories of the Stein Workshop

Steins’s Method revisited

In tune with new ideas: Charles and Margaret Stein

A break in a brain storm: (From left) Gesine Reinert and Susan Holmes


xvii

The proof is in the eating!: (From left) Louis Chen, Friedrich Gotze (foreground), Mike Steele (background), Persi Diaconis, Vladimir Rotar, Andrew Barbour, Luc Devroye, Sungchul Lee, Gesine Reinert (back), and Peter Eichelsbacher

Qi-Man Shao and Charles Stein

Wei-Liem Loh and Charles Stein

xviii


A magical moment with Stein at IMS: (From left) Louis Chen, Persi Diaconis, Charles Stein and Zhidong Bai

Celebrating a rare event with Stein: (From left) Charles Stein, Vladimir Rotar, Luc Devroye and Andrew Barbour


In Choi's company: (From left) Adrian Rollin, Kwok-Pui Choi, Narumol Chaidee and Martin Raic

Feasting over Stein's Method: (From left) Pat Steele, Mike Steele, Joseph Yukich and Mathew Penrose

xix

xx


Captivated by the magic of mathematics

Third generation of Stein's Method: (Prom Left) Gesine Reinert, Margaret Stein, Torkel Erhardsson, Charles Stein and Aihua Xia Captions by Y. K. Leong

Zero biasing in one and higher dimensions, and applications

Larry Goldstein and Gesine Reinert Department of Mathematics, University of Southern California Los Angeles CA 90089-2532, USA E-mail: [email protected] and Department of Statistics, University of Oxford 1 South Parks Road, Oxford 0X1 3TG, UK E-mail: [email protected] Given any mean zero, finite variance a2 random variable W, there exists a unique distribution on a variable W* such that EWf(W) = a Ef'iW*) for all absolutely continuous functions / for which these expectations exist. This distributional 'zero bias' transformation of W to W*, of which the normal is the unique fixed point, was introduced in Goldstein & Reinert (1997) to obtain bounds in normal approximations. After providing some background on the zero bias transformation in one dimension, we extend its definition to higher dimension and illustrate with an application to the normal approximation of sums of vectors obtained by simple random sampling.

1. Introduction Stein (1972), in his seminal paper, proved the following fundamental characterization of the univariate normal distribution: Z ~ A/"(0, a2) if and only if EZf(Z)=a2Ef'{Z)

(1.1)

for all absolutely continuous / with E\f'(Z)\ < oo. Normal approximations through the use of identity (1.1), first provided by Stein (1972), have since been obtained by other authors using variations in abundance (see e.g. Stein (1986), Chen & Shao (2004), Raic (2003) and references therein). In Goldstein & Reinert (1997), the authors introduced and studied the zero bias transformation in one dimension; further development is continued in 1

2

Larry Goldstein and Gesine Reinert

Goldstein (2004, 2005). Here, we extend the study of zero biasing to R p , and illustrate with an application. If W is a mean zero variance cr2 variable then generally EWf(W) does not equal a2Ef'(W), the normal being the unique distribution satisfying the Stein characterization (1.1). Asking that identity (1.1) be satisfied by some transformation of the W distribution leads to the following definition. Definition 1.1: For a mean zero, finite variance a2 random variable W, we say that W* has the W-zeio biased distribution if EWf(W) = a2Ef{W*),

(1.2)

for all absolutely continuous functions / for which the expectations above exist. In Goldstein & Reinert (1997), the distribution W* was shown to exist for all mean zero, finite variance W. In particular, W* is always absolutely continuous, and one can verify that p*(w) = a~2E[Wl(W

> w)}

(1.3)

is a density function for a distribution which satisfies (1.2). Though Definition 1.1 is stated in terms of random variables, it actually defines a transformation on a class of distributions, and we will use the language interchangeably. The normal distribution being the unique fixed point of the zero bias transformation, one can guess that when the transformation maps a variable W to a W* which is close by, then W itself is close to being a fixed point, and must therefore be close to normal. Equation (1.5) indicates one way in which this intuition can be quantified. Let W be a mean zero, variance 1 random variable, Z a standard normal and Nh = Eh(Z) for a given test function h. Based on the characterization (1.1), Stein's method obtains information on Eh(W) — Nh, the distance from W to the normal on a given test function h, by solving the differential equation f'(w) - wf(w) = h{w) - Nh

(1.4)

for / and considering the expectation of the left hand side. For instance, when W and W* are jointly defined, from (1.4) and Definition 1.1 we obtain

\Eh(W) - Nh\ = \E [f'{W) - Wf{W)\ | = \E [/'(WO - f{W*)\ |.

Zero biasing in one and higher dimensions

3

By Stein (1986), for h absolutely continuous, we have ||/"|| < 2\\h'\\ and hence, with || • || the supremum norm, \Eh{W)-Nh\f(X')-X>'f(X>)) = Ê (X^(X') - XY'f(X'))

by (4.3).

Now, taking expectation over Vji and noting that fji = fij, we have

j=l

1=1

= a2J2 EfviX") + ^J2 E { ^ « ( X " ) + £/*(X")}

+ SEE £; (^(x')-Ay7/(x')) r> 2

P

= -^-E £ (^( x ')-^7i(x')) n

P

E

(

P

X

P

X

y

+ ^ Y, { E^/'( ') - 'M*) - AE '//(X') + AF'/,(X') U=l

3=1

= 2^TAilE

1=1

(XftW-XY'MX'))

3=1 Okr

P

+ Ê

E

3=1

( P

P

}

£*;/"

and any three times differentiate test function h, lEhCL-WW) - Nh\

< ^(minpSjj)-^\\D3h\\ ±±E fajX^ + ^ | 3 ) ,

16


and when Xjti, i = 1,..., n are identically distributed with variance 1 for allj = l,...,p,

lEhp-WW) - Nh\ < ^||£> 3 /i|| J2E\XjA?In our next example we consider vectors having non-independent components. Example 4.5: Simple Random Sampling. Let X' g R p be a vector whose values are obtained by taking a simple random sample of size p from a population having characteristics A, \A\ = N > p, with J2aeA a ~ ®- Taking one additional observation X" we form an enlarged vector that satisfies (4.1). In the notation of Theorem 4.1, (4.8)

and (4.2) is satisfied with K the value (4.8), and

WI X ') = W±-PY'^ so (4.3) is satisfied with A = —l/(N — p). Hence the hypotheses of Theorem 4.1 hold, and the construction so given can be used to produce a collection with the X' zero biased distribution. We now arrive at Theorem 4.6: Let (Xj)i=i...in be an independent collection of mean zero random vectors in R p obtained by simple random sampling as in Example 3.4. Suppose \a\ < M for all a e A%, i'• = 1,..., n, and let n

W = ^ X,

and sjt = Cov(Wj ,W{).

»=i

Then for a three times differentiable test function h,

\EhCs-^2w)-Nh\ < ^ M / I I S - 1 / 2 ! ! 3 ! ! ^ ! ! E l^'l3,1=1

Proof: As shown in Example 3.4, the collection of vectors have constant sign covariance, and hence Theorem 3.5 applies. Using the construction given Theorem 4.1 and the bound from Remark 4.2 we have ||X7j, - X * | | < 2 M ,

Zero biasing in one and higher dimensions

and the conclusion follows.

17

•

In typical situations, £ and Sji will have order n, resulting in a bound of the correct order, n~ 1//2 . Acknowledgement: The authors would like to thank the Institute for Mathematical Sciences for their support during the program 'Stein's method and applications: A program in honor of Charles Stein', where this work was partially completed. GR was also supported in part by EPSRC grant no. GR/R52183/01. We also wish to thank a referee for very helpful remarks. References 1. A. D. BARBOUR (1990) Stein's method for diffusion approximations. Probab. Theory Rel. Fields 84 297-322. 2. T. CACOULLOS (1982) On upper and lower bounds for the variance of a function of a random variable. Ann. Appl. Probab. 10, 799-809. 3. T. CACOULLOS & V. PAPATHANASIOU (1992) Lower variance bounds and a new proof of the central limit theorem. J. Multiv. Analysis 43, 173-184. 4. T. CACOULLOS, V. PAPATHANASIOU & S. UTEV (1994) Variational inequalities with examples and an application to the central limit theorem. Ann. Probab. 22, 1607-1618. 5. L. H. Y. CHEN & Q.-M. SHAO (2004) Stein's method and normal approx-

6. 7. 8. 9. 10. 11. 12.

imation. In: An introduction to Stein's method, Institute for Mathematical Sciences Lecture Notes Series No. 4, ch. 1. World Scientific Press, Singapore. F. GOTZE (1991) On the rate of convergence in the multivariate CLT. Ann. Probab. 19, 724-739. L. GOLDSTEIN (2004) Normal approximation for hierarchical structures. Ann. Appl. Probab. 14, 1950-1969. L. GOLDSTEIN (2005) Berry Esseen bounds for combinatorial central limit theorems and pattern occurrences, using zero and size biasing. J. Appl. Probab. 42 (to appear). L. GOLDSTEIN & G. REINERT (1997) Stein's Method and the Zero Bias Transformation with Application to Simple Random Sampling. Ann. Appl. Probab. 7, 935-952. L. GOLDSTEIN & G. REINERT (2004) Distributional transformations, orthogonal polynomials, and Stein characterizations. J. Theoret. Probab. (to appear). L. GOLDSTEIN & Y. RINOTT (1996) On multivariate normal approximations by Stein's method and size bias couplings. J. Appl. Probab. 33, 1-17. N. PAPADATOS & V. PAPATHANASIOU (2001) Unified variance bounds and a Stein-type identity. In Probability and Statistical Models with Applications, eds. Ch. A. Charalambides, M. V. Koutras & N. Balakrishnan, pp. 87-100. Chapman and Hall/CRC, New York.

18


13. M. RAIC (2003) Normal approximations by Stein's method. In Proceed-

ings of the Seventh Young Statisticians Meeting, ed. A. Mvrar, pp. 71-97. Metodoloski zveski, 21, Ljubljana, FDV. 14. C. STEIN (1972) A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. Proc. Sixth Berkeley Symp. Math. Statist. Prob. 2 583-602. Univ. California Press, Berkeley. 15. C. STEIN (1986) Approximate Computation of Expectations. IMS, Hayward, CA.

Poisson limit theorems for the appearances of attributes

Ourania Chryssaphinou, Stavros Papastavridis and Eutichia Vaggelatou Department of Mathematics, University of Athens Panepistimioupolis, GR-157 84, Athens, Greece E-mail: [email protected], [email protected], [email protected] We examine the number of appearances of certain types of events in a sequence of independent (but not necessarily identical) trials. We introduce a concept of Renewal which, in this context, in certain ways generalizes Feller's Theory. Under quite general conditions, Poisson limit theorems are obtained, which extend known results. Key tools in that are Stein-Chen method and probabilistic arguments.

1. Introduction In the past, many authors have studied various questions around the following general situations: we have a sequence of independent trials (usually identical), each trial producing a letter from certain alphabets. We have a set of words from these alphabets, under consideration, and we examine the number of appearances of these words etc. In this paper, we consider a general setup for the case of independent and identical experiments. We study attributes in the sense given by W. Feller (1968, p.308) and under quite general conditions we present limit theorems to the Poisson distribution. This work is organized as follows: In Section 2 we give the basic definitions and notations with examples concerning attributes and the related random variables under consideration. In Section 3 we examine the case of renewal appearances of one attribute and we present a Poisson limit theorem (Theorem 3.1) for the number of such appearances. Finally, in Section 4, applications of the above results are given. 19

20

O. Chryssaphinou, S. Papastavridis and E. Vaggelatou

2. Preliminaries and notation Let us consider the triangular array of probability spaces

and (f2n,fn,Pn),

n>\

be the product probability space i.e., fin = ^n,i x . . . x nn,n, Fn = Fn,l ® • • • ® ?n,n *n

-*n,l X . . . X / n , n «

~

Let also ln be a positive integer such that ln < n . We consider the event En,i i

En,iQ

(f2nJ x . . . x

\J

flnii)

(2.1)

i=max(l, i-i n + l)

and we define Sn = (EnA,...,En,n),

(2.2)

which we call the £n attribute. As W.Feller wrote, a£n itself is a label rather than an event". Attributes may be runs of symbols (events, patterns, words etc.) or sequences of numbers. In what follows we consider independent and identical trials. The model studied by Guibas & Odlyzko, ("String Overlaps, Pattern Matching and Nontransitive Games", (1981)), is the case where we have totally i.i.d. trials, i.e. i? nii = Q and Pn ^21 n + ll, 2(n+6>

^ 2 / n + ll, 2/ n +7-

23


For example we have —D3

•pA

£/

2in+n, in+2 ~

i i

x{{S}x{S}x{S}...{S}x{S}x{S})

x nl-+9 u n2 x{{F}x{F}x{F}...{F}x{F}x{F}) In

and rpA _Oin+7 2ln + ll, 2ln+7 —il

a

x {{F} x {F} x {F} ... {F} x {F} x {F})

x n4 u oln+8 x {{5} x {5} x {S}... {S} x {S} x {S}} x /?4 i n -i

In our example we have that the sequence belongs to ^2ln + U,ln+2

a n

^2ln + ll,ln+2

an

since R

R

R

R

{F 'SFS 'SFFS FF SFS FF 'SFS FS} and to -^20,4 >

^20,8'

since Q

Q

{FSFSSFFSFFSFSFFSFSFS}. 3. One attribute We define the ^-measurable function Rn : Qn —* R, which counts the number of appearances of the £n as Renewal, i.e. if {w\,...,ujn) € fin, then the value Rn(u>i,..., u)n) is the cardinality of the set {i:li,..., ujn) is the cardinality of the set {i:ln) is the cardinality of the set {i:l +00. The term b\. It is n

6

1= E »6/»

E ^(Xn,i)£7(Xnii) = £ j6B n ,i

J2

P(En,i)P(Enj)

i=l j=max(l, i-2I n +2)

< 4 E P < ) SUp I i=1

min(n, i+2ln-2)

P E

J2

( nM'

l> I (in + 1)!' "' Then by applying Theorem 3.3, we come to a similar conclusion. Due to symmetry, the previous results also hold true for the case of decreasing sequences of the above type. We note that Pittel (1981) proved a similar result for the number of decreasing sequences with length larger or equal to ln using the method of moments. Theorems 3.1 and 3.3 give us the rate of convergence, as well as the possibility to study some more relevant aspects as the next problem. -

4.2.2. Increasing or decreasing sequences of real numbers The basic model is the same and let us consider the attribute £n which corresponds to both an increasing and decreasing sequence of length ln. We are interested in the number of renewal appearances of either an increasing or a decreasing sequence of length ln. For that purpose we define En,i

: = { ( « i , a 2 , •••,ain) /- Tain t IK

• a\ < a2 < • •• < ain o r a\ > o 2 > ... >

ain)

and therefore,

o r uJi-i,,+i

> Uiîn+2

> ...

>

UJi}.

Obviously P(E^)

= P{Xi_ln+1

...>Xi)

=

^-

for i > ln.

The calculation of E(Rn) is not possible, but we can estimate the mean E(Qn). The event E®'i occurs whenever an increasing or decreasing sequence of length /„ takes place at the i-th trial and no other suchlike sequence has occurred at the ln - 1 former trials. We have P(EQ

)_(

°.

i < l

n

34

O. Chryssaphinou, S. Papastavridis and E. Vaggelatou

Furthermore, for /„ + 1 Xi-ln+l)n(Xi-ln+1 < Xi.ln+1)

...>Xi)]

2ln

~(Zn + l)!' Finally for i > 2ln — 1 we get that = p[(x i _ 2 I n + 2 > • • • > Xi-,JC n (Xi_«n > x,_ / n + 1 ) n (X,_ (n+1 < ... < Xi)] + P[(Xt-2in+2 < < Xi-in)c n (Xi_jn < Xi_, B+ i) n ( x ^ ^ + i > > Xi)] = 2P{(Xi-2ln+2

>•••

> . X i _ « J c n ( X i _ , n > X t _ , n + 1 ) n (Xi-ln+1

{^TTji-(a,-i)[k-w}-

Supposing that 1. limn ln = oo 2. limn E(Q n ) = A it can be verified that conditions A, B are satisfied, thus by a straightforward application of Theorem 3.3 we deduce that the random variable Rn converges in distribution to Po(A) with rate of order

The above result has been proved by Wolfowitz (1944) using the method of moments. Later, David & Barton (1962) proved an analogous result concerning sequences with length greater or equal to ln. In both cases the rate of convergence was not obtained. Acknowledgement: This work was fully supported by the Research Account of the University of Athens, Greece.

X^}


35

References 1. D. J. ALDOUS (1989) Probability approximations via the Poisson clumping heuristic. Springer, New York. 2. R. ARRATIA , L. GOLDSTEIN & L. GORDON (1989) Two moments suffice for

Poisson Approximations: The Chen-Stein method. Ann. Probab. 17, 9-25. 3. R. ARRATIA, L. GOLDSTEIN & L. GORDON (1990) Poisson Approximation

and the Chen-Stein method. Statist. Science 5, 403-434. 4. A. D. BARBOUR, L. HOLST & S. JANSON (1992) Poisson Approximation.

Oxford University Press. 5. G. BLOM & D. THORBURN (1982) How many random digits are required until given sequences are obtained? J. Appl. Probab. 19, 518-531. 6. L. H. Y. CHEN (1975) Poisson approximation for dependent trials. Ann. Probab. 3, 534-545. 7. O. CHRYSSAPHINOU & S. PAPASTAVRIDIS (1988) A limit theorem for the number of nonoverlapping occurrences of a pattern in a sequence of independent trials. J. Appl. Probab. 25, 428-431. 8. O. CHRYSSAPHINOU, S. PAPASTAVRIDIS & E. VAGGELATOU (1999) On the number of appearances of a word in a sequence of i.i.d. trials. Meth. Comput. Appl. Probab. 1:3, 329-348. 9. O. CHRYSSAPHINOU, S. PAPASTAVRIDIS & E. VAGGELATOU (2001) On the number of nonoverlapping appearances of several words in a Markov chain. Combin. Probab. Comp. 10, 293-308. 10. F. N. DAVID & D. E. BARTON (1962) Combinatorial Chance. Hafner, New York. 11. W. FELLER (1968) An Introduction to Probability Theory and its Applications. New York, Wiley, 3nd edition. 12. L. GuiBAS & A. ODLYZKO (1981) String Overlaps, Pattern Matching, and Nontransitive Games. J. Comb. Theory Ser. A 30, 183-208. 13. B. G. PlTTEL (1981) Limiting behavior of a process of runs. Ann. Probab. 9, 119-129. 14. J. WOLFOWITZ (1944) Asymptotic distribution of runs up and down. Ann.

Math. Statist. 15, 163-172.

Normal approximation in geometric probability

Mathew D. Penrose and J. E. Yukich Department of Mathematical Sciences, University of Bath Bath BA2 7AY, United Kingdom E-mail: [email protected] and Department of Mathematics, Lehigh University Bethlehem PA 18015, USA E-mail: [email protected] We use Stein's method to obtain bounds on the rate of convergence for a class of statistics in geometric probability obtained as a sum of contributions from Poisson points which are exponentially stabilizing, i.e. locally determined in a certain sense. Examples include statistics such as total edge length and total number of edges of graphs in computational geometry and the total number of particles accepted in random sequential packing models. These rates also apply to the 1-dimensional marginals of the random measures associated with these statistics.

1. Introduction In the study of limit theorems for functionals on Poisson or binomial spatial point processes, the notion of stabilization has recently proved to be a useful unifying concept (Baryshnikov & Yukich (2005 [BY2]), Penrose & Yukich (2001 [PY1], 2003 [PY3]). Laws of large numbers and central limit theorems can be proved in the general setting of functionals satisfying an abstract 'stabilization' property whereby the insertion of a point into a Poisson process has only a local effect in some sense. These results can then be applied to deduce limit laws for a great variety of particular functionals, including those concerned with the minimal spanning tree, the nearest neighbor graph, Voronoi and Delaunay graphs, packing, and germ-grain models. Several different techniques are available for proving general central limit theorems for stabilizing functionals. These include a martingale ap37

38

Mathew D. Penrose and J. E. Yukich

proach [PY1] and a method of moments [BY2]. In the present work, we revisit a third technique for proving central limit theorems for stabilizing functionals on Poisson point processes, which was introduced by Avram & Bertsimas (1993 [AB]). This method is based on the normal approximation of sums of random variables which are 'mostly independent of one another' in a sense made precise via dependency graphs, which in turn is proved via Stein's method: see the monograph Stein (1986). It has the advantage of providing explicit error bounds and rates of convergence. We extend the work of Avram & Bertsimas in several directions. First, whereas in [AB] attention was restricted to certain particular functionals, here we derive a general result holding for arbitrary functionals satisfying a stabilization condition which can then be checked rather easily for many special cases. Second, we consider non-uniform point process intensities and do not require the functionals to be translation invariant. Third, we improve on the rates of convergence in [AB] by making use of the recent refinement by Chen & Shao (2004) of previous normal approximation results for sums of 'mostly independent' variables. Finally, we apply the methods not only to random variables obtained by summing some quantity over Poisson points, but to the associated random point measures, thereby recovering many of the results of [BY2] on convergence of these measures, and without requiring higher order moment calculations. We add to [BY2] by providing information about the rate of convergence, and relaxing the continuity conditions required in [BY2] for test functions and point process intensities. A brief comparison between the methods of deriving central limit theorems for functionals of spatial point processes is warranted. Only the dependency graph method used here, to date, has yielded error bounds and rates of convergence. On the other hand, our method requires bounds on the tail of the 'radius of stabilization' (i.e., on the range of the local effect of an inserted point). The martingale method, in contrast, requires only that this radius be almost surely finite, and for this reason is applicable to some examples such as those concerned with the minimal spanning tree, for which no tail bounds are known and which therefore lie beyond the scope of the present work. The moment method [BY2] and martingale method (Penrose (2004 [Pe2])), unlike the dependency graph method, provide information about the variance of the Gaussian limits. Whereas the moment method requires exponential tail bounds for the radius of stabilization, one of our central limit theorems (Theorem 2.5) requires only that this tail r(t) decay as a (large) negative power of t.


39

With regard to ease of use in applications, the dependency graph method and method of moments require checking tail bounds for the radius of stabilization, which is usually straightforward where possible at all. The method of moments requires a more complicated (though checkable) version of the bounded moments condition (2.5) below (see [BY2]). The dependency graph method requires some separate calculation of variances if one wishes to identify explicitly the variance of the limiting normal variable. The martingale method requires the checking of slightly more subtle versions of the stabilization conditions needed here [Pe2, PY1]. 2. General results Let d > 1 be an integer. For the sake of generality, we consider marked point processes in Rd. Let (M, FM^M) be a probability space (the mark space). Let £((x, s); X) be a measurable R-valued function defined for all pairs ((x, s),X), where X c Rd x M is finite and where (x, s) € X (so l e i 1 1 and s G M). When (x, s) G (Rd x M)\X, we abbreviate notation and write £((x, s); X) instead of £,((x, s); X U {(x, s)}). Given X C RdxM, a > 0 andy G Rd, we let aX := {(ax,i) : (x,t) e X} and y + X := {(y + x,t) : (x,t) G X}; in other words, translation and scalar multiplication on Md x Ai act only on the first component. For all A > 0 let £x({x, s); X) := i{{x, s); x + X1/d(-x

+ X)).

We say £ is translation invariant if £((y + x, s); y + X) = £((x, s); X) for all y G Rd, all (x,s) £ Md x M and all finite X C Md x M. When £ is translation invariant, the functional £A simplifies to £\((x,s);X) = Let K be a probability density function on Rd with compact support A C Rd. For all A > 0, let V\ denote a Poisson point process in Rd x M with intensity measure {\n{x)dx) x ^M{ds). We shall assume throughout that K is bounded with supremum denoted ||K||OOLet (A\, A > 1) be a family of Borel subsets of A. The simplest case, with A\ = A for all A, covers all examples considered here; we envisage possibly using the general case in future work. The following notion of exponential stabilization, adapted from [BY2], plays a central role in all that follows. For x G W1 and r > 0, let Br(x) denote the Euclidean ball centered at x of radius r. Let U denote a random element of M with distribution FM , independent of V\.

40


Definition 2.1: £ is exponentially stabilizing with respect to K and (AA)A>I if for all A > 1 and all x € Ax, there exists an a.s. finite random variable R := R(x, A) (a radius of stabilization for £ at x) such that for all finite X C (A \ Bx-\/dR{x)) x M, we have 6 ((x, U); [Vx n (B A -./d fl (i) x M)] U A") = £A ((x, £/); PA n (BA-i/-fl(x) x A*)),

(2.1)

and moreover the tail probability r(t) denned for t > 0 by r(i) := satisfies

sup

P[R{x, A) > t]

(2.2)

A>1, x€Ax

limsupfMogr^) < 0 .

(2.3)

t—>oo

For 7 > 0, we say that £ is polynomially stabilizing of order 7 if the above conditions hold with (2.3) replaced by the condition limsup£'l'T(£) < 00. t-KX>

Condition (2.1) may be cast in a more transparent form as follows. Each point of X is a pair (x,U), with x G Rd and (7 € .M, but for notational convenience we can view it as a point x in Rd carrying a mark U := Ux. Then we can view X as a point set in Rd with each point carrying a mark in M.. With this interpretation, (2.1) stipulates that for all finite (marked) X C A \ Bx-i/dji(x), we have & (x; (Vx n B A -i/- fl (x)) U #) = & (z; Vx n B A - V - H W ) •

(2-4)

Roughly speaking, .R := i?(x, A) is a radius of stabilization if the value of ^x^'jVx) is unaffected by changes to the points outside Bx-i/dR(x). Functionals of spatial point processes often satisfy exponential stabilization (2.1) (or (2.4)); here is an example. Suppose M = [0,1] and PM is the uniform distribution on [0,1]. Suppose that A is convex or polyhedral, and K is bounded away from zero on A. Suppose a measurable function (q(x),x 6 A) is specified, taking values in [0,1]. Adopting the conventions of the preceding paragraph, for a marked point set X C Md let us denote each point x G X as 'red' if Ux < q(x) and as 'green' if Ux > q(x). Let £(x; X) take the value 0 if the nearest neighbor of x in X has the same color as x, and take the value 1 otherwise. Note that £ is not translation invariant in this example, unless q(-) is constant. For x G A let R := R(x, A) denote the distance between A ^ x and its nearest neighbor in \l^dVx • Then

41


stabilization (2.4) holds because points lying outside Bx-i/dR(x) will not change the value of £\{x; V\), and it is easy to see that R has exponentially decaying tails. This example is relevant to the multivariate two-sample test described by Henze (1988). See Section 3 for further examples. Definition 2.2: £ has a moment of order p > 0 (with respect to K and (Ax)x>i) if (2.5) sup E[|6((z,t/);V]1,

x£Ax

For A > 0, define the random weighted point measure p,x on Rd by (x,s)eTxri{AxxM)

and the centered version Jrx := /x| — E [fJ-x]Let B(A) denote the set of bounded Borel-measurable functions on A. Given / 6 B(A), let (f,fy := / A fd,i{ and 2. Let q G (2,3] with q < p. Let f e B(^4) and put T\ := (/, f/x). Then there exists a finite constant C depending on d,^, K, p, q and f, such that for all X > 2, SU

P P f m ? w a ^ f l " $ W ^ C ( lQ g A)d9A(VarTA)-«/2. (2.6)

Separate arguments are required to establish the asymptotic behavior of the denominator (Var(TA))1/2 in (2.6). When A\ = A for all A, it is typically the case for polynomially stabilizing functionals satisfying moments conditions along the lines of (2.5) that there is a constant 0 such that lim A^Varf/, /*{} = a2(f, $, «).

(2.7)

A—»-oo

For further information about 0, showing that the case q = 3 of (2.6) yields a rate of convergence O((log A)3rfA~x/2) to the normal distribution. In other words, we will make frequent use of: Corollary 2.4: Suppose ||K||OO < oo. Suppose that £ is exponentially stabilizing and satisfies the moments condition (2.5) for some p > 3. Let f G B(A) and put T\ := (f,fJ,{). If (2.7) holds with 0, then there exists a finite constant C depending on d, £, n, p and f, such that for all A > 2,

tern

[{V&TTX)1'2

J

Our methods actually yield normal approximation and a central limit theorem when the exponential decay condition is replaced by a polynomial decay condition of sufficiently high order. We give a further result along these lines. Theorem 2.5: Suppose ||K||OO < oo. Suppose for some p > 3 that £ is polynomially stabilizing of order j with 7 > d(15Q + 6/p), and satisfies the moments condition (2.5). Let f e B(A) and put T\ := (f,^x). Suppose that (2.7) holds for some a2 > 0. Tiien (2.8) holds and if a2 := <x2(/,£, K) > 0 there exists a finite constant C depending on d, £, K, p and f, such that for all A > 2,

~ E J )1/2 sup P [ ^Var tem. L( -'A)

< t] - $(t) < CA< 150pd+6d -^/ 2 ^- 6d ). J

(2.9)

Remarks:

(1) Our results are stated for marked Poisson point processes, i.e., for Poisson processes in Rd x M where M is the mark space. These results are reduced to the corresponding results for unmarked Poisson point processes in Rd by taking Ai to have a single element (denoted m, say) and identifying Rd x M with Rd in the obvious way by identifying (x, m) with x for each x G M.d. In this case the notation (2.4) is particularly appropriate. Other treatments, such as [BY,Pe2,PYl] and Penrose & Yukich (2002 [PY2]), tend to concentrate on the unmarked case with commentary that the proofs carry through to the marked case; here we spell out the results and proofs in the more general marked case, which seems worthwhile since examples such as those in Section 3.3 use the results for marked point processes. Our examples in Sections 3.1, 3.2,


(2)

(3)

(4)

(5)

(6)

43

and 3.4 refer to unmarked point processes and in these examples we identify Rd x {m} with Rd as indicated above (so that V\ is viewed as a Poisson process in Rd). We are not sure if the logarithmic factors can be removed in Theorem 2.3 or Corollary 2.4. Avram & Bertsimas [AB] obtain a rate of 0((logA) 1+3/ ( 2d )A- 1/4 ), for the length of the fc-nearest neighbors (directed) graph, the Voronoi graph, and the Delaunay graph (see Sections 3.1 and 3.2). Our method for general stabilizing functionals is based on theirs, but uses a stronger general normal approximation result (Lemma 4.1 below). If (2.7) holds with }ldV\ when K is the uniform distribution on the unit cube. Since V^ is strictly positive (Theorem 6.1 of [PY1]), it follows that (2.7) holds with (T2(f,£k,n) > 0. We thus obtain via Corollary 2.4 the following rates in the CLT for the total edge length of Nk(\1/dV\) improving upon Avram & Bertsimas [AB] and Bickel & Breiman (1983). A similar CLT holds for the total edge length of the fc-nearest neighbors directed graph. Theorem 3.1: Suppose A is convex or polyhedral and n is bounded away from 0 on A. Let Nx := Nk{\l/dV\) denote the total edge length of the l d k-nearest neighbors graph on \ ^ V\. There exists a finite constant C depending on d,£k, and K such that

p

(1) SC(logA)3 1/2

d [wM-* l s

^ -

(31)

Similarly, letting £ (:r; X) be one or zero according to whether the distance between x and its nearest neighbor in X is less than s or not, we


45

can verify that £s is exponentially stabilizing and that the variance of SxeA'/dpA €s(x': ^d/P\) is bounded below by a positive multiple of A. We thus obtain rates of convergence of O((logA)3dA~1/2) in the CLT for the one-dimensional marginals of the empirical distribution function of k nearest neighbors distances on >}IAV\, improving upon those implicit on p. 88 of Penrose (2003 [Pel]). Using the results from section 6.2 of [PYl], we could likewise obtain the same rates of convergence in the CLT for the number of vertices of fixed degree in the k nearest neighbors graph. Finally in this section, we re-consider the non-translation-invariant example given in Section 2, where a point at x is colored red with probability q(x) and green with probability 1 - q(x), and £(x; X) takes the value 0 if the nearest neighbor of x in X has the same color as x, and takes the value 1 otherwise. We can use Corollary 2.4 to derive a central limit theorem, with O((log A)3dA~1^2) rate of convergence, for J2xev /( x )£( x ! ^M> where / is a bounded measurable test function. 3.2. Voronoi and sphere of influence graphs We will consider the Voronoi graph for d = 2 and the sphere of influence graph for all d. Given a locally finite set X C M2 and given x € X, the locus of points closer to x than to any other point in X is called the Voronoi cell centered at x. The graph consisting of all boundaries of Voronoi cells is called the Voronoi graph generated by X. The sum of the lengths of the finite edges of the Voronoi graph on X admits the representation J2xex £( x ' ^)i w n e r e £(#; X) denotes one half the sum of the lengths of the finite edges in the Voronoi cell at x. If K is bounded away from 0 and infinity and A is convex, then geometric arguments show that there is a random variable R with exponentially decaying tails such that for any x € V\, the value of £(x;V\) is unaffected by points outside Bx-i/dR(x) [BY2,PY1,PY3]. In other words, £ is exponentially stabilizing and satisfies the moments condition (2.5) for all p > 1. Also, the variance of the total edge length of these graphs on \lldV\ is bounded below by a multiple of A. We thus obtain O((log A)3dA~1/2) rates of convergence in the CLT for the total edge length functional of these graphs on \1/ldP\, thereby improving and generalizing the results of Avram & Bertsimas [AB]. Given a locally finite set X C M.d, the sphere of influence graph SIG(A') is a graph with vertex set X, constructed as follows: for each x € X let Bx be a ball around x with radius equal to mm.yex\{x){\y ~ XW- Then Bx

46


is called the sphere of influence of x. We put an edge between x and y iff the balls Bx and By overlap. The collection of such edges is the sphere of influence graph (SIG) on X. The total number of edges of the sphere of influence graph on X admits the representation Ylxex £(x> %), where £(x; X) denotes one half the degree of SIG at the vertex x. The number of vertices of fixed degree 8 admits a similar representation, with £(x; X) now equal to one (respectively, zero) if the degree at x is 8 (respectively, if degree at x is not 5). If K is bounded away from 0 and infinity and A is convex, then geometric arguments show that both choices of the functional £ stabilize (see sections 7.1 and 7.3 of [PY1]). Also, the variance of both the total number edges and the number of vertices of fixed degree in the SIG on X^dV\ is bounded below by a multiple of A (sections 7.1 and 7.3 of [PY1]). We thus obtain O((log A)3dA-1/2) rates of convergence in the CLT for the total number of edges and the number of vertices of fixed degree in the sphere of influence graph on V\. 3.3. Random sequential packing models The following prototypical random sequential packing model is of considerable scientific interest; see [PY2] for references to the vast literature. With N(X) standing for a Poisson random variable with parameter A, we let B\,i,B\t2, • ••, BA,AT(A) be a sequence of d-dimensional balls of volume A"1 whose centers are i.i.d. random [0, oo). Without loss of generality, assume that the balls are sequenced in the order determined by marks (time coordinates) in [0,1]. Let the first ball £A,I be packed, and recursively for i = 2,3,..., let the i-th ball B\i be packed iff B\ti does not overlap any ball in BA,I, •••,#A,i-i which has already been packed. If not packed, the i-th ball is discarded. Packing models of this type arise in diverse disciplines, including physical, chemical, and biological processes [PY2]. Central limit theorems for the number of accepted (i.e., packed) balls are established in [PY2,BY2], whereas laws of large numbers are given in [PY3]. Let M. = [0,1] with P x being the uniform distribution on the unit interval. For any finite point set X c Md x [0,1], assume the points (x,s) of X represent the locations'and arrival times. Assume balls of volume A"1 centered at the locations of X arrive sequentially in an order determined by the time coordinates, and assume as before that each ball is packed or discarded according to whether or not it overlaps a previously packed ball.


47

Let £((x, s); X) be either 1 or 0 depending on whether the ball centered at x at times s is packed or discarded. Consider the re-scaled packing functional £${x, s); X) = £{{\1/dx, s); \l'dX), where balls centered at points of \l'dX have volume one. The random measure N{$ i=l

is called the random sequential packing measure induced by balls with centers arising from n. The convergence of the finite dimensional distributions of the packing measures fj,^ is established in Baryshnikov & Yukich (2003 [BY1]),[BY2]. £ is exponentially stabilizing [PY2,BY1] and for any continuous / e B([0, l]d) and K uniform, the variance of (/, fi^) is bounded below by a positive multiple of A [BY2], showing that (/, /u^} satisfies a CLT with an O((log A)3dA~1/'2) rate of convergence. It follows easily from the stabilization analysis of [PY2] that many variants of the above basic packing model satisfy similar rates of convergence in the CLT. Examples considered there include balls of bounded random radius, cooperative sequential adsorption and monolayer ballistic deposition. In each case, the number of particles accepted satisfies the CLT with an O((logA)3dA~1/2) rate of convergence. The same comment applies for the number of seeds accepted in spatial birth-growth models. 3.4. Independence number, off-line packing An independent set of vertices in a graph G is a set of vertices in G, no two of which are connected by an edge. The independence number of G, which we denote /3(G), is denned to be the maximum cardinality of all independent sets of vertices in G. For r > 0, and for finite or countable X C Rd, let G(X, r) denote the geometric graph with vertex set X and with edges between each pair of vertices distant at most r apart. Then the independence number P(G(X, r)) is the maximum number of disjoint closed balls of radius r/2 that can be centered at points of X; it is an 'off-line' version of the packing functionals considered in the previous section. Let b > 0 be a constant, and consider the graph G{V\,b\~lld) (or equivalently, G(X1/'d'P\,b)). Random geometric graphs of this type are the subject of the monograph [Pel], although independence number is considered only briefly there (on page 135). A law of large numbers for the independence number is described in Theorem 2.7 (iv) of [PY3].

48


For u > 0, let 7iu denote a homogeneous Poisson point process of intensity u on Rd, and let H^ be the point process ~HU with a point inserted at the origin. As on page 189 of [Pel], let Ac be the infimum of all u such that the origin has a non-zero probability of being in an infinite component of G(Hu,l). If 6d||«;||oo < Ac, we can use Corollary 2.4 to obtain a central limit theorem for the independence number /3{G{V\, b\~l^d)), namely

™S

P

\P(G(VX,bX-W))

[

-E(3(G(Vx,b\-1/d))

(Var/J(G(PAl6A-V-)))Va

^ .1

* *J "

a,,.

$W

< C(logA)3dA-1/2.

(3.2)

We sketch the proof. For finite X C Kd and x £ X, let £(x; X) denote the independence number of the component of G(X, b) containing vertex x, divided by the number of vertices in this component. Then Ylxex £( x ' ^) is the independence number of G(X,b), since the independence number of any graph is the sum of the independence numbers of its components. Also, our choice of £ is translation-invariant, and so we obtain

(3{G{Vx,\-l/db)) = P{G{\1'dVx,b))= E

i{^fdx\\lldVx)

= J2 &(*;PA) = , x€Vx

where we here take the test function / to be identically 1 and take A\ to be A for all A. Thus a central limit theorem holds for P(G{V\,\-1/db)) by application of Corollary 2.4, if £ and K satisfy the conditions for that result. We take R(x, A) to be the distance from \lldx to the furthest point in the component containing \l/dx of G(\1/dV\,b), plus 26. Since £\{x;V\) is determined by the component of G^^Vx^) containing A1/dx, and this component is unaffected by the addition or removal of points to/from V\ at a distance greater than \~l/dR(x, A) from x, it is indeed the case that R(x, A) is a radius of stabilization. The point process \1/dV\ is dominated by H^w^ (in the sense of [Pel], page 189). Hence, P[R(x,\) > t] is bounded by the probability that the component containing x of G(7i\\K^!x> U {\l^dx),b) has at least one vertex outside Bt-2b(^l^dx). This probability does not depend on x, and equals the probability that the component of G(H®d,.,, ,1) containing the origin includes a vertex outside i?(i/(>)_2- By exponential decay for subcritical continuum percolation (Lemma 10.2 of [Pel]), this probability decays exponentially in t, and exponential stabilization of £ follows. The moments


49

condition (2.5) is trivial in this case, for any p, since 0 < £(z; X) < 1. Thus, Corollary 2.4 is indeed applicable, provided that (2.7) holds in this case, with a2 > 0. Essentially (2.7) follows from Theorem 2.1 of [BY2], with strict inequality a2 > 0 following from (2.10) of [BY2]; in the case where K is the density function of a uniform distribution on some suitable subset of Kd, one can alternatively use Theorem 2.4 of [PY1]. We do not go into details here about the application of results in this example, but we do comment further on why the distribution of the 'add one cost' (see [PY1,BY2]) of insertion of a point at the origin into a homogeneous Poisson process is nondegenerate, since this is needed to verify a1 > 0 and this example was not considered in [PY1] or [BY2]. The above add one cost is the variable denoted A(oo) in the notation of [PY1], or A?(u) in the notation of [BY2]. It is the independence number of the component containing the origin of G(Hu;b) minus the independence number of this component with the origin removed (we need only to consider the case where bdu is subcritical). This variable can take the value 1, for example if the origin is isolated in G(TCu;b), or zero, for example if the component containing the origin has two vertices. Both of these possibilities have strictly positive probability, and therefore A(oo) has a non-degenerate distribution. 4. Proof of theorems 4.1. A CLT for dependency graphs We shall prove Theorem 2.3 by showing that exponential stabilization implies that a modification of (/, Jjrx) has a dependency graph structure, whose definition we now recall (see e.g. Chapter 2 of [Pel]). Let Xa, a £ V, be a collection of random variables. The graph G := (V,£) is a dependency graph for Xa, a 6 V, if for any pair of disjoint sets A\,Ai C V such that no edge in £ has one endpoint in A\ and the other in A2, the sigma-fields a{Xa,a G Ai}, and cr{Xa,a e A2}, are mutually independent. Let D denote the maximal degree of the dependency graph. It is well known that sums of random variables indexed by the vertices of a dependency graph admit rates of convergence to a normal. The rates of Baldi & Rinott (1989) and those in Penrose [Pel] are particularly useful; Avram & Bertsimas [AB] use the former to obtain rate results for the total edge length of the nearest neighbor, Voronoi, and Delaunay graphs. In many cases, the following theorem of Chen & Shao (2004, Theorem 2.7) provides superior rate results. For any random variable X and any

50

p>0,\et\\X\\p


= (E[\X\P})1/P.

Lemma 4.1: Let 2 < q < 3. Let Wi, i G V, be random variables indexed by the vertices of a dependency graph. Let W = ^ i 6 V Wj. Assume that E [W2] = l,E[Wi] = 0, and \\Wi\\q < 0 for alii e V and for some 6 > 0. Then sup \P[W 1 throughout. Let (p\,X > 0) be a function to be chosen later, in such a way that px —> co and \~1ldp\ —> 0 as A —* oo. Given A > 0, let s\ := \~1ldp\, and let V := V(A) denote the number of cubes of the form d

Q = JJiJiSx, (ji + !)**),

w i t h a11

i=l

U e Z,

such that fQn(x)dx > 0; enumerate these cubes as Qi,Q2, • • • ,Qv(\)Since n is assumed to have bounded support, it is easy to see that V(A) = O(\pid) as A -> oo. For all 1 0, choose an ordering on the points of V\ n (Qj x A'l) uniformly at random from all (A^)! possible such orderings. Use this ordering to list the points as (Xiti, C/j,i), ••-, (Xi,Nn î.AfJ, where conditional on the value of Ni, the random variables Xij, j = 1,2,... are i.i.d. on Qi with a density Kj(-) := K ( - ) / / Q . n{x)dx, and the f/jj are i.i.d. in A^ with distribution P x , independent of {Xjj}. Thus we have the representation V\ = ^^{(Xij, Uij)}^. For all 1 < i < V(X), let Vi :— V\ \ {(XijjUij)}^ and note that Vi is a Poisson point process on Rd x M with intensity \K(x)l^d\Q.(x)dx x Pyn(ds).

51


We show that the condition (2.5), which bounds the moments of the value of £ at points inserted into V\, also yields bounds on ÎMiXijiUi^VxWlA^Xi^l^N,}. More precisely, we have: Lemma 4.2: If (2.5) holds for some p > 0, there is a constant C := C(p) such that for all X > 1, all j > 1 and 1 q > 1, then there is a constant forl,il ^=1

< Cpfp+1)/p.

(4.4)

52


Proof: Fix i < V(\) and write ^ for £itj. Clearly, with N := Ni and v •=

i/u

oo

oo

i=l

/

V

i=l

g

oo

OO

\

/

fc=O

OO

q

OO

j=lfc=O

J= l

q

g

Since a.s. only finitely many summands in the double sum are non-zero, by subadditivity of the norm, the above is bounded by oo

oo

M

fc=O j = l oo fe=0

^

l

J=

+1

|2* "J

?

H

j=l

g

oo L2fc+1>/j

j=l

q

M

< E E llo-i^îi^ + Ei^- 1 ^!!,' fc=O j = l

j= l

(4-5)

where here and elsewhere \x\ denotes the greatest integer less than or equal to x. With rj := (1/q) — (1/p), Holder's inequality followed by (4.3) yields

Ifo • lN>2*v\\q < IIOIIP • (P[N > 2kv]T < Cpdx/P(P[N

> 2kv\y.

(4.6)

Substituting into (4.5) we obtain oo

oo

i=i

\v\

< Cpf P Z)i/2 fc+1 • (P[N > 2fcH)" + £ |& • l;v 2kv])r> < (exp (-2 fc - 1 i/log(2 /c ))) r ' = exp(-fc2 fc - 1 ^log2), Hence, (P[N > 2kv])r> < 2"2fc,

k > max(3,2 - log^rji/)).

k > 3.

53


Hence, since i) is a constant, for v > 0 we have oo

Y^ 2fe+i • {p\N > 2ku])r> < k=0

fc<max(3,2-log2(T)i/))

2k+1

Y^

+

Yl

2l k

fc>max(3,2-log2(t)i/))

~

? . =A /

P[R{x,\)>px}K{x)dx < AT(PA).

(4.11)

For each A, and x € Rd, set /A(X) := /(ar)lAA(a0- Recalling the representation V\ = U i=1 {î,j}jli, we have V(\)

Ni

T*=Y, T,M(Xi,j,Ui,i)\'Px) • h{Xi,i). To obtain rates of normal approximation for T\, it will be be convenient to consider a closely related sum enjoying more independence between terms, namely VW Ni T

A := E X > ( f e Ui,j);Vx) • IEV • fx{Xitj).

For all 1 < i < V(X) define Ni

Si := SQi := (VarTA)-x/2 ^ ^ ( ( X i J ; ^ , i ) ; ^ ) • 1^,, • h(Xij) j=\

and put 5 := (VaiTxyx/2(Tx - ETJ[) = E ^ ^ i - E5 4 ). It is immediate that Var5 = E 5 2 = 1. We define a graph G\ := (V\,£\) as follows. The set VA consists of the subcubes Q\, —,Qv(\), and edges (Qi,Qj) belong to the edge set £x if d(Qi,Qj)

< 2a\-l/dpx,

where d{Qi,Qj)

:= inf{ja: - y\: x e Qt, y e Qj}.

By definition of the radius of stabilization R(x, A), the value of Si is determined by the restriction of Vx to the A~1/dpA-neighborhood of the cube Qi. By the independence property of the Poisson point process, if Ai and At are disjoint collections of cubes in VA such that no edge in £x has one endpoint in A\ and one endpoint in A2, then the random variables {SQ1,Q1 & AI} and {SQ^QJ € A2} are independent. Thus Gx is a dependency graph for To prepare for an application of Lemma 4.1, we make five observations: (i) V(X) := |VA| - O(Xpxd) as A ^ oo.


55

(ii) Since the number of cubes in Qi,.-.,Qv distant at most 2\~1/dp\ from a given cube is bounded by 5d, it follows that the maximal degree D satisfies D := Dx < 5 d . (iii) The definitions of Si and £jj and Lemma 4.3 tell us that for all 1 < i < V(X)

115,11, < CCVarTl)-1^ £ | £ . . |

< C{Y^T'xr^pfp+l)/p.

(4.12)

(iv) We can bound Var[T{] as follows. Observe that T'x is the sum of V(A) random variables, which by the case q = 2 of Lemma 4.3 each have a second . Thus the variance of moment bounded by a constant multiple of px each of the V(X) random variables is also bounded by a constant multiple of px ( p + 1 ) / p . Moreover, the covariance of any pair of the V(\) random variables is zero when the indices of the random variables correspond to non-adjacent cubes. For adjacent cubes, by the Cauchy-Schwarz inequality the covariance is also bounded by a constant multiple of ^ ( P + I J / P rp m g shows that Var[Tj;]=O(pf p + 2 ) / p A).

(4.13)

(v) Var[T{] is close to Var[2\] for A large. We require more estimates to show this. Note that \T'X - T\\ = 0 except possibly on the set Ex. Lemma 4.3, along with Minkowski's inequality, yields the upper bound V(X)

Ni

EEi^^'^);^)!1^^)

2 by changing C if necessary. Trivially, (2.6) holds for large enough A when Var[T\] < 1, and so without loss of generality we now assume Var[T\] > 1. To establish the error bound (2.6) in this case, we apply the bound (4.1) to Wi-.= Si-ESi: l< V\, with 6 :=

C{V&iT'x)-1'2pdx{p+1)/p.

Our choice of 8 is applicable by (4.12). We clearly have E [Wi] = 0 and E [(EH(iA) wi)2} = 1- W i t h S = Y^~i] Wi, Lemma 4.1 along with observation (i) above yields sup \P[S 0 is a constant and Z any random variable then by (4.18) we have for all t e R P[Z < t] < P[S < t + j3] 4- P[\Z < $(* + (3) + CX{Va.xTxyq/2pdxq

-S\>0\

+ P[\Z

< $(t) + C0 + CA(VarTA)-"/2p*> + P[\Z

-S\>0\ -S\>0\

by the Lipschitz property of $. Similarly for all ( e l , P[Z $(«) -CP-

CA(VarT A )- 9/2 / 9f - P[\Z - S\ > /?].

In other words s u p \P[Z

/?]. (4.19)

57


Now by definition of 5, |(VarTjO-1/2(7A -ETX)-S\

= \{V&xT'x)-l'2{(Tx - ETA) - (TA - ET'X)}\ - T'x\ + E [\TX - TA|]}

< {\*vT'x)-^{\Tx

which by (4.16) is bounded by C\~3 except possibly on the set Ex which has probability less than CX~2 by (4.11) and (4.8). Thus by (4.19) with Z = (VarTj;)-1/2(rA _ E T A ) and 0 = C\~3 sup |P[(Varr A )- 1/2 (T A - ETA) < t] - $(*)| t

< C\{Va.xTx)-q/2pdxq + C\-2.

(4.20)

Moreover, by the triangle inequality sup IPKVarT;,)- 1 / 2 ^ - ETA) < t] - $(t)

-"+[^t-/H]-('\/il) + (()

0 the right hand side of (4.22) is bounded by C\~l^2p^, and since in this case we set p\ = Aa with a given by (4.10), some elementary algebra yields (2.9). Our assumption that 7 > d(150 + 6/p) then yields the central limit theorem behavior (2.8), which is also trivially true in the case where cr2(f, £, K) = 0. This completes the proof of Theorem 2.5. • Acknowledgements: We began this work while visiting the Institute for Mathematical Sciences at the National University of Singapore, and continued it while visiting the Isaac Newton Institute for Mathematical Sciences at Cambridge. We thank both institutions for their hospitality. JEY's research was supported in part by NSA grant MDA904-01-1-0029 and NSF grant DMS-0203720. References 1. F. AVRAM & D. BERTSIMAS (1993) [AB] On central limit theorems in geometrical probability. Ann. Appl. Probab. 3, 1033-1046. 2. P. BALDI & Y. RINOTT (1989) On normal approximations of distributions

in terms of dependency graphs. Ann. Probab. 17, 1646-1650.

3. Yu. BARYSHNIKOV & J. E. YUKICH (2003) [BY1] Gaussian fields and ran-

dom packing. J. Statist. Phys. I l l , 443-463.

4. Yu. BARYSHNIKOV & J.E. YUKICH (2005) [BY2] Gaussian limits for random

measures in geometric probability. Ann. Appl. Probab. 15, 213-253. 5. P. J. BICKEL & L. BREIMAN (1983) Sums of functions of nearest neighbor distances, moment bounds, limit theorems and a goodness of fit test. Ann. Probab. 11, 185-214.

6. L. H. Y. CHEN k. Q.-M. SHAO (2004) Normal approximation under local

dependence. Ann. Probab. 32, 1985-2028. 7. N. HENZE (1988) A multivariate two-sample test based on the number of nearest-neighbor type coincidences. Ann. Statist. 16, 772-783. 8. M. D. PENROSE (2003) [Pel] Random Geometric Graphs. Oxford University Press. 9. M. D. PENROSE (2004) [Pe2] Multivariate spatial central limit theorems with applications to percolation and spatial graphs. Ann. Probab. (to appear). Electronically available via http: //arxiv. org 10. M. D. PENROSE & J. E. YUKICH (2001) [PY1] Central limit theorems for some graphs in computational geometry. Ann. Appl. Probab. 11, 1005-1041. 11. M. D. PENROSE & J. E. YUKICH (2002) [PY2] Limit theory for random sequential packing and deposition. Ann. Appl. Probab. 12, 272-301. 12. M. D. PENROSE & J. E. YUKICH (2003) [PY3] Weak laws of large numbers in geometric probability. Ann. Appl. Probab. 13, 277-303. 13. C. STEIN (1986) Approximate Computation of Expectations. IMS, Hayward, CA.

Stein's method, Edgeworth's expansions and a formula of Barbour

Vladimir Rotar Department of Mathematics and Statistics, San Diego State University San Diego, CA 92182-7720, USA and Central Economics and Mathematics Institute, Russian Academy of Sciences Nakhimovsckii prospect - 47, Moscow 117418, Russia E-mail: [email protected] This paper concerns an approach to providing rates and asymptotic expansions in the CLT, based on the representation r

E{Wf(W)}= J2 :l~kE{f{rn\w)}

+ R,

m=0

where / is an arbitrary r times differentiable function, f^m' is its mth derivative, W is a r.v. with finite first r + 1 moments, 7m is the m-th cumulant of W or a characteristic close to it, and R is a remainder which may be small under suitable conditions. If W is a sum of r.v.'s, a good bound for the remainder should reflect the dependency structure between the summands. We review and discuss here known results on such bounds, and provide a new result that essentially widens possibilities of applications.

1. Introduction Starting from the pioneering paper by C. Stein (1972), there was a great deal of interest to Stein's method in limiting theorems, extended, developed and modified in many directions by C. Stein himself and many others during these thirty years. Most of these modifications were reflected, to a certain extent, in monographs or surveys by Stein (1986), Barbour, Hoist & Janson (1992), Barbour (1997), Chen (1998), Reinert (1998,1999), Rinott & Rotar (2000), Raic (2003). It is worthwhile also to mention here the approach from Tikhomirov (1980) which was not covered in the surveys above but which, 59

60

Vladimir Hotar

though involves characteristic functions, is close to Stein's method. Recently this approach was applied to limit theorems for quadratic forms in Gotze & Tikhomirov (1999, 2002). The present paper has mainly a methodological character and concerns one particular approach to the CLT and asymptotic expansions in it. Namely, we study the following representation, the possibility of which was first pointed out and justified in a certain case in Barbour (1986). Let / be an r times differentiate function, and W be a r.v. with finite first r + 1 moments. Then the representation mentioned is r

E{Wf(W)} = J2 -Ê{f(m\W)}

+ R,

(1.1)

m=0

where -ym is the m-th cumulant of W, /( m ) is the m-th derivative of / , and R is a remainder. If f(w) = wr, then R = 0 (see also (2.5) below), and (1.1) gives just the well known relation between moments and cumulants. In the general case, R does not vanish but may be small, as we will see, under suitable conditions. If E{W} = 0, and E{W2} = 1, then (1.1) implies that r

E{Wf(W)} - E{f'(W)} = ^ 7m±LE{f(m\W)} + R. m=2

m

(1.2)

"

For a given function h, denote by S(h) the Stein function / solving the differential equation f'(w) — wf(w) = h(w) — $(/i) (see, e.g., Stein (1972, 1986)), that is,

S{h)(x) = ^JX

[h(t)-*(h)Mt)dt,

where ip is the standard normal density, and #(/i) = / ^ h(t)ip(t)dt. For / = S(h), from (1.2) it follows that r

E{h(W)} - *(/i) = - J2 ^-E{f(m\W)} m=2

m

- R.

(1.3)

-

The main term in (1.3) specifies the proximity to normality in terms of cumulants which are small under very mild requirements, provided W is close to normal. If the class of functions h under consideration is closed with respect to <S, regarding the main term one can use induction when establishing asymptotic expansions for E{h(W)}. This was shown first in Barbour (1986), and later in a more general situation in Rinott & Rotar

Stein's method for Edgeworth's expansions

61

(2003). In Section 4 we consider it in more detail, and right now we just note that, if one wants to write the Edgeworth expansion for E{h{W)} of a length of r, it suffices to write for E{f^m\W)} in (1.3) the expansion of the length r — m, which allows to use induction in r. Certainly, it presupposes a smoothness of h, though as was shown in Rinott & Rotar (2003), this requirement may be replaced, for some dependency structures, by a requirement on the smoothness of the distribution of W. Thus, the main problem concerns the remainder. The essential difficulty here lies in the fact that, as a rule, it is not possible to estimate R efficiently in terms of cumulants or some other characteristics of W itself as a non-decomposable r.v. (see Section 2 for details), so R should involve the "interior structure" of W. Considering independent summands, Barbour (1986) wrote down the representations (1.1) for each summand separately, and then combined them adroitly, using independence in a crucial way. For dependent summands the problem becomes essentially more complicated. For a generalized local dependency scheme the representation (1.1) was considered in Rinott & Rotar (2003). This paper has two aims. First, we systematize and discuss some known results concerning (1.1), second, we prove a new result that essentially widens possibilities of applications. In Section 2 we review results considering W as & non-decomposable random variable. Section 3.1 concerns the case when W is the sum of independent r.v.'s. In Sections 3.2-3.4 we consider dependent summands with a general dependency structure. Sections 3.2-3.3 concern results that use the so called ^-mixing characteristics. Section 3.4 contains a new result involving the so called strong mixing dependence measure a. Conditions on dependency structures in terms of the latter measure are, as a rule, much weaker than those in terms of -mixing; see Sections 3.2 and 3.4 for detail. As was noted already, this paper is of a methodological character, so we did not consider asymptotic expansions themselves in detail. Nevertheless, Section 4 contains some general remarks on how expansions may be provided with making use of (1.1). In Section 5 we give a proof of the main Proposition 3.12. 2. A representation for a non-decomposable W Let / be any sufficiently smooth function, not necessarily Stein's, and um(f;x)=

sup | / ( m ) ( t ) - / ( m ) ( 0 ) | . |t|<W

62

Vladimir Rotar

Proposition 2.1: (Barbour, 1986). Suppose f is r times differentiable, and E{\W\r+1ur(f;W)}

< oo.

(2.1)

Then (1.1) is true with a remainder R = Rr such that ^\-ya+1\E{\Wr-Ur(f;W)}

l^l^2^-î

(7^7)\

v

s=0

where

does. On the other hand, if ||/ (m+1) ||oo and E{\W\r+2} are finite, since «m(/;z)l)\y,u)dy

< /r+l-mVW Hence, in this case (cf. (3.11)) IV+lU) < /r+l-mVW The same integration by parts may be carried out in the terms E{f^m\W)} in (3.12) too. Formally it does not mean yet that we have reduced the necessary order of the finite derivatives of / , since when integrating by parts we still used the boundedness of all derivatives, but to make the reduction under discussion rigorous it suffices to use standard smoothing argument: to smooth / , to write the expansion (3.12) for the smoothed / , and then to come back to the original / . In the last step it suffices to use, for example, the smoothing Lemma 11.4 from Bhattacharya & Ranga Rao (1987), requiring - to meet conditions of this lemma - that sup f (fe+{x + z)- / e _(x + z)) (1 + |z| 3r+1 ) 0, as e -> 0, (3.16) where fe+{x) := sup| y |< £ /(:r + y), and / e _(z) :=mi\y\<ef{x + y). So, in particular, if T — 0, all densities qsi{x,w) are r + 1 times differentiable in x, and their derivatives are absolutely integrable, then |£r(/)| 2.

73


Consequently, /

fc'(r-l)/(r+2)

K(r) = O M ^ < r + 1 > / ( r + 2 ) + M2Kr^r+V +M 4

K(r-2)/(r+2)

+ M3-

=

Rl/(r+2)

+ • • • + Mr+1

,

\

) ,

and if M is uniformly bounded in n, /

R(r) = o (vfar

(p+1)/(r+2 +

K"(r-l)/(r+2)

r+2

> + ir/( > + - — = —

K(r-2)/(r+2)

— n —

+

+

^l/(r+2) \

--- ^m)-

(3-22)

Thus, we have switched to the measure a at a cost that K (which is a counterpart of T from (3.4)), has been raised to a power less than one. On the other hand K > 1}, are known signed measures involved in asymptotic expansions, namely, Qv(dx) = £ where Lm(dx) =

M

p^Lv+zsidx),

(4.1)

(-l)mip^(x)dx, V

I

/

O

m=ikj\^T^.)

\

*!m

'

(4 2)

-

74

Vladimir Rotar

s = s(k) =fci+ ... +fcy,the summation in ]C(K) *S o v e r a u vectors of nonnegative integers k = (&!,...,/;„)such that k\ + 2k2 + ••• + vkv = v (see, e.g., Bhattacharya & Ranga Rao (1986), Petrov (1995)) Coefficients /?; in (4.2) are either cumulants of W, or modified characteristics 7J defined in Section 3.3.1 and in Section 5.1 below, depending on what characteristics we decided to use. Let again r denote an integer. Assume that for a function h we have proved already that for 1 < t < r — 1 t-i

Eh(W) = J2

(4.3)

h(x)Qv(dx) + Rt(h),

v=0J

where (4.4)

\Rt(h)\ < Dt(h),

for an appropriate bound Dt(h). Our goal is to prove (4.3)-(4.4) for t = r. Suppose that we have at our disposal the representation

E{Wf(W)} = J2 £=±l£{/M(W)} + Br(f),

(4.5)

m=l

where / is a sufficiently smooth function, Br(f) is a remainder, and j3m is again either the m-th cumulant of W or the modified characteristic 7 m . Let now / = S(h), the Stein function. Then, since /?i = 0, and /52 = 1 for both choices of /?'s,

Eh(W) - *(/*) = - ]T ^±E{f^(W)} 771.

- Br(f).

(4.6)

By (4.6) and the induction hypothesis, applied to the function / ( m ' and taking t — 1 = r — m, Eh{W) - *(A) r a fr-m .

im

M

\

= - E ^ T E / f Hx)QAdx) + Rr-m+l(f ) 771 = 2

r

'

\l/=0 J

- Br(f) )

.

r-m

= -E^rEE ( /-// ( m ) ww^) R

m-2

•

i^=0

J

-E^T^- + i(/ ( m ) )-5 r (/). m=1

(4.7)

75

Stein's method for Edgeworth 's expansions

Thus, regarding the remainder, one should prove that ]

J2

-^Dr-m+1(fW) + \Br(f)\ < Dr{h).

m=2

This is possible only if one manages to represent Dt(-) in a special form, in terms of the function h itself. including a possibility to estimate Dt(f^) In the case of dependence measure 4>, and for dependency-neighborhoods structures, it has been done in RR, which required long calculations. Next consider the main term in (4.7), that is, r

„

r-m

£%r££(W

m=2

-

/(m)(x)L

v=0

^ («**)•

Using the formula (see, e.g., Barbour (1986)) / f<m\x)Lv(dx) = J

- i — / h(x)Lv+m+1(dx), m +v+1 J

(4.8)

it is straightforward to verify that r

„

r-m

.

" £ ^ T £ E M « - f(m)(x)L»+2s(dx) m=2

(4.9)

i/=0

- E _E d t i ^ r E M "-TT^ny / "(l)L'« 1, and for m > 1

?lm = Tlm(3) =

£

(5-2)

E - E

Jl+--.+Jm=iPl=l

Pm = l

Xh (s,Pl)Xh(s + pi,p 2 ) • • • XJm(s + Pi + ... + p m -l,p m ). For I < r set [1/2]

[1/2]

c(s,I) = E (-l) Tlm(s), c(s,0 = E IT'-(S)Im

m=0

( c ( s , 0 ) = 1, c ( s , l ) = 1,

jn=O

s>0).

(5-3)

77


Recalling that the above quantities depend on the suppressed index i, define i V~~* V ^ o

Tm+l,t

E

—^T=

i \ /

, •. i\

2^^+i,i(s)ci(s + l,t),

t+l-m,l>l s=l n

Im+i = ^lm+\,h

m = l,...,r.

(5.4)

The last formula gives a formal definition of the quantities 7 m introduced in Section 3.3.1.

5.2. Proof of Proposition 3.12 Below we use the following inequality from Davydov (1970), see also, for example, Bradley (2002): \E{XY}-E{X}E{Y}\ 1 are independent. We use CP(A) to denote the distribution of Z, with A = (Ai, A2, • • • )• We assume EZ = X ^ i iî < 00 in the paper. Barbour, Chen & Loh (1992) established the following Stein equation for the compound Poisson approximation: 00

J2^l9(n + l)-ng(n) = f(n)-CP(X)(f)

(1.1)

1=1

for suitable class of test functions / on Z + := {0,1,2, • • • } , where CP(A)(/) := E / ( Z ) . It is shown in Barbour, Chen & Loh (1992) that for every bounded function / we can find a solution g = g/ of (1.1) that is also bounded and this g is then used to investigate the errors of the compound Poisson approximation as follows. Let W be any random variable on Z + . Suppose that it can be shown that

lElf2lXig(W + l)-Wg(W)\<eo\\g\\+s1\\Ag\\

(1.2)

for all bounded g : Z+ —> R, where, for any function h, we define \\h\\ := sup t u e Z + \h(w)\ and Ah(w) :— h(w + l) - h(w). Then it follows that for any class of bounded test functions Q, sup \TEf(W) - CP(A)(/)| < £0 sup || 5 / || + El sup \\Agf\\. feg /es /eg The class of test functions TTV '•= {/ = 1A '• A c Z + } defines the total variation metric for probability measures Qi and Q2 on Z + : drv(Qi,Q2)=

sup

fe^rv

/ / d Q i - [ fdQ2

J

J

= sup |Qi(^4) - Q 2 (4)|, ACZ+

and the corresponding Stein factors for the solution to the Stein equation (1.1) are ||A5/|| 1, a\k — a^k = • • • = ank, the total variation distance should be not bigger than the total variation distance between the distribution of Y = £"=1 l[X;>i] and Po(E(y)), the general result here is too crude. Although, for this particular case, it is possible to use Stein's method proposed here to tackle the problem directly, we acknowledge that in order to apply the method to various types of dependent structures, much more studies are needed to perfect the approach. 2. Stein's equation for the compound Poisson approximation Suppose X is a fixed locally compact second countable Hausdorff topological space. If J is a finite or infinitely countable index space, one can use the discrete topology. To achieve the idea of "grouping", we define an artificial carrier space F = {ao, a\, • • • } with the discrete topology. Let Tij and Wr be the spaces of all finite point measures on X and F respectively. For each point measure £, we use |£| to stand for the total mass of £. We recall that E is a point process on I if it is a random element on a probability space with values in "Hx [see Kallenberg (1983)]. Our aim is to estimate the errors of the compound Poisson approximation to the distribution of |H|, denoted by£(|2|). We partition X into Borel sets {Ui, i S N} and for each £ £ TCi, we group the points of £ into clusters by denning an associated member oo

(2.1)

where 5X is the Dirac measure at x. Hence, for a point process H, the associated member S is a point process denned on F and |H| = |H|. The grouping into clusters depends on the choice of {Ui, i € N} and is reflected in the upper bound of the errors of the compound Poisson approximation. The idea of grouping and laying the grouped random variables in a new carrier space extends the declumping idea in Arratia, Goldstein & Gordon (1990, Section 3.1), which was later refined by Chen (1998, Section 5) and formulated into a concept of lifting in Chen & Xia (2004). Another way of viewing E, is through a marked point process by treating ct=(Ui) a s locations and E(Ui) as marks [Daley & Vere-Jones (2003), pages 194-210], then apply the theories in the Poisson process approximation [Barbour & Brown (1992), Brown & Xia (2001) and Chen & Xia (2004)] to the

90

Aihua. Xia

ground process Ng = X ^ i ^ s ( u . y Noting that all of these approaches will inevitably throw away lots of information and this paper aims to make use of some information neglected in these approaches. Define a generator on the bounded functions on Tir as oo

oo

Ah(0 = J2 J W £ + «««) - M0] + ! > ( £ - «««) - h(Oie{«i}- (2-2) (=1

1=1

Then A is the generator of an immigration-death process oo

Zt(t) = 52z®(t)8ai,t>0,

(2.3)

*=i

which is constructed as follows. For each fixed I, Z^ (•) is the number of particles at location ai and it is an immigration-death process with initial state Zi (0) = £{«;}, the immigration rate l\i for batches of I immigrants (coming to location a;), and the death rate I for every batch of I particle^ to die together, independently of the others. The processes Z^ , I > 1 are independent. It is possible to argue directly as in Ethier & Kurtz (1986) [Chapter 4 and Problem 15 of page 263] that Z^Kt) constructed above satisfies Z^(0) = £, is Markovian and is the unique solution to the generator A with stationary distribution CP(A) [cf Preston (1975)]. We omit the proof as it is a routine exercise. The immigration-death process used here is to be compared with the previous studies [e.g., Barbour, Chen & Loh (1992)] where the death can only occur with unit per capita rate. When F isfiniteand 5 is a simple point process, i.e., P(H{x} = 0 or 1, for all x) = 1, then the choice of Ux = {x} results in a unit per capita death rate. Thus, in essence, our interpretation is more flexible and likely to have more applicability. Moreover, there is always a probabilistic interpretation using Markov immigration-death process for such an approach without any restrictive conditions imposed, hence enabling us to get sharper bounds for the Stein factors [cf Barbour, Chen k Loh (1992)]. The reason for using an immigration-death process with deaths occurring independently for each cluster and births arriving independently of everything is that we can construct such a process effectively according to our needs, as shown in the proofs of Propositions 3.2 and 3.3. If we use a marked Poisson approximation to tackle the compound Poisson approximation by looking at a Poisson process approximation of the ground process Ng only, then lots of valuable information in the marks will

Stein's method for compound Poisson approximation

91

be lost. It is more hopeful that working on the actual values of the random variables proposed here will get us better estimates. We now establish the Stein equation and find its solution. Lemma 2.1: Under the assumption that ^ i > 1 i\i < oo, for each bounded function f on HT, the integral

h(0 = - f°° E[/(Z£(t)) - CP(A)(/)]dt Jo

(2.4)

is well-defined and satisfies

Ah(O = f(0 - CP(X)(f).

(2.5)

Proof: We write Z£(t) = Dt(t) + Z0(t) and ZCP{t) = D(t) + Z0(t), where Dj and D are two pure death processes (no immigration) with death rates as described above and initial distribution D^(0) = £ and .D(O) ~ CP(A), respectively, and ZQ is a copy of the above immigrationdeath process with initial state 0 and independent of D^ and D. Then ZCp{t) ~ CP(A) for all t > 0. Define Si = inf{i: Ds(t) = 0} and S2 = inf It : D(t) = o | , then we have |E[/(Z € (t)) - CP(A)(/)]| < E|/(Z £ (t)) - f(ZCp(t))\ t), so that r°°

/ |E[/(Z£(t))-CP(A)(/)]|dt < 2||/||E(5iVS2) < 2||/||E(5i+5 2 ) Jo

< 2||/||f]e{a,}A2 + 2||/||f;E(iV0// < oo, i=i

i=i

where Ni ~ Po(A(). To show Stein's equation (2.5), conditional on the first time of change of states r = inf {t : Z^(t) ^ £}, it follows from the strong Markov property

92

Aihua Xia

that, for each 0 < u < oo,

- f[E/(Z £ (t))-CP(A)(/)]dt Jo = - E [U{f(Zs(t)) - CP(X)(f)]dt Jo

= - [ / ( 0 - CP(A)(/)]E[r Au] - E jf" [/(Zc(t)) - CP(A)(/)]d* = -[/(O-CP(A)(/)]E[TAU]

- r £ ^ ° ° a ! , . . . r E[/(z€+i,Q; w) - cp(A)(/)]dtP(T e d«) /•u °°

~

Ev°°

Pin \

pu—v

x 4.1.1 /

E[/(Z € _ W a i (i))-CP(A)(/)]dtP(T€d«).

Letting u —+ oo gives

HO = -[/(O - CP(A)(/)]

1

°°

+ 5]h(£ + I6ai)

which implies Stein's equation (2.5) by some reorganisation.

1\

'

•

To estimate the Stein factors, there are two ways to couple the immigration-death processes starting from different states [see Remark 3.4]. Erom Poisson random variable approximation we learned that the best coupling for estimating the Stein factors is in terms of the time for one Markov chain to run until it reaches the initial state of the other [Xia (1999), Brown and Xia (2001)], and since the Markov chain used for Poisson random variable approximation has single birth and single death, it is possible to represent the mean of the time in terms of the stationary distribution of the Markov chain. For the compound Poisson approximation, when j\j J, 0 as j —> oo, Barbour, Chen & Loh (1992) introduced a Markov chain with multiple births and single death to estimate the Stein factors. Although it seems impossible to have any explicit expression for the mean time of the Markov chain to travel from one state to another, when the test functions are of the form / = l[o,fc], k € Z + , Barbour & Xia (2000) managed to exploit this coupling technique to obtain some neat bounds. The difficulty here is multiple births and multiple deaths. For such immigration-death

93


processes, little is known about effective couplings [Mufa Chen (2000), personal communication]. 3. The compound Poisson approximation in total variation As in the Poisson random variable approximation case, the most valuable part, which is also the most difficult part, is to obtain the best possible Stein factors. To achieve this, we first need a technical lemma. Lemma 3.1: For any two positive integers j , I with j < I, we have

xj-xlt, Zi+jSaj (t) = Zg(t) + j6aj l T l > t -

94

Aihua Xia

Now, if T\ < t, then f(Zz+lsai(t))-f(Zi+jSa.(t))=O, hence, by conditioning on T\ > t, then on T^ < t and T2 > t, we get nf(zi+iSai{t))

-

f{zMSa.{t))\

= e-*E[/(Z c (t) + lSai ln>t) - f(Z€(t) + j8aj)] = e-"E[/(Z £ (t) + I6ai) - f{Zt{t) +e-*(l

(3.3)

+j8aj)\

_ e-('-^)[/(^(*)) - f(Zdt) +JSaj)].

(3.4)

It follows from (3.2) and (3.3) that f°° 1 \HZ + I6ai) - h(£+j5aj)\ < / e~jtdt < -. Jo J

(3.5)

Noting that f(rj + iSai) = f(r] + iSai),

for all rjeHr, ie Z+,

(3.6)

we have from (3.4) and Lemma 3.1 that

\nf(zi+iSai(t))-f(zMSoj(t))}\ < e~lt \nf(Zdt) + lK)-f(Zdt) +{e-jt

+jSai)]\

__ e-it} |E[/(Z£(t)) - fXZs(t)+J8ai)]\

lt

< e~ |E[/(Z € (t) + WQ1) - /(Z £ (t) + jSai)}\ + i ^ l e - * |E[/(ZC(*)) - /(Z € (t) +j6ai)]\.

(3.7)

Now, bearing the representation (2.3) in mind, we can decompose Z^\t) = D^\t) + Z^1](t), t > 0, where D^ is a pure death process with Di (0) = ^{ai} and unit per capita death rate, Zg

is a birth-

death process with Zg (0) = 0, birth rate Ai and unit per capita death rate, D[ , ZQ and {Z\ , I > 2} are independent. Hence, if we define

Rdt) = TZ2 Zf\t)8ai + D^(t)Sai, then Ze(t) = Z£\t)8ai + Rt(t), t > 0.

(3.8)


95

Noting that Z^\t) ~ Po(Ai(l - e"*)), we obtain |E[/(Z£(t) + W a i )-/(Z c (t)+j6 a i )]|

< £ nRdt) = v)\m(v + (Z^(t) + l)Sai)--Ef(r,+ (Z^(t)+j)Sai) < £ nJk(t) = ri)drv(C(Z^(t)),C{Z^(t) + l-j)) i)£Hr

< (i - ^drvmzît^cizît)+1)) < (/ - j) m a x P ^ 1 ^ ) = k) < (I - j) min ( l , - _ L = \ ,

(3.9)

where Ait = Ai(l — e~l) and the last inequality is from Proposition A.2.7, Barbour, Hoist &. Janson (1992). Similarly, |E[/(Z { (t))-/(Z C (*)+#a 1 )]l

< Y. F(^(*) = V) |E/(T, + Z^W^) - E/fa + (4J)(t) + j)*ai) < ^ p(^(t) = v)dTV(c(z^(t)),c(z^(t) +j)) V<EHr

<jdTV(C(Z^(t)),£(Z^(t) + l)) (3.10)

<jm^T(Z^(t) = k)<juan\l,-^==). Combining (3.2), (3.7), (3.9) and (3.10) yields, for Ai > i ,

|Mf + ««,) - Ht +J8ai)\ 1, and the bounds will be a function of YliLi ^< rather than Ai. Hence, there is some room to improve on the bounds if the sizes of the clusters satisfy more conditions. Proof of Proposition 3.3: It follows from (2.4) that | h(£ + jSaj + I8ai )-h(£ + jSaj )-h(£ + lSai) + h(01

(3.12)

/-OO

= / nf(Zi+iSai+jsa.(t)) - f(Zi+l5ai(t)) - f(Zi+jS {t)) + f(Z^t))]dt • Jo Let Si ~ exp(i) and 5 2 ~ exp(j) be independent and independent of Z^(t), then we can write % w O | (*) = Z^(t) + Zi+j6aj(t)

=

l5ailSl>t,

Zi(t)+jSajlS2>u

z

i+isai+jsaj (t) = Zt(t) +

lSailSl>t+jSajls2>t-

Hence, if either Si < t or S2 < t, then f{zi+lSai+j5aj

(t)) - f(zi+lSai

(0) - f(z€+jSa.

(t)) + /(z € (t)) = o.

This, together with (3.6), implies 1E\f(Zs+i5ai+j5aj (t)) - f(Zi+lSai (

99


Combining (3.12), (3.17) and (3.18) gives | KZ + j6aj + I5ai) - h(£ + jSaj )-h(£ + I5ai) + h{£) |

(3.19)

Kjlfe-^^j^L^y It is clear that the integral in (3.19) is bounded by e 2t 2 A

/

~ (

T77^ n) dt = A 1 - s) f2 A T-) ds

Jo \ î(l-e"*)/ Jo V - W = I"1 2(1 - s)ds + f (1 - s)/(XlS)ds = i - fln(2Ai) + - M , ô

JJ-

Ai \

4Ai/

valid for Ai > 0.5. However, because of (3.15), direct verification ensures that (3.11) is true for Ai < 0.5 as well. • To state the main theorem, we need the Wasserstein distance between two probability measures Qi and Q2 on Z + defined as < M Q i , Q a ) = sup \ffdQife?w \J

J

f fdQ2

=inf|Yi-y2|,

where Tw — {f '• Z + >-> R, \f(x)-f(y)\ < \x — y\} and the second equation is due to the Kantorovich-Rubinstein duality theorem [Rachev (1991), Theorem 8.1.1, p. 168] and the infimum is taken over all couplings of (Yi,Y2) such that Yi ~ Q,, i = 1, 2. We also need the Palm distributions of a point process [Kallenberg (1983)]. For a point process H on X with finite mean measure fj,, we define probability measures {Px, x € J}, called the Palm distributions, on the Borel sets of Tlj equipped with the weak topology, satisfying Px(.) =

L[

~;j ;

fi{dx)

n

,

x e I H - almost surely.

When H is a simple point process, namely, P(E{x} = 0 or 1 for all x € 1) = 1, the Palm distribution Px can be heuristically interpreted as the conditional distribution of the process H given H{x} = 1. A point process Hx on X is called a Palm process of S at location x if its distribution equals Px. T h e o r e m 3 . 7 : Assume S is a point process on T with finite mean measure \i, {Ex, x e X} are the Palm processes of E, A; = £ ~ i P(H(C/j) = l),l> 1,

100

Aihua Xia

and Ex is the associated member ofEx, Ex = Ex - Ex(Ui)5aSx{U), oo

then

.

dTV(£(\E\),CP(\)) < ^(Ai)^; / EfllS-SillJEpS,^)]/!^) 7=1. ^ oo

.

+ ci(Ai)V;/ ^ ^ ( H ^ ) ^ ) , ^ ^ ) ) ) / ^ * ) .

(3.20)

Remark 3.8: When F is a finite set and S is a simple point process, the choice of Ux = {x} gives a unit per capita death rate and the second term of (3.20) simply vanishes. Proof of Theorem 3.7: We write H* = E - E{Ui)SasiUi), be independent copies of H and Ex respectively, then

let H' and E'x

EAh(E) oo

oo

oo

= E £ l\t [h(E + I5ai) - h(E)} + E ^ ^ [ ^ ( 5 - Wai) - /.(S)]/!^^)^ (=i x=i

i=i

= £ { £ [ ^ ( 5 + H'(I/t)<W(Ui)) " ^(H)]S'(^) i=l

+ E[/i(3') - /i(H' + 2(^)*a a(l/i) )]3(y i )} =£

/" E[/i(3 + 3;(C/0*aa,(l/i)) - ft(S)]/i(di) OO

.

+ 5 ] / E[/i(3i) - ft(3i + 3x(I/0Jas,(l/i))]**( 0, and for sufficiently large d > do(e) and n > no(e), P(N(GL,1\\

-^(Iogd-loglogd-log2 + l) < ^ ) - 1 .

In this direction, Boucheron, Lugosi & Massart (2000) used a powerful tool of log-Sobolev inequality and got a sharp concentration inequality which is independent of d. In the case of random geometric graphs, Penrose (2003), Penrose & Yukich (2005) verified the exponential stabilization property for some random geometric graphs and obtained a law of large numbers and the central limit theorem. In this paper we prove the central limit theorem for the independence number of an MST when the random points are from the Poisson point process. Our results are as follows: Theorem 1.1: Suppose that V\ is a Poisson point process of intensity 1 in R 2 . Let fn be the MST on Vx n [-n 1 / 2 /2, n 1 / 2 /2] 2 and denote by N(fn) the corresponding independence number. Then there are two positive finite constants c\ and C2 such that cxn < VarN(fn) < c2n and N(Tn) is asymptotically normally distributed, that is, asn->oo

N(fn)-EN(fn)^ (VariV(Tn))i/2 in distribution. Following the framework of Kesten & Lee (1996), we write the quantity N(Tn) — EN{Tn) as a sum of martingale differences, and then apply the

106

Sungchul Lee and Zhonggen Su

Levy martingale central limit theorem. In this way, the proof of the central limit theorem for N(Tn) is reduced to a kind of weak law of large numbers estimate for certain conditional variances. To get a weak law we need some independence and this required independence is usually obtained by the truncation method using the stabilizing property (see Lee (1997), Penrose & Yukich (2001), Baryshnikov & Yukich (2005), Penrose & Yukich (2005), for the stabilizing property and it applications). However, for our random variable N(Tn) this stabilizing property does not hold or at least we don't know how to prove it. So, we need a new idea; instead of studying the original random variable N(Tn) we artificially create an approximating random variable, say N^. By its construction this approximating random variable N'n has the stabilizing property so we can follow the steps of Kesten & Lee (1996) and get the central limit theorem for N'n. Furthermore, the approximation error turns out to be small enough to dig out the central limit theorem for N{Tn) from the central limit theorem for N'n. One may hope to prove that Theorem 1.1 holds for the non-Poisson case. Unfortunately, since we don't have the stabilization property for the independence number N(Tn), we cannot follow the de-Possionization step of Kesten & Lee (1996) and even worse we cannot prove the existence of ~VaiN(Tn). Our main tool is the percolation technique regarding the existence of the open and closed circuit at criticality. So, our approximation method works only for d = 2 and hence the problem is still open for high dimensions. In Section 2, we review MSTs and continuum percolation for further use. Since there are a large number of literatures on MSTs and continuum percolation, we will omit the details. In Section 3, we prove Theorem 1.1. In this paper, there are lots of strictly positive but finite constants whose specific values are not of interest to us. We denote them by a, C(9). 2. Minimal spanning trees and continuum percolation In this section, we recall several facts regarding minimal spanning trees and continuum percolation for further use. Lemma 2.1: Let G — (V, E, w) be a connected weighted graph satisfying \V\ < oo, \E\ < oo. If there exists a path -K = {v\,V2,- • •,vn) in G with (VJ,VJ+I) € E, 1 < j < n — 1, from vi e V to vn e V, such that also Vj - Vj+i\ < A, 1 < j < n - 1, then for any MST T on G there exists a path 7r' = (v[,v'2 ...,v'm) in G with ( v ^ + 1 ) G T, 1 < j < m - 1, from vi = v[ to vn = v'm such that \v'j - v'j+11 < A, 1 < j < m - 1.

The independence number for minimal spanning trees

Proof: See Lemma 2 of Kesten & Lee (1996).

107

•

Lemma 2.2: There exists a finite constant Dd, which depends only on the dimension d, such that, for any MST T(A) on any finite subset A of Kd, T(A) has maximum vertex degree bounded by D&. In particular, D2 — 6. Proof: See Lemma 4 of Kesten & Lee (1996).

•

Now, we review continuum percolation. For a set W of points of R 2 and for r > 0, x € R 2 , denote by {x \V\) = / > rc, P(0 oo [0 iir < rc.

3. Proof of Theorem 1.1 Let us start with the proof of the statement that the variance Var7V(Tn) is asymptotically the same order as n. By a standard block argument like that of Kesten & Lee (1996) Theorem 2, pp.525-527, we easily see that VariV(fn) > Cln.

(3.1)

Suppose that {Xi : i > 1} are i.i.d. uniform points on [-n 1 / 2 /2,n 1 / 2 /2] 2 . Let an be a Poisson random variable with mean n, then X\, X2, • • •, Xan is identical in distribution to V\ n [-n 1 / 2 /2, n 1 / 2 /2] 2 . Now Let Tn be the MST on {Xi : 1 < i < n} and denote by N(Tn) the corresponding independence number. To stress the dependency to the points, we denote by N(S) the independence number of the MST through S. By Steele, Eddy & Shepp (1987) Lemma 2, it follows for any m > 1 and X\, X2, • • •, xm \N(x1,x2,...,xm)

-N(x1,x2,..-,xm-i)\

n1/2/2}2 \ Q(x)) n W), the big jump point does not change, the graph structure outside the big jump point does not change, and the vertices outside the big jump point which are included in the modified independence set also do not change. Therefore, the point configuration outside the big jump point has no influence on DL(X;W). Of course, if Gn(x;W) does not happen, by definition DL(X;W) completely depends on the point configuration nfe=i[xfe - L,Xk + L]n W. Therefore, in either case DL(X;W) completely depends on the point configuration

113


nLifr* - L,xk+L]nW and if Gn(x;W) happens, then DL(x;W) = Dn([-n^2/2, nll2/2]2 n W, Q(s) n W). Now, to proceed, we write AfciL for the expression arising in (3.9) when Ai(_[Uikai\,Ak) and Dn({Uifcaj],afc) are replaced by DL(v{k); [L>ifcai]) and DL{v{k); [UiL(u(fc); [U< fc A] U [UixtOi]) - DL{v(k); [\Ji 0.

(3.18)

Thus we have

k:An(v(k))nR2\{-n1/2/2,n1/2/2]29i0

k=l

k:An(v(k))c[-nV2/2,nl/2/2]2

-> 0.

(3.19)

Thus, since cgn < a'^ < cin, it follows that hm -~ = 1 and

cgn < o'^L < CIQU.

Now (3.17) follows from Chebyshev's inequality using the fourth moment. Since the events Afc^ and Afcî, are independent as long as An(v(k)) n A^(v(k')) = 0, and since for each k the number of k' with K(v(k)) n A°n(v(k')) ^ 0 is of order n2/3, E(J2(ÎL it=i

~ £Afc,L))4 < cn[n + n 1+2/3 + n1+i0 + n1+20} = O(n1+6^)

and hence

p(\-^y:AlL-EAlL > £ ) < £ i ^ - , 0 .

^- M4

Now, Theorem 1.1 follows. Acknowledgements: SL was supported by the BK21 project of the Department of Mathematics, Yonsei University, the interdisciplinary research program of KOSEF 1999-2-103-001-5 and Com2MaC in POSTECH. ZS was supported by NSFC 10371109, 10071072.

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References 1. D. J. ALDOUS k. J. M. STEELE (2003) The objective method: Probabilistic combinatorial optimization and local weak covergence. In: Probability on Discrete Structures, Vol. 110 of Encyclopaedia of Mathematical Sciences, ed: H. Kesten, pp. 1-72. Springer, New York. 2. K. S. ALEXANDER (1996) The RSW theorem for continuum percolation and the CLT for Euclidean minimal spanning trees. Ann. Appl. Probab. 6, 466494. 3. F. AVRAM & D. BERTSIMAS (1993) On central limit theorems in geometric probability. Ann. Appl. Probab. 3, 1033-1046. 4. Y. BARYSHNIKOV & J. E. YUKICH (2005) Gaussian limits for random measures in geometric probability. Ann. Appl. Probab. 15, 213-253. 5. J. BEARDWOOD, J. H. HALTON & J. M. HAMMERSLEY (1959) The shortest

path through many points. Proc. Cam. Philos. Soc. 55, 299-327. 6. S. BOUCHERON, G. LUGOSI & P. MASSART (2000) A sharp concentration inequlaity with applications. Rand. Struct. Alg. 16, 277-292. 7. A. FRIEZE (1990) On the independence number of random graphs. Discrete Math. 81, 171-176. 8. G. R. GRIMMETT (1999) Percolation, 2nd edition. Springer, New York. 9. P. HALL & C. C. HEYDE (1980) Martingale Limit Theory and Its Application. Academic Press, New York. 10. H. KESTEN & S. LEE (1996) The central limit theorem for weighted minimal spanning trees on random points. Ann. Appl. Probab. 6, 495-527. 11. S. LEE (1997) The central limit theorem for minimal spanning trees I. Ann. Appl. Probab. 7, 996-1020. 12. S. LEE (1999) The central limit theorem for minimal spanning trees II. Adv. Appl. Prob. 31, 969-984. 13. P. LEVY (1937) Theorie de I'addition des variables aleatoires. GauthierVillars, Paris. 14. D. L. MCLEISH (1974) Dependent central limit theorems and invariance principles. Ann. Probab. 2, 620-628. 15. A. MEIR & J. W. MOON (1973) The expected node-independence number of random trees. Nederl. Akad. Wetensch. Indag. Math. 76, 335-341. 16. M. D. PENROSE (1996) The random minimal spanning tree in high dimensions. Ann. Probab. 24, 1903-1925. 17. M. D. PENROSE (2003). Random Geometric Graphs. Oxford University Press. 18. M. D. PENROSE & J. E. YUKICH (2001) Central limit theorems for some graphs in combinatorial geometry. Ann. Appl. Probab. 11, 1005-1041. 19. M. D. PENROSE & J. E. YUKICH (2005) Normal approximation in geometric probability. In: Stein's method and applications, Eds: A. D. Barbour & L. H. Y. Chen, pp. 37-58. IMS NUS Lecture Notes Series Vol. 5, World Scientific, Singapore. 20. B. PlTTEL (1999) Normal convergence problem? Two moments and a recurrence may be the clues. Ann. Appl. Probab. 9, 1260-1302.


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21. C. REDMOND & J. E. YUKICH (1994) Limit theorems and rates of convergence for Euclidean functionals. Ann. Appl. Probab. 4, 1057-1073. 22. J. M. STEELE (1988) Growth rates of Euclidean minimal spanning trees with power weighted edges. Ann. Probab. 16, 1767-1787. 23. J. M. STEELE (1997) Probability Theory and Combinatorial Optimization. SIAM. 24. J. M. STEELE. L. A. SHEPP & W. F. EDDY (1987) On the number of leaves of a Euclidean minimal spanning tree. J. Appl. Probab. 24, 809-826. 25. J. E. YUKICH (1998) Probability Theory of Classical Euclidean Optimization Problems. Lecture Notes in Mathematics no. 1675, Springer, New York.

Stein's method, Markov renewal point processes, and strong memoryless times

Torkel Erhardsson Department of Mathematics, KTH S-100 44 Stockholm, Sweden E-mail: [email protected] We show (leaving out most of the details) how Stein's method can be used to obtain bounds for the total variation distance between the distribution of the number of points in (0, t\ of a stationary finite-state Markov renewal point process, and a compound Poisson distribution. Two bounds are presented, of which the first makes use of an embedded renewal-reward process, where visits to a particular single state serve as renewals. For the second bound, the original point process is first embedded into a process of the same kind, but with an added state, representing "loss of memory" in the original process. An explicit expression for the second bound is given for the case when the original process is a renewal process.

1. Introduction

Here we shall describe an application of Stein's method to the following problem. Let $ be a stationary Markov renewal point process on a finite state space S. (By such a process we mean a special kind of marked point process on the real line, with mark space S. The definition is given in Section 2.) Let B be a subset of S. Let W = *((0,t] x B), i.e., W is the number of points of \P in the interval (0, t] with marks in B. Some special cases of interest are the number of points of a stationary renewal process in (0,t], and the number of visits by a stationary irreducible Markov chain on the state space 5 to the set B during (0,t\. We want to find an explicit bound for dTV(J?(W),POIS(ir)), 119

(1.1)

120

Torkel Erhardsson

where drv{-r) is the total variation distance, and POIS(TT) is a suitably chosen discrete compound Poisson distribution. The reason why we want to bound (1.1) is that we expect that, under an additional condition, J£{W) is close to a compound Poisson distribution. The additional condition, which should be kept in mind although it is not necessary for any of the theorems to hold, is that the set B is rare in the sense that points of ^ with marks in B are rarely found. These points can then be partitioned into clumps, the sizes of which are approximately I.I.D., and the number of which is approximately Poisson distributed, an idea which is known as the Poisson clumping heuristic; see Aldous (1989). Before proceeding, we recall a few basic definitions. POIS(TT), where n is a finite measure on the positive integers, is defined as _2f (%2i=i î)> where all variables are independent, _2?(Tj) = TT = 7r/||7r||, and U ~ Po(||7r||). TT is called the compounding measure. The total variation distance (ITV{', •) is a metric on the space of probability measures on the nonnegative integers, denned by drv(j/i,(/2) = sup \vi(A) - v2{A)\. ACZ+

We use the following notation: K = the real numbers, Z = the integers, R+ = [0,oo), R'+ = (0,oo), R'_ = (-oo,0), Z+ = {0,1,2,...}, and Z'+ = {1,2,...}. The distribution of any random element X in any measurable space (S, y) is denoted by _S?(X). The Borel cr-algebra of any topological space S is denoted by 38$• Finally, we point out that the results presented here can be found also in Erhardsson (2000, 2004), together with detailed proofs. 2. Marked point processes We first recall the definition of a Markov renewal point process on a finite state space. Let {((f i^+i);* e Z} be a stationary Markov chain taking values in M'+ x S, where S = {l,...,n}, with a transition probability p which only depends on the second coordinate. Assume that {Vf;i G Z} is irreducible, and that E(£jf) < oo. For each A C S, let {(Cf^^î e Z} have the distribution ^((Cf >^+i);» G Z|VOS € A), and define {UtA;i £ Z} by

feoCf, _V*

ift>l; CA

if ? < - 1

121

Strong memoryless times

Define a point process Â on R x 5 by * A (-) = E i e z / { ( c / / 1 ' V%A) e }• This point process is called a Palm version with respect to marks in A of a Markov renewal point process. By definition, tyA is a marked point process on R with marks in S, and the basic theory for such processes (see Pranken et al. (1982) or Rolski (1981)) tells us that there exists a unique stationary version \P(-) = Eiez-MX^*' ^*) e '}> satisfying the Palm inversion formula: for each measurable function g : 9t —» R + (where 9t is the space of locally finite counting measures on R x 5),

E

,

m

_ E ( J ^ g(ft(**))*)

M «Tt is the shift operator. Some important special cases deserve mentioning. If S = {1}, then SP1 and \Er are Palm and stationary versions, respectively, of a renewal process. Likewise, the stochastic process {r?t;i e M} defined by r/t = Vmax{i€z;UiR+,

E(S(O) -

E(X?)

•

The point of introducing the renewal-reward process is that for this process, a useful total variation distance bound exists, due to Erhardsson (2000). The result is contained in the following theorem.

122

Torkel Erhardsson

Theorem 2.1: Let £(•) = Xêz-MX-^>^) € •} be a stationary renewal€ reward process, and £°() = JîezÛî^) "} tne corresponding Palm version. Then dTV(J?([

S

l W

J(O,t}xZ+

E(X°)

l

ydt;(x,y)),POIS(Tr))

E(*°) 2

E(X°)

>'

0

where irk = f^njPCô = fc) for fc > 1, and '(lA^-Je'^ll,

always;

+ log + (2(7ri-27r 2 ))J, 1 . (1-20)A '

where X = X ^ i ^

if ITT, - (i + l)7ri+i > 0 Vi > 1; if fl •e' i ^ 2'

an
i

%>1;

) - S (0,2/),

i = 0;

»ri(u,v) + y

J{-x,t-x]xZ+

< Yo + f

J(t-x-X[°'y\t-x]xZ+

+ I

vdt(u,v) vdZ{0'y\u,v)

J (U'__n(-x,~x+\X0\])xZ+

+ f

vd^°>y\u,v).

i(E' + n((-i-x 1 ,t-i])xz +

The terms on the right-hand side can be bounded using the Palm inversion formula and the independence assumptions. For details, see Erhardsson (2000). • 3. A partial solution Using the renewal-reward process defined in the previous section, a solution can be found for the problem considered in Section 1. The solution does not cover all the cases that we are interested in. We assume that the subset Bc is non-empty, and pick a £ B c . As in the previous section, let $ a (-) = J2 i g Z /{([/", V?) 6 •} be a Palm version of the Markov renewal point process with respect to marks in {a}. Let ... < X'L1 < Xft = 0 < Xf < ... be the ordered random variables {f/f; V? = a,i & Z}, and define Y? = \{k e Z;X? < U£ < X?+1,Vka 0

^ r f ' ^ w ^

(exp( 7 ) X /l)(i4)

Vt€M+, +

'

(4.1) V

'

and such that cr(t)e'Y* is nondecreasing and right-continuous, there exists a strong memoryless time £ such that P(C 0 and let a : R + —> [0,1] satisfy the conditions of Theorem 4.2. E.g., we might take <j(t) = e"7*ess inf xe(tiOo) / c (a;)

Vt G R+,

where fe is the Radon-Nikodym derivative ofJ£(£) with respect to exp(j). Define c0 = 7 /0°° a(x)dx and c\ = 7 /0°° J^° a(x)dxdu. Then, drv(JSf(*((0, 1; V° = 0}, P(r o ° = k)= F{T° = A: + 1) = e(l - co)*" 1 ^

VA; > 1.

Strong memoryless times

129

In particular, E(?o°) = E/CQ. Also, define h : {0,1} -> R + by

Ms) = E(£c s IKf = *)> »=o where T? = min{i > 1; Vf = 0}. Then, E(X°1) = h(0)=e/1

+ eh(l),

and /i(l) = E(CoW = 1) + (1 - co)/i(l) = E(C) - co/7 + (1 - co)/i(l). It follows that E(Xj') = eE(C)/co. The other quantities are treated similarly. We finally let e -» 0. • As a last example, we consider the case when the interrenewal time has a mixed exponential distribution. This is relevant e.g. when studying the number of visits to a single state by a reversible Markov chain. Example 6.2: Let / c (x) = £ " = 1 ai7ie~ 7iI , where 0 < 71 < ... < j n , o.i € [0,1] for i = l , . . . , n , and Yî=i ai — *• Choosing 7 = 71, we get a(t) = ct\e nt, Co = a.\ and c\ = ai71~1. Moreover, in this case E(£) = If C ~ exp(7), then Co = 1 and c\ = 7""1, making the bound 0. Remark 6.3: In a forthcoming paper, bounds for (1.1) in the case when W is the number of visits to a rare set by a stationary irreducible finite state Markov chain, will be computed using both the partial solution in Section 3 and the more general solution in Section 5. Numerical comparisons can then be made easily. References 1. D. J. ALDOUS (1989) Probability approximations via the Poisson clumping heuristic. Springer, New York. 2. T. ERHARDSSON (2000) On stationary renewal reward processes where most rewards are zero. Probab. Theory Rel. Fields 117, 145-161. 3. T. ERHARDSSON (2003) Stein's method for Poisson and compound Poisson approximation. In: An introduction to Stein's method, Institute for Mathematical Sciences Lecture Notes, Vol. 4, ch. 2. World Scientific Press, Singapore. 4. T. ERHARDSSON (2004) Strong memoryless times and rare events in Markov renewal point processes. Ann. Probab. 32, 2446-2462.

130

Torkel Erhardsson

5. P. FRANKEN, D. KONIG, U. ARNDT & V. SCHMIDT (1982) Queues and point processes. Wiley, Chichester. 6. T. ROLSKI (1981) Stationary random processes associated with point processes. Lecture Notes in Statistics 5, Springer, New York.

Multivariate Poisson—binomial approximation using Stein's method

A. D. Barbour Angewandte Mathematik, Universitat Zurich Winterthurerstrasse 190, CH-8057 Zurich, Switzerland E-mail: [email protected] The paper is concerned with the accuracy in total variation of the approximation of the distribution of a sum of independent Bernoulli distributed random d-vectors by the product distribution with Poisson marginals which has the same mean. The best results, obtained using generating function methods, are those of Roos (1998, 1999). Stein's method has so far yielded somewhat weaker bounds. We demonstrate why a direct approach using Stein's method cannot emulate Roos's results, but give some less direct arguments to show that it is possible, using Stein's method, to get very close.

1. Introduction The Stein-Chen method for Poisson approximation (Chen, 1975) has found widespread use, being easy to apply, even for sums of dependent indicators, and yet typically delivering bounds for the approximation error which are accurate at least as far as the order is concerned. For instance, considering the very simplest setting of a sum W of independent Bernoulli random variables (Xj, 1 < j < n), with Xj ~ Be (pj), the distance in total variation between £(W) and the Poisson Po(A) distribution with the same mean A := Y?j=i Pj 1S bounded by n

dTV(C(W),Po(X)) <min(\-l,l)J2pj,

(L1)

which is of the correct order; in particular, if all the pj are equal to p, the bound is just min{p, np2}, and if n tends to infinity while p remains fixed, the bound does not become progressively worse, but stays at the value p. 131

132

A. D. Barbour

For the multivariate analogue, take a sequence of independent Bernoulli random d-vectors (Xj, j > 1), with T[Xj = ^)=pf,

l i n I A " 1 ^ ( V {pf}^ ,p$\ ,

(1.3)

where v :— C((Z\,..., Zd)) and the (Zi, 1 Mi : = A"1 n = 1 j ^ ' ' and cA:=i+log+(2A).

(1.4)

This bound is in many ways rather similar to that of the one dimensional version, but not quite as good. For instance, for identically distributed Xj, reduction to the one dimensional case shows that dTV(C(W),v)<min{P,np2}

(1.5)

is a valid upper bound in the multivariate case, too (Le Cam (1960), Michel (1988)). In contrast, the bound (1.3) translates to min{cnpp,np2}; for np large, this has an extra factor cnp x log(np), making the bound less accurate than that of (1.5), even if only logarithmically with np, and, for large enough n and fixed p, eventually destroying the bound altogether. This suggests that (1.3) could perhaps be improved in general, and replaced with a bound which would not become larger just because the size of the problem increased. Using two quite different methods, based on generating functions, Roos (1998, Corollary 1; 1999, Theorem 1) proved two bounds which demonstrated that this is indeed possible. His first bound was derived from a multivariate Charlier asymptotic expansion, and looks a little different from (1.3):

dTV(£(W),v) c O i i ^ r ^ A - M o g A for any j such that (ji, j 2 ) = (wii,m2), where mi := [_A/x»J, i = 1,2, and

Remark: The lower bound for the A-dependence in (2.6) now follows by considering the choices a = e^ + e^2\ a — e^ and a = s^K Proof: We evaluate A12/MC?) by using the formula (2.1), realizing copies of processes V starting in the states j , j + £(1), j + e(2) and j + £(1) + £(2) simultaneously by setting y(D(i) : = y(°)(i) + J[£i > t]ew,

W(t)

:= V^(t)+I[E2

and y(12)(f) : = y(»)(t) + I[E! > t]eW + I[E2 > t]e{2\

> t]e(2)

136

A. D. Barbour

where V^ is started in j , and E\ and £ 2 are independent of each other and of V^°\ and have standard negative exponential distributions. This then gives A12MJ) = - | ° ° E [lA{V^2\t))

- lA(V{2\t)) - lA(V{1\t)) + lA(V^(t))}

= - I"0 e- 2t E {lA(V^(t)

+ eW + eM) - lA(V^(t)

dt

+ s^)

-lA(V + log ( i ^ ) } for c as defined above, if logA > -2hogm\n(ti1,fj,2)+iog(-J-^J

>.

• Although the direct argument by way of (2.5) and a bound such as (2.6) thus cannot yield an estimate of d,Tv{£(W), v) that is as good as those of Roos, alternative ways of exploiting (2.5) can be effective. Our next theorem uses (2.5) to give an analogue of Roos's bound (1.6). To state it, we define

A 2 «:=I>?} 2 ; j=\

^=maxpf(l- P f). -

3

-

n

Theorem 2.2:

dTV{c{w),V) < 2 y (1 A x} = I— >x[———-) \. n lVn \n + xz — 1 / J Hence we state our main result in terms of the self-normalized sum. Theorem 1.1: We have sup \P(Sn/Vn < z) - $(z)| < 10.202 + 25/33, z

where

n

p2 = B-2ÊXfl{\Xi>0.5Bn}

n

andfa= B~3 ^

(1.2)

E\Xi\3I{lXi\ 0.1 or /33 > 0.04. So we can assume that (32 < 0.1 and & < 0.04.

(2.1)

Recall a Hoeffding (1963) type inequality for non-negative independent random variables (see, e.g., the proof of Lemma 3.1 in Chen & Shao (2001)). Let {Yi,l 0

< 0.015 + 0.585 = 0.6.

(2.4)

146

Qi-Man Shao

Thus, (1.2) holds if /32 > 0.6/10 = 0.06 or (33 > 0.6/25 = 0.024. Therefore we can further assume /?2 < 0.06 and /?3 < 0.024.

(2.5)

Let Xi = XiI{\Xi\ 0.5) + P(Vn < 0.8) l<j i (e-II^H + e-l^l") ; the conditional probability that both yi and y 2 are maximal is less than ( l - - (V l|yi11 + e - | | y 2 l l ) V

< e-§(«-2)(e-»yiii+e-»"ii)_

We thus have E

[-^n( S a)] = E [X/, = 1 ^y* is maximal and ||yi||l

- E E E(Z"liV- = m]m2ê-P" v m>l

< 9 V V EfZ^lAT, = m]ê-P" + V V m 3 ê-P" f

m>l

'

« 7n>4

(1) = 1/2. The triangle graph K3 is more interesting. Here the associated matrix is (011\ / 1 1 \ A = I 1 0 1 I , which we abreviate by its upper triangle I j . \110/ ^ ' From the recursion (8.2) we then find

EU^[LMST(K3)]

= ^ ( X J ) = [ 2 + 30(2)]/4 = 3/4,

179

Expected length of an MST

a result that is already quite familiar. The analogous calculation for K4 is only a little more complicated. One application of the recursion (8.2) gives

= ( l l ) = [3 + 6 4 > ( 2 j ) | / 7 ,

Vu[0

so in the end we find E ^ ' ^ L M S T C - K ^ ) ] = 31/35, which fortunately agrees with the table of Section 7. For K5 the computation is not so brief, but the pattern is familiar. To begin we have /I 1 1 1\

V

1/

and one application of the recursion (8.2) brings us to 1 F Ev[0'1][LMsT(/f5)] = YJ 4 + 1 0 0

/222V 11 .

The second application now gives

s+

'C")-M «*O"(")]and we also have / 3 3 \ _ 2 + 6^>(4) + (6) _ 2 + (6/5) + (1/7) _ 117 ^ \ l) ~ 8 ~ 8 ~280 and / 4 2 \ __ 2 + 40(4) + 40(6) _ 2 + (4/5) + (4/7)

^ V 2J ~

9

~

9

118

~ 315'

from which we find , /

2 2 2

\

*{ ' J J

3 + (351/140)+ (118/105)

=

io

=

557

Wo-

180

James Allen Fill and J. Michael Steele

After feeding these intermediate quantities into our first identity, we find at last ,„» 4 + (557/84) 893 vu[o,ihT 9. Concluding remarks Neither the elementary recursion (8.2) nor the integral formula (1.1) seems to provide one with an easy path to the qualitative features of E[LMST(-Kn)]i though they do provide tantalizing hints. The sequence E[LusT(Kn)] is quite likely monotone and concave, but a proof of these properties would seem to call for a much clearer understanding of either the Tutte polynomials T(Kn; x, y) or the intermediate quantities (A) = E^ 0 ' 1 ' [LMST(G(A))] associated with the recursion (8.2). Ira Gessel kindly provided an extension to n = 15 of the table of Tutte polynomials for Kn given in Gessel & Sagan (1996). The Tutte polynomial for the Petersen graph was obtained from the Web page of S.C. Locke at http://www.math.fau.edu/locke/TPol.htm. JAF was supported by NSF Grant DMS-0104167 and by The Johns Hopkins University's Acheson J. Duncan Fund for the Advancement of Research in Statistics. ACKNOWLEDGEMENTS:

References 1. N. ALON, A. FRIEZE & D. WELSH (1994) Polynomial time randomized approximation schemes for the Tutte polynomial of dense graphs. In: Proceedings of the 35th Annual Symposium on the Foundations of Computer Science, ed: S. Goldwasser, pp. 24-35, IEEE Computer Society Press. 2. F. AVRAM & D. BERTSIMAS (1992) The minimum spanning tree constant in geometric probability and under the independent model: a unified approach. Ann. Appl. Probab. 2, 113-130. 3. A. M. FRIEZE (1985) On the value of a random minimum spanning tree problem. Discrete Appl. Math. 10, 47-56. 4. I. M. GESSEL & B. E. SAGAN (1996) The Tutte polynomial of a graph, depth-first search, and simpicial complex partitions. Electron. J. Combinatorics 3, Research Paper 9, (36 pp.). 5. J. M. STEELE (2002) Minimal Spanning Trees for Graphs with Random Edge Lengths. In: Mathematics and Computer Science II: Algorithms, Trees, Combinatorics and Probabilities, eds: B. Chauvin, Ph. Flajolet, D. Gardy, and A. Mokkadem, Birkhauser, Boston. 6. D. WELSH (1999) The Tutte polynomial. Rand. Struct. Alg. 15, 210-228.

Limit theorems for spectra of random matrices with martingale structure

F. Gotze and A. N. Tikhomirov Fakultdt fur Mathematik, Universitat Bielefeld D-33501 Bielefeld, Germany E-mail: [email protected] and Faculty of Mathematics, Syktyvkar State University Oktjabrskyi prospekt 55, 167001 Syktyvkar, Russia E-mail: [email protected] We study two classical ensembles of random matrices introduced by Wigner and Wishart. We discuss Stein's method for the asymptotic approximation of expectations of functions of the normalized eigenvalue counting measure of high dimensional matrices. The method is based on differential equations for the densities of the limit laws.

1. Introduction The goal of this note is to illustrate the possibilities of Stein's method for proving convergence of empirical spectral distribution functions of random matrices. We consider two ensembles of random matrices: real symmetric matrices and sample covariance matrices of real observations. We shall give a simple characterization of both semicircle and Marchenko-Pastur distributions via linear differential equations. Using conjugate differential operators, we give a simple criterion for convergence to these distributions. Furthermore we state general sufficient conditions for the convergence of the expected spectral distribution functions of random matrices. As applications we recover some well-known theorems for bounded ensembles of random matrices (Corollaries 1.7 -1.8 and 2.7-2.8 below). For a more detailed discussion of the literature we refer the readers to surveys by Pastur (2000, 2004), Bai (1999) and the publications of the authors, Gotze & Tikhomorov (2003a; 2004a). The mentioned results are based on the paper Gotze & Tikhomorov (2003b, 2004b). 181

182

F. Gotze and A. N. Tikhomirov

1.1. Real symmetric matrices Let Xjk, 1 < j < k < oo, be a triangular array of random variables with EXjk

= 0 a n d EXfk

= a2jk,

a n d l e t Xkj = Xjk,

forl<j 00.

(1.10)

184

F. Gotze and A. N. Tikhomirov

Then An := sup \EFn(x) - G(x)\ -> 0, as n -> oo.

(1.11)

X

Here and throughout the paper Tr denotes the trace of an n x n matrix. A sufficiently rich class of differentiable functions for which it suffices to check the condition in the above Theorem is given by the imaginary part of x —> l/(x — z), z € C \ R. In the following we will study this class in detail. 1.4. The Stieltjes transform of a semicircle law Introduce the Stieltjes transform of a random variable £ with distribution function F(x), for any z = u + iv, v / 0, via

T(z) = E-i- = r ?-•*

—dF{x).

(1.12)

J-^x-z

Note that T(z) is analytic for non-real z and satisfies the conditions ImT-Imz>0,

ImzÔ,

supj/|T(iy)| = 1.

(1.13)

v>i

It can be shown that for any continuous function ip(\) with compact support /•OO

-I

/'OO

/ 0.

(1-24)

Then A n := sup |EFn(a:) - G{x 00.

(1.25)

Remark: Note that condition (1.23) implies that lim •iE||W n || 2 = (7 2 oo n

where we denote by ||W|| the Frobenius norm of a matrix W:

HW||2 = f > / = J2 WttM2-

(1-27)

Usually one considers only the case cr2; = 0, Ln(r) -> 0,

as n -> oo.

(1.28)

Then the expected spectral distribution function of a matrix W converges to the distribution function of the semicircle law, that is An :=sup|EF n (x)-G(x<J- 1 )| -> 0,

as n -> oo.

(1.29)

Another application of Theorem 1.5 concerns the distribution of spectra for the unitary invariant ensemble of real symmetric nx n matrices W n = (Xij) induced by the uniform distribution on the sphere of the radius %/iV in RN with TV = "lit!), that is

J2 (4 n) ) 2 = iV-

l 0, as n -» oo, n 1 uniformly on compacts sets in C\R. Then A n -> 0, a s n - > oo.

(2.14)

(2-15)

This result allows to prove convergence in case of dependent matrix entries as well. 2.5. Convergence of random spectral distributions to the Marchenko-Pastur law We shall assume that EXjL = 0 and a2jt := EX?;, for 1 < j < n and 1 oo.

(2.27)

X

Another application of Theorem 2.5 concerns the distribution of spectra for the n x m matrices X n = (X^) induced by the uniform distribution on the sphere of the radius y/N in M.N with iV = nm , that is n

m

E E ( 4 n ) ) 2 = iV-

3=11=1

(2-28)

Here, we obtain from Theorem 2.5: Corollary 2.7: Assume (2.17). Let X^\ 1 < j < n, 1 1 and m > 1. Then An

:= sup \Fn(x) - Ga(x)\ -> 0 as n -> oo.

(2.29)

X

We may again consider the ensemble of real nx m matrices determined by the uniform distribution in the ball of the radius y/N in RN with N = nm, that means n

m

EEPO2^0 = 1 1=1

(2.30)

Spectra of random matrices with martingale structure

193

Using Theorem 2.5, we arrive at the following result. Corollary 2.8: Assume (2.17). Let xfr\ 1 1. Then A n -> 0,

as n -> oo.

(2.31)

Acknowledgements: Part of the work was done while FG was visiting the Institute for Mathematical Sciences, National University of Singapore in 2003. The visit was supported by the Institute. ANT was partially supported by Russian Foundation for Fundamental Research Grant N 02-0100233, and both FG and ANT were partially supported by INTAS grant N 03-51-5018. References 1. Z. D. BAI (1999) Methodologies in spectral analysis of large dimensional random matrices: a review. Statistica Sinica, 9 611-661. 2. B. V. BRONK (1964) Topics in the theory of random matrices. Ph.D. thesis, Princeton University. 3. F. GOTZE & A. N. TIKHOMIROV (2003a) Rate of convergence to the semicircular law. Probab. Theory Rel. Fields 127, 228-276. 4. F. GOTZE & A. N. TIKHOMIROV (2003b) Limit theorems for spectra of random matrices with martingale structure. Bielefeld University, Preprint 03-018. www.mathemathik.uni-bielef eld.de/fgweb/preserv.html 5. F. GOTZE k. A. N. TIKHOMIROV (2004a) Rate of convergence in probability to the Marchenko-Pastur law. Bernoulli, 10, 503-548. 6. F. GOTZE & A. N. TIKHOMIROV (2004b) Limit theorems for spectra of positive random matrices under dependence. Zap. Nauchn. Sem. POMI v. 311. Probability and Statistics 7, 92-124. 7. P. HALL & C. C. HEYDE (1980) Martingale limit theory and its application,

Academic Press, New York. 8. M. L. MEHTA (1991) Random Matrices, 2nd ed., Academic Press, San Diego. 9. L. A. PASTUR (2000) Random matrices as paradigm. Math. Phys. 2000, 216-265. 10. L. A. PASTUR (2004) Random matrices: asymptotic eigenvalues statistics. Seminaire de Probability XXXVI, 135-164. Lecture Notes in Math. 1801, Springer. 11. N. ROSENZWEIG (1962) Statistical mechanics of equally likely quantum systems. In: Statistical physics (Brandeis Summer Institute, Vol. 3), Benjamin, New York, 12. C. STEIN (1986) Approximate computation of expectations. IMS Lecture Notes Series 7, Institute of Mathematical Statistics, Hayward, CA.

Characterization of Brownian motion on manifolds through integration by parts

Elton P. Hsu Department of Mathematics, Northwestern University Evanston, IL 60521, USA E-mail: [email protected] Inspired by Stein's theory in mathematical statistics, we show that the Wiener measure on the pinned path space over a compact Riemannian manifold is uniquely characterized by its integration by parts formula among the set of probability measures on the path space for which the coordinate process is a semimartingale. Because of the presence of the curvature, the usual proof will not be readily extended to this infinite dimensional setting. Instead, we show that the integration by parts formula implies that the stochastic anti-development of the coordinate process satisfies Levy's criterion.

1. Introduction The basis of Stein's approach to the central limit theorem is the fact that the equation (1.1)

Ef'(X) = EXf(X)

characterizes the standard normal distribution N(0,1). More precisely, for a real-valued random variable X, if the above equality holds for all realvalued functions / such that both xf(x) and f'{x) are uniformly bounded, then X has the standard normal distribution: F{X <x} = - 1 = / X V27T J-oo 195

e-y2/2dy.

196

Elton P. Hsu

(1.1) is an integration by parts formula for the standard Gaussian measure /x because it moves the differentiation away from the function / . Let D

=7T' ax

D

* = ax ~T+X-

Then D* is the adjoint of D with respect to fi:

(Dg,f)fl = (g,D*f)^, and (1.1) can be simply written as MD*f(X) — 0. In this article, we consider an infinite dimensional extension of this characterization of the standard Gaussian distribution. Consider a real-valued Brownian motion W = {Wt, 0

~ *{Z))>

&{x) \ * ( z ) ( l - # ( x ) ) ,

X Z]

~

x>z.

It is easy to verify that both xf(x) and f'(x) are uniformly bounded. Using this / in Ef'(X) = EXf(X) we have immediately F{X Ez+ defined by

$(W) - {Xn, neZ+} = If

endWt, n e Z+j .

Here {e n , n & R+} is an orthonormal basis for L2[Q, 1], which we may take to be eo(t) = 1 and en(t) = v2cos7rnt,

n = l,2,....

200

Elton P. Hsu

Here we assume that W = {Wt, 0 < £ < l } i s a semimartingale so that the stochastic integration makes sense. Let

MO = / en(s)ds. Jo

The inverse

Stein's Method and Applications

Stein's method and applications

Delphi Method: Techniques and Applications

Die Macht des Steins

Die Macht des Steins

The method of equivalence and its applications

The method of equivalence and its applications

The State Space Method: Generalizations and Applications

The state space method: generalizations and applications

Applications of the Expansion Method

Dynamical Systems Method and Applications: Theoretical Developments and Numerical Examples

Dynamical Systems Method and Applications: Theoretical Developments and Numerical Examples

Stability by Liapunov's direct method with applications

Fast Multipole Boundary Element Method: Theory and Applications in Engineering

Normal Approximation by Stein's Method (Probability and Its Applications)

Lattice Boltzmann Method: Fundamentals and Engineering Applications with Computer Codes

The Cauchy method of residues: theory and applications

The Generalised Phase Contrast Method: Theory and Applications

The finite element method and applications in engineering using ANSYS

Ineffability, Ontology, and Method

Truth and Method

Truth and method

Descartes and Method

Externalism, Naturalism and Method

Truth And Method

Intentionality and Method

Science and Method

Science and Method

Beginnings: Intention and Method

Science and method (Nelson)

Stein's Method and Applications