Probability Measures on Semigroups: Convolution Products, Random Walks and Random Matrices

Probability and Its Applications Published in association with the Applied Probability Trust

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

For other titles published in this series, go to www.springer.com/series/1560

About the Authors: ˚ Akademi G¨oran H¨ogn¨as is Professor of Applied Mathematics at his alma mater Abo University, from which he received his Ph.D. in Mathematics in 1974. Professor H¨ogn¨as is the co-author of a text on applied mathematics as well as co-editor of the Proceedings of the Third Finnish-Soviet Syposium on Probability Theory and Mathematical Statistics. He serves on the editorial board of the Journal of Theoretical Probability. Professor H¨ogn¨as was the director of the Finnish Graduate School in Stochastics and Statistics (1998–2009). Arunava Mukherjea is Professor of Mathematics at the University of Texas–Pan American, and prior to 2007, was Professor at the University of South Florida. He received his Ph.D. in Mathematics in 1967 from Wayne State University. As visiting professor, he spent sojourns at various academic institutions including Tata Institute of Fundamental Research in Mumbai, India and Universite of Paul Sabatier in Toulouse, France. He was a Fulbright scholar. He is an associate editor (2005– present) of the Journal of Theoretical Probability, and was its editor-in-chief prior to 2005.

G¨oran H¨ogn¨as  Arunava Mukherjea

Probability Measures on Semigroups Convolution Products, Random Walks, and Random Matrices

ABC

G¨oran H¨ogn¨as ˚ Akademi University Abo F¨anriksgatan 3B ˚ 20500 Abo Finland [email protected]

Arunava Mukherjea University of Texas – Pan American Department of Mathematics 1201 West University Drive Edinburg, Texas 78539 USA [email protected]

Series Editors: Søren Asmussen Department of Mathematical Sciences Aarhus University Ny Munkegade 8000 Aarhus C Denmark [email protected] Joe Gani Centre for Mathematics and its Applications Mathematical Sciences Institute Australian National University Canberra, ACT 0200 Australia [email protected]

ISSN 1431-7028 ISBN 978-0-387-77547-0 DOI 10.1007/978-0-387-77548-7

Peter Jagers Mathematical Statistics Chalmers University of Technology and G¨oteborg (Gothenburg) University 412 96 G¨oteborg Sweden [email protected] Thomas G. Kurtz Department of Mathematics University of Wisconsin - Madison 480 Lincoln Drive Madison, WI 53706-1388 USA [email protected]

e-ISBN 978-0-387-77548-7

Library of Congress Control Number: 2010938438 c Springer Science+Business Media, LLC 2011  All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Cover design: WMXDesign Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

From the Preface to the First Edition

“A Scientific American article on chaos, see Crutchfield et al. (1986), illustrates a very persuasive example of recurrence. A painting of Henri Poincar´e, or rather a digitized version of it, is stretched and cut to produce a mildly distorted image of Poincar´e. The same procedure is applied to the distorted image and the process is repeated over and over again on the successively more and more blurred images. After a dozen repetitions nothing seems to be left of the original portrait. Miraculously, structured images appear briefly as we continue to apply the distortion procedure to successive images. After 241 iterations the original picture reappears, unchanged! Apparently the pixels of the Poincar´e portrait were moving about in accordance with a strictly deterministic rule. More importantly, the set of all pixels, the whole portrait, was transformed by the distortion mechanism. In this example the transformation seems to have been a reversible one since the original was faithfully recreated. It is not very farfetched to introduce a certain amount of randomness and irreversibility in the above example. Think of a random miscoloring of some pixels or of inadvertently giving a pixel the color of its neighbor. The methods in this book are geared towards being applicable to the asymptotics of such transformation processes. The transformations form a semigroup in a natural way; we want to investigate the long-term behavior of random elements of this semigroup. To be more specific, let us consider a sequence of independent and identically distributed random variables X0 ; X1 ; X2 ; : : : taking values in a set of affine maps from Rd into Rd , that is, maps of the form f .x/ D Ax C B, where B and x are d 1 column vectors and  A is a  d d real matrix. Since f can be identified with the AB .d C 1/  .d C 1/ matrix , the random variables Xi s can also be regarded as 0 1 .d C 1/  .d C 1/ random matrices; thus, , the distribution of X i , is aprobability AB measure on the set of .d C 1/  .d C 1/ matrices of the form . Let S be 0 1 the closed (with usual topology) multiplicative semigroup generated by the support of . Then the study of the random walks Yn ; Yn D X0 X1 : : : Xn with values in S and distribution n (the nth convolution power of ), and the set of recurrent states

v

vi

From the Preface to the First Edition

of .Yn / become relevant in the context of the so-called iterated function systems introduced by Barnsley and his colleagues [see Barnsley (1988)]. Let us briefly discuss another example. Suppose that we are monitoring a random system with two states denoted 0 and 1. Let 010001101001100010011001111000 and 101110010110011101111001111000 be observed time series of the successive states of the system. The observations seem rather like a record of independent coin tosses, with 0 for heads and 1 for tails, say. Viewed as a Markov chain on the two-state state space X D f0; 1g our process would have the transition probability matrix   1=2 1=2 P D : 1=2 1=2 Let us assume, however, that the above time series are concurrent. Then another interpretation imposes itself: the state space is subjected to a succession of random transformations. (The first two transformation are transpositions, 0 and 1 just trade places. At the third and fourth steps the identity map is at work. A sequence of transpositions and identities then follows, but at step 19 everything is mapped onto the state 1. From then on the two paths are identical.) The transformations are the four possible mappings of X into itself, the identity , the transposition , and the two constant mappings 0 and 1. The transition matrix P is then a convex combination of matrices representing those transformations:         1 0 0 1 1 0 0 1 P Da Cb Cc Cd ; 0 1 1 0 1 0 0 1 where a, b, c, d are nonnegative numbers with a C b C c C d D 1. Thus, a natural way to analyze our observed time series is to think of them as emanating from an independent, identically distributed sequence of mappings of the state space into itself, or, in other words, a random walk on the transformations of X . A Markov chain on a finite state space can always be regarded in this way. Its transition matrix P is a convex combination of 0-1 matrices representing mappings of the state space X into itself. (If P is doubly stochastic we can write it uniquely as a convex combination of permutation matrices, this is the celebrated Birkhoff theorem.) The corresponding result is true even for a large class of Markov chains on a topological space X , see Kifer (1986), Chapter 1. To consider an example in the context of particle systems, let V be an arbitrary countably infinite set (with discrete topology), and let  denote the semigroup of

From the Preface to the First Edition

vii

functions f W V ! V under composition. We can then identify each f in  with an infinite 0-1 stochastic matrix Af such that .Af /ij D 1

if and only if

f .i / D j:

By a configuration  of V , we mean a nonnegative integer valued function on V such that X .x/ < 1: x2V

The idea is that .j / is the number of particles that occupy the site j 2 V , and that when we apply the mapping f W V ! V , all these particles move to the site f .j /, and the configuration changes to   Af , where .  Af /.x/ D

X

.y/  ıf .y/ .x/:

y

The new configuration has at site x all the particles that the map f has sent to the site x from the sites of the original configuration. Thus, to study the random motions of finite systems of particles on V , without births or deaths, where each site may be occupied by a finite number of particles, and all particles at a particular site move together, one needs to study the random transformations F (that is, the infinite random stochastic matrices AF ). Instead of studying the different configurations, we study a sequence of independent identically distributed countably infinite stochastic matrices, and among other things, will be interested in gaining some insights in the limiting laws of products of these matrices. To mention yet another context where probability measures on countable semigroups have been found useful, we mention the paper of Hansel and Perrin (1983), where the authors utilized the structure of an idempotent probability measure on a semigroup in order to have some insights in certain problems in coding theory. It is also relevant to mention that Ruzsa (1994) utilized his results on weak convergence of the sequence 1  2  : : :  n , where the i s are probability measures on a countable semigroup, in proving a generalization of a result in number theory due to Davenport and Erd¨os (1936). This last mentioned result simply says that every multiplicative ideal A of the set N of positive integers has a logarithmic density, that is, X 1 1 .A/ D lim ; n!1 log n a a2A an exists. Note that for a set A  N , its logarithmic density  may exist while its asymptotic density d , given by 1 X 1; n!1 n a2A an

d.A/ D lim

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From the Preface to the First Edition

may not exist. [It is well known, however, that .A/ exists whenever d.A/ does, and then .A/ D d.A/.] Ruzsa’s result says the following: if f is a homomorphism from the multiplicative semigroup of integers to a commutative semigroup H , then for every h 2 H , the set fn 2 N W f .n/ D hg has a logarithmic density. Let us finally mention, before we go to the text proper, that abstract semigroup theory was of crucial importance in developing the methods used in H¨ogn¨as and Mukherjea (1980) to study the set of recurrent states of a random walk taking values in n  n real matrices.”

References Barnsley, M. F., Fractals Everywhere, Academic Press, Orlando (1988). Crutchfield, J. P., J. D. Farmer, N. H. Packard, and R. S. Shaw, “Chaos,” Scientific American 255, No. 6, 38–49 (1986). Gelbaum, B. R. and G. K. Kalisch, “Measures in Semigroups,” Canadian J. Math. 4, 396–406 (1952). Hansel, G. and D. Perrin, “Codes and Bernoulli partitions,” Math Systems Theory 16, 133–157 (1983). Hennion, H., “Limit theorems for products of positive random matrices,” The Annals of Probability 25, No. 4, 1545–1587 (1997). H¨ogn¨as, G. and A. Mukherjea, “Recurrent random walks and invariant measures on semigroups of n  n matrices,” Math Zeitschrift 173, 69–94 (1980). Kifer, Y., Ergodic Theory of Random Transformations, Birkh¨auser, Boston–Basel–Stuttgart (1986). Raugi, A., “Une demonstration du th´eoreme de Choquet-Deny par les martingales,” Ann. Inst. Henri Poincar´e Statist. Probab. 19, 101–109 (1983). Ruzsa, I. Z., Logarithmic density and measures on semigroups, Manuscripta Mathematica 89, 307– 317 (1996).

Preface to the Second Edition

In this new edition, we have corrected all errors (mathematical and otherwise) that we could find or were brought to our attention from the book’s first edition by our colleagues and students. We have also updated the references, all notes and comments at the end of each chapter, and more importantly, added exercises at the end of each section in each chapter. We have also added new results often in the main text and in the appendices. The book, we feel, can be a useful reference for courses in the area of probability on algebraic structures. It can also serve as a text for graduate or senior undergraduate students for an one semester course on “Probability measures on semigroups” designed around the following four core topics: (i) Completely simple semigroups: their structure and the Rees-Suschkewitsch theorem (see Chapter 1) (ii) Convolution products of probability measures on locally compact semigroups, their weak convergence, and their weak limits which are idempotent probability measures, with supports always closed completely simple subsemigroups, and structure theorem for idempotent probability measures (see Chapter 2) (iii) Recurrent random walks on semigroups: characterization of the set of recurrent states as closed and completely simple subsemigroups (see Chapter 3) (iv) Tightness and convergence in distribution of products of d by d i.i.d. random nonnegative matrices: The Kesten-Spitzer theory and various generalizations to random real matrices (see Chapter 4) The first chapter of the book can also serve as a text for an one semester course on Semigroups for senior undergraduate or graduate students. Sufficiently many exercises of varying levels of difficulty have been included to help the instructor of such a course. Semigroups are very general structures and we often come across them in various contexts in science and engineering. (See the preface to the first edition.) The results that we have presented here on weak convergence or random walks or random matrices using semigroup ideas are for the most part complete and best possible. Still

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Preface to the Second Edition

there are other results that remain to be completed. These are all mentioned in the notes and comments at the end of each chapter, and this, we hope, will keep the readership of this book enthusiastic and interested for some time to come. ˚ Akademi University, Abo, ˚ Abo Finland UTPA, Edinburg, Texas 78539, USA August 30, 2010

G¨oran H¨ogn¨as Arunava Mukherjea

Acknowledgements ˚ We want to thank Abo Akademi University student Susanna Nyberg for technical assistance and help with the exercises in Chap. 1. Also we are much indebted to student Andreas Anckar who made the LaTeX compilations for us in a very competent way. Let us also thank many of our colleagues and friends for assisting us in various ways in the completion of the present edition. We are specially indebted to Greg Budzban, Edgardo Cureg, S.G. Dani, Phil Feinsilver, Yves Guivarc’h, Herbert Heyer, Karl Hofmann, B.V. Rao, Ricardo Restrepo, Todd Retzlaff, M. Rosenblatt, Imre Ruzsa, T.C. Sun and Nicolas Tserpes. Last but not least, we must thank Vaishali Damle (the Springer editor) who very patiently and efficiently helped us through different stages of the production of this book.

Contents

1

Semigroups . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 1.2 Homomorphisms, Quotients, and Products .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . 1.3 Semigroups with Zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 1.4 The Rees–Suschkewitsch Representation Theorem .. . .. . . . . . . . . . . . . . . . . 1.5 Topological Semigroups .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 1.6 Semigroups of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 1.7 Semigroups of Infinite Dimensional Matrices . . . . . . . . . .. . . . . . . . . . . . . . . . . 1.8 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .

1 1 6 10 12 22 33 53 60

2 Probability Measures on Topological Semigroups . . . . . . . .. . . . . . . . . . . . . . . . . 63 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 63 2.2 Invariant and Idempotent Probability Measures .. . . . . . .. . . . . . . . . . . . . . . . . 64 2.3 Weak Convergence of Convolution Products of Probability Measures 83 2.4 Weak Convergence of Convolution Products of Nonidentical Probability Measures . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .137 2.5 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .168 3 Random Walks on Semigroups .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .171 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .171 3.2 Discrete Semigroups .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .180 3.3 Locally Compact Groups .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .198 3.4 Compact Semigroups .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .219 3.5 Completely Simple Semigroups . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .241 3.6 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .247 4 Random Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .253 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .253 4.2 Recurrent Random Walks in Nonnegative Matrices . . .. . . . . . . . . . . . . . . . .254 4.3 Tightness of Products of I.I.D. Random Matrices: Weak Convergence 284 4.4 Invariant Measures for Random Walks in Nonnegative Matrices: Laws of Large Numbers .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .333 4.5 Notes and Commments .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .355 xi

xii

Contents

A Products of I.I.D. Random Stochastic Matrices: Their Skeletons and Convergence in Distribution . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .359 B An Example Due to Chamayou and Letac . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .365 C Asymptotic Behavior of kXn Xn1 : : : X0 uk for I.I.D. Random Nonnegative Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .369 D Continuous Singularity of the Limit Distribution of Products of I.I.D. Stochastic Matrices . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .383 E Rate of Decay of Concentration Functions on Discrete Groups, by T.M. Retzlaff .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .397 E.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .397 E.2 Irreducible Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .404 E.3 Adapted Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .408 References .. . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .413 Index . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .423

Chapter 1

Semigroups

1.1 Introduction Chapter 1 contains the basics of semigroups: definitions, elementary concepts, and fundamental examples. We assume some familiarity with standard notions of point-set topology (see [120, 163]); the algebraic portions of Chap. 1 are, however, completely self-contained. Without going into any details whatsoever, it is perhaps prudent to remark at this point that our main interest centers around asymptotics, invariance questions, etc. Our treatment is a reflection of this. We concentrate on algebraic concepts corresponding to such phenomena as absorption, stability, and invariance: zeros, simple semigroups, minimal ideals, maximal subgroups, and so on. We strive to keep the digressions at a minimum. Clifford and Preston [43] offers a wealth of information on all aspects of algebraic semigroups, and this text is recommended to any reader interested in a much more elaborate treatment of this fascinating subject. Sections 1.1–1.5 contain basic material necessary for the development of all subsequent chapters, while Sects. 1.6 and 1.7, which deal with more specific applications, can be skipped at first reading. Arguably, the most important notion in mathematics is that of a mapping (or function or transformation). The ultimate goal of the research presented in this book is to describe the long-term behavior of random transformations of some set. Transformations of a set form a semigroup in a natural way. Indeed, we see that any semigroup is (algebraically) a transformation semigroup in a canonical way. Linear transformations of a vector space form another family of fundamental examples. We devote considerable effort to those semigroups, which incidentally may just as well be viewed as semigroups of matrices. Let S be a set. If S is endowed with an associative binary operation [which we call multiplication and denote by a dot, ./ or simply by juxtaposition], then the S is called a semigroup. Strictly speaking, the semigroup is the pair .S; /, but the intended operation is usually quite clear from the context. When we are dealing with a specific application, we of course use the established notation. If s is an element of a semigroup S and A and B are subsets of S , then we denote by sA the set fsa 2 S ja 2 Ag and by AB the set fab 2 S ja 2 A; b 2 Bg.

G. H¨ogn¨as and A. Mukherjea, Probability Measures on Semigroups: Convolution Products, Random Walks and Random Matrices, Probability and Its Applications, c Springer Science+Business Media, LLC 2011 DOI 10.1007/978-0-387-77548-7 1, 

1

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1 Semigroups

(As is, of course, defined analogously.) Note that products with more than two factors, such as abc and aBcS , are well defined due to the associativity of the multiplication. aa; aaa; : : : are usually written a2 ; a3 ; : : : A nonempty subset T of S is called a subsemigroup if it is stable under multiplication, i.e., if T T  T . If T is also a group, we call it a subgroup of S . As we see later, it is important in many applications to identify the subgroups of a given semigroup S . A subsemigroup L of S is called a left ideal, if SL  L; right ideals are similarly defined. A nonempty subsemigroup I is a bilateral or two-sided ideal or just ideal of S if it is both a right and a left ideal: SI  I; IS  I . S is said to be left (right) simple if it contains no proper left (right) ideal. Similarly, S is simple if the only ideal of S is S itself. A left (right) ideal is a principal left (right) ideal if it is of the form fag [ Sa (fag [ aS ) for some a 2 S . Note that S is left simple if and only if for any given a; b 2 S , the equation xa D b is soluble. (Sa is a left ideal of S for all a 2 S . On the other hand, any left ideal L contains a subset of the form Sa.) An element e 2 S is called a left (right) identity element of S if es D s .se D s/ for every s 2 S . e is a two-sided identity element of S , or simply identity of S , if it is both a right and a left identity of S . It is easy to see that the identity is unique if it exists. An element z of S is called a left (right) zero element of S if zs D z .sz D z/ for all s 2 S . If z is both a left zero and a right zero of S , we simply call it a zero of S . A semigroup has at most one zero. The semigroup S is said be left (right) cancellative if for any a 2 S , the equation ax D ay .xa D ya/ in S implies x D y. An element a 2 S is idempotent if a2 D a. Zeros and identities are idempotent. An idempotent is in a trivial fashion the identity element of a subgroup of S . Elements a and b of S are said to commute if ab D ba. If all elements of S commute with each other, S is called a commutative or abelian semigroup. In abelian semigroups, the operation is often called addition and denoted by C, the identity element by 0 and the inverse of a by a. To put the preceding concepts into perspective, we now investigate some of their relationships to a group structure. Most textbooks define a group as a nonempty set G with an associative binary operation with identity element e and inverses; i.e., for all a 2 G, there is a b 2 G such that ab D ba D e, (see, for example, [62] or [217]). There is, however, a multitude of alternative, seemingly weaker, but in fact equivalent definitions, see [43], Chap. 1. In our context, Proposition 1.1 is a convenient characterization of a group. Proposition 1.1. A semigroup is a group if and only if it is both left and right simple. Proof. A group is clearly both left and right simple. Conversely, let S be a semigroup which is both left and right simple. For any a 2 S , the equation ae D a has a solution. On the other hand, Sa is all of S ; hence, e is a right identity of S . In the same way, we can produce a left identity f which turns out to be equal to e. (We have f D f e D e.)

1.1 Introduction

3

It is just as straightforward to obtain the two-sided inverse for an element a 2 S . Let b and c be the solutions of the equations ab D e and ca D e, respectively. Then b D eb D .ca/b D c.ab/ D ce D c: Thus, S is a group. t u If A is a subsemigroup of S we denote by hAi all the elements in S that can be expressed as finite products of the elements in A. In the case hAi D S , we call A the generator of S . If a semigroup is generated by a single element, we call it monothetic or cyclic. Remark 1.1. In a finite semigroup S , hxi contains a subgroup for every x 2 S , cf. Exercise 1.6. Example 1.1 (The semigroup of transformations of a set). Let X be any finite or infinite nonempty set. If f and g are mappings from X to itself, we define as usual the composition f ı g of f and g by .f ı g/.x/ D f .g.x//; x 2 X . (The domain of the mappings is always understood to be all of X .) The composition of mappings is an associative operation; hence, the set of all mappings from X into X forms a semigroup with composition as multiplication. This semigroup is called the full transformation semigroup on X and denoted by TX . A more complete treatment of TX is given in [43], Chap. 2.2. TX has an identity element, the identity mapping : .x/ D x; x 2 X . The constant mappings are left zeros, since c ı f D c for all f if c.x/ D x0 ; x 2 X (where x0 is a particular element of X ). On the other hand, TX does not admit any right zeros unless of course X is a singleton. Define the range R.f / of an f 2 TX to be the set f .X /  ff .x/ j x 2 X g (where  means equal by definition). Clearly, R.f ı g/  R.f /. If R0 is a subset of X , then mappings with range inside R0 , ff 2 TX jR.f /  R0 g, form a right ideal of TX . The partition .f / of X generated by an element f 2 TX is the equivalence relation on X defined by x.f /y ” f .x/ D f .y/; x; y 2 X . In other words, elements are .f /equivalent if and only if they have the same image under f . The .g/-equivalence implies .f ı g/- equivalence, .f ı g/  .g/. Consequently, if 0 is a given equivalence relation on X , then those f 2 TX with .f /  0 form a left ideal of TX . Define the rank of f to be the cardinality of R.f /, which we denote by jR.f /j. Note that the cardinality of R.f / is the same as that of the quotient space X=.f / [the number of .f /-equivalence classes]. Mappings with rank no larger than a given cardinality r, ff 2 TX j jR.f /j  rg, form a two-sided ideal of the semigroup TX . The constant mappings, i.e., the mappings of rank 1, form the minimal two-sided ideal of TX . This subsemigroup is a left zero semigroup, since all its elements are left zeros. Let us now identify the subgroups of TX . Since any subgroup has an idempotent as its identity element, our first task is to determine idempotents in TX . Let e be idempotent with range R and partition , whence e.x/ D x; x 2 R. As before jRj D jX=j. This is possible if and only if R is a complete set of representatives of

4

1 Semigroups

the -equivalence classes, i.e., every equivalence class contains exactly one element of R. In the terminology of Clifford and Preston [43] R is a cross section of . Suppose f belongs to a subgroup G of TX and the identity element of G is the idempotent e previously discussed. We can immediately conclude from the relation e ı f D f ı e D f that the range of f is R and the partition corresponding to f is . Furthermore, f is a one-to-one mapping from R to R precisely because R is a cross section of . We can then construct a g belonging to TX with the following properties: g has range R and partition , and the restriction of g to R is the inverse of the restriction of f to R. It is then clear that g ı f D f ı g D e, in other words, the inverse of f in the subgroup G is g. Our result is thus the following, see [43], Theorem 2.10: f belongs to a subgroup of TX if and only if R.f / is a complete set of representatives of .f /. The sets ff jR.f / D R; .f / D g where jRj D jX=j are groups if and only if R is a complete set of representatives of . If there is an f whose range is not a cross section of .f /, then f ı f has a smaller range than f : R.f ı f / is a proper subset of R.f /. Clearly, such an f cannot belong to a group. The preceding discussion holds almost verbatim for any transformation semigroup on X ; i.e., any subsemigroup S of TX . The conditions are necessary only in the general case. For example, f 2 S can belong to a subgroup of S only if R.f / is a cross section of .f /. For a transformation semigroup S on a finite set X we have the converse: If elements of a subsemigroup of S have common range R and partition , where R is a cross section of , then it is a group. (Each element f is of finite order, i.e., some power of f equals the identity mapping e on R with the given partition .) Let X be countably infinite and G a subgroup of infinite rank of TX . If e is the identity element of G, then we can construct a mapping ˛ with the properties that ˛ restricted to the range of e is a bijection onto all of X (practically the definition of an infinite subset of X ) and ˛ has the same partition as e. If we denote the inverse of ˛ restricted to the range of e by ˇ, then ˇ is injective, and has the same range as e. Hence, ˛ ı ˇ D  (the identity mapping on X ) and ˇ ı ˛ D e. For any g 2 G, ˛ ı g ı ˇ is a bijection on X . Conversely, the elements of G can be written ˇ ı h ı ˛, where the h’s are bijections on X . The semigroup BX of relations on the set X consists of all subsets of the Cartesian product X  X equipped with the composition operation ı: For any two  and  in X  X define .x; y/ 2  ı  if and only if there is a z 2 X such that .x; z/ 2  and .z; y/ 2 .

Section 1.1 Exercises Exercise 1.1. Show that a semigroup with more than one element contains a subsemigroup S 0 ¤ S .

1.1 Introduction

5

Exercise 1.2. Prove that a finite semigroup that is cancellative is in fact a group. Exercise 1.3. Show that if a semigroup S contains a left identity element e, such that there for any x 2 S exists a y 2 S such that yx D e, then S is a group. Exercise 1.4. Prove that the following are groups: (i) A left simple semigroup that contains a left identity. (ii) A cancellative and simple semigroup containing an idempotent Exercise 1.5. Let S be a semigroup so that for every x 2 S there exists an a 2 S such that x D xax. Show that the following are equivalent: (a) S has exactly one idempotent (b) S is cancellative (c) S is a group Exercise 1.6. Let S be a finite semigroup generated by a single element x. Prove that there exists m; r 2 N, such that x mCr D x r and hxi D fx; x 2 ; : : : ; x mCr1 g: Also show that the set fx r ; x rC1 ; : : : ; x mCr1 g is a subgroup of S . Exercise 1.7. Let S be a finite commutative semigroup. Show that S can be partitioned into maximal subsemigroups fP .e.i //ji D 1; 2; : : : rg, where fe.1/; e.2/; : : : e.r/g is the set of all idempotents of S , and x is in P .e.i // iff x k D e.i / for some positive integer k. Further, show that there are smallest positive integers K and D such that for all x in S , x K D x KCD . Exercise 1.8. Verify that the definition of the operation ı defined above on the semigroup BX of relations on the set X coincides with the composition on TX defined in Example 1.1 in case the relations are transformations of X . (f 2 TX can be viewed as a relation f 2 BX when we identify f with its graph, a subset of X  X : .x; y/ 2 f if and only if y D f .x/.) Show that if a relation  is reflexive and transitive, then it is idempotent, i.e.,  ı  D . Exercise 1.9. A d  d real matrix is called circulant and denoted by circ(x.0/; x.1/; : : : ; x.d  1/) if its first row is x.0/; x.1/; : : : ; x.d  1/, its second row x.d  1/; x.0/; x.1/; : : : x.d  2/, the third row x.d  2/; x.d  1/; x.0/; : : : ; x.d  3/ and so on: 0

x.0/ Bx.d  1/ B @ ::: x.1/

1 x.1/ x.2/ : : : x.d  1/ x.0/ x.1/ : : : x.d  2/C C: ::: ::: ::: ::: A x.2/ x.3/ : : : x.0/

Prove that the circulant matrices form a commutative semigroup with respect to matrix multiplication.

6

1 Semigroups

Exercise 1.10. Let x be a d d real matrix. Show that the following are equivalent: (i) (ii) (iii) (iv)

x is circulant; x T , the transpose of x, is circulant; xP D P x, where P is the permutation matrix circ(0; 1; 0; : : : ; 0); x D f .P /, where f is a polynomial of order less than d .

1.2 Homomorphisms, Quotients, and Products A mapping  between two semigroups .S; / and .T; / is called a semigroup homomorphism (antihomomorphism) if .a  b/ D .a/ .b/ ..a  b/ D .b/ .a//; a; b 2 S: The  is said to be a semigroup isomorphism (anti-isomorphism) if it is bijective (i.e., onto and one-to-one) as well. We however usually suppress the explicit reference to semigroups when it is clear from the context that we are dealing with a semigroup structure. The S and T are isomorphic (as semigroups) if there exists a semigroup isomorphism between them. Using the antiisomorphism a  b 7! b a we can always convert an antihomomorphism to a homomorphism, if need be. As an example, consider the set of left translations on a given semigroup S : a ; a 2 S , where a is defined by a .x/ D ax; x 2 S . Clearly, ab D a ı b . The  is thus a homomorphism from S to the full transformation semigroup on S , TS . Right translations define in a similar way a semigroup antihomomorphism from S to TS . If in addition ax D bx for all x 2 S implies a D b (the left translations act effectively on S), then  is injective and S is isomorphic to a subsemigroup of TS . In particular this is the case when S has a right identity element. If we extend the idea of left translations slightly, we obtain the useful result that any semigroup S is isomorphic to some transformation semigroup. We need only take X D S [ 1 and define a1 D a; a 2 S . Left translations a ; a 2 S on X define an injective homomorphism from S into TX . An equivalence relation  on a semigroup S compatible with the multiplication is called a congruence on S . More formally,  is a congruence if for all a; b; s 2 S , ab implies asbs and sasb. If  is a congruence on S , then the multiplication of equivalence classes in the natural way will be a well-defined operation on S=. (For a; b 2 S , we take Œa Œb D Œab , where Œa is the equivalence class containing a). The semigroup thus obtained is called the quotient or factor semigroup of S mod . The discussion in the preceding paragraphs shows that if  is the congruence on S defined by ab ” ax D bx for all x 2 S; then S= is isomorphic to a subsemigroup of TS . Definition 1.1 presents another useful congruence in semigroup theory.

1.2 Homomorphisms, Quotients, and Products

7

Definition 1.1. Let I be an ideal of S . If we define a relation  by ab ” a D b or else both a and b 2 I; then  is a congruence on S . The corresponding factor semigroup is usually written S=I , and it is called the Rees factor (or quotient) semigroup of S mod I . The intuitive idea behind the Rees factor semigroup is to lump all elements of I together into a single zero. Take a semigroup S and view it as transformation semigroup on X . For f 2 S , define a matrix Bf .x; y/ indexed by X according to the following prescription: Bf .x; y/ D ıf .y/ x, i.e., Bf .x; y/ D 1 if x D f .y/ and 0 otherwise, x; y 2 X . Multiplying the matrices according to the usual rules of matrix multiplication, we see that indeed Bf ıg D Bf Bg , [see [53] where the matrices Af have the antihomomorphism property instead]. Hence, any semigroup can be described as a semigroup of 0–1 matrices, transformation matrices, with exactly one 1 in each column. If there is exactly one 1 in each row as well, we obtain the familiar permutation matrices corresponding to bijections on the set X . For semigroups .S; / and .T; /, we obtain a new structure on their cartesian product by the rule .s; t/ ? .s 0 ; t 0 / D .s  s 0 ; t t 0 /: The resulting semigroup .S  T; ?/ is called the direct product of .S; / and .T; /. Direct products with several factors are defined analogously. Let G be a group and E be a right zero semigroup (where ee 0 D e 0 ; e; e 0 2 E). Consider the direct product of G and E. Multiplication in G  E is given by .g; e/.g 0 ; e 0 / D .gg 0 ; e 0 /. The usefulness of this structure is due to the fact that any right group has this representation. Definition 1.2. A right group (left group) is a semigroup that is right simple (left simple) and left cancellative (right cancellative). Alternative characterizations are given in Proposition 1.2 Proposition 1.2. For a semigroup S , the following statements are equivalent: (i) (ii) (iii) (iv)

S is a right group; For any a; b 2 S , the equation ax D b has one and only one solution; S is right simple and contains an idempotent; S is isomorphic to the direct product of a group G and a right zero semigroup E.

Proof. The equivalence of (i) and (ii) follows immediately from the definitions and the characterization of right simplicity in Sect. 1.1. Let S satisfy (ii). Then for any a 2 S the equation ax D a has a solution e, say. We have aee D ae; by left cancellativity, ee D e: Thus (ii) implies (iii).

8

1 Semigroups

Assume (iii) and let ax D ay for some a; x; y 2 S . There is an idempotent e in S , e is a left identity of S D eS . There is a b 2 S such that ab D e. .ba/.ba/ D b.ab/a D bea D ba, so ba is an idempotent too and consequently a left identity of S . We finally obtain ax D ay H) bax D bay H) x D y; i.e., (i) holds. The equation .g; e/.x; y/ D .g 0 ; e 0 / in the direct product G  E has the unique solution .x; y/ D .g 1 g 0 ; e 0 /. This shows that (iv) implies (ii). Suppose now that S satisfies (iii), and, equivalently, (i) and (ii). Let E be the set of idempotents in S . We saw above that any idempotent is a left identity, so ee 0 D e 0 ; e; e 0 2 E. In other words, E is a right zero (sub)semigroup (of S ). Take an e 2 E. We show next that Se is a subgroup of S . Se is right simple: seSe D Se. Se is also left simple. To see this, take an element se 2 Se and let t 2 S be such that set D e. Then se.te/ D .set/e D ee D e and se.tese/ D .sete/se D ese D see (since e is a left identity of S ), so that tese D e by left cancellability. For an arbitrary ue 2 Se; the equation .xe/.se/ D ue can be solved, namely by x D uet 2 S . By Proposition 1.1, Se is a group. Take a particular idempotent e0 and let Se0 D G. Consider the map .g; e/ 7! ge; g 2 G; e 2 E: Call it . We prove that  is the desired isomorphism. The map  is a homomorphism: For g; h 2 G and e; f 2 E, gehf D ghf because each idempotent is a left identity.  is injective. To see this, let ge D hf . Then g D ge0 D g.ee0 / D .ge/e0 D .hf /e0 D h.f e0 / D he0 D h. Left cancellativity then yields e D f .  is surjective, since any a 2 S can be written in the form ae for some idempotent e [see the proof of (ii) H) (iii) above]. e0 is a left identity of S , so a D a.e0 e/ D .ae0 /e. t u To understand right groups concretely, let us look at the full transformation semigroup TX and its subsemigroups. The possible idempotents and subgroups were characterized in Sect. 1.1. Note that the subgroups consist of bijections (permutations) of the set R. It is not difficult to see that subgroups are isomorphic to a permutation group on R (i.e., a subgroup of the symmetric group GR on R consisting of all the bijections from R to R). It is evident that a right group S  TX consists of mappings with a common range R0 ; otherwise, it could not possibly be right simple. It follows from the assumption that S is closed under multiplication that the common range R0 is a cross section of all partitions generated by elements of S . Indeed, we obtain Proposition 1.3 in the case of a finite X : Proposition 1.3. A subsemigroup S of TX is a right group if and only if the elements of S have common range. Proof. The only if statement is immediate, so let us concentrate on the sufficiency, the if statement. Any f 2 S is bijective on the common range, so left cancellativity follows. Since X is finite, the powers f n ; n D 1; 2; 3 : : : form a group whose identity is a left identity of S . The equation f ı g D h thus has a solution g D f r1 ı h (where r is the order of the group generated by f ). In other words, S is right simple and hence a right group. t u

1.2 Homomorphisms, Quotients, and Products

9

To see how the concepts work for countably infinite X , let us return to the last paragraph of Example 1.1. If H is a subgroup of the symmetric group GX , then ˇ ı H ı ˛ is an infinite-rank subgroup of TX . Conversely, any such subgroup is isomorphic to ˇ ı H ı ˛ for some H; ˛, and ˇ. Right groups are then obtained by varying the partition in the construction of ˛: Any right group of infinite rank is isomorphic to ˇ ı H ı E, where E is some set of mappings constructed exactly as ˛ but with the partition varying; of course, the fundamental property of the range as a cross section of the partition must be maintained. In particular for an ˛ 0 2 E, ˛ 0 ı ˇ D  and ˇ ı ˛ 0 is the identity of some group of mappings with the same range as e. Definition 1.3. We conclude this section by introducing another important product structure, the Rees product. Let E be a left zero semigroup, F a right zero semigroup, G a group, and  a function from F  E to G. Define a multiplication on E  G  F by .e; g; f /.e 0 ; g 0 ; f 0 / D .e; g.f; e 0 /g 0 ; f 0 /: Note that this product is direct if the sandwich function  maps everything onto the identity element of the group G. Such a  is termed trivial. We emphasize at this point that  is a completely arbitrary function from F  E to G. Different choices of  may produce isomorphic semigroups. If, for example,  maps everything onto a constant c 2 G, then the resulting Rees product is isomorphic to the direct product of E; G; and F . We will return briefly to this question in Sect. 1.4 (Proposition 1.10). Remark 1.2. The cylinder subsets of the form feg  G  ff g , called cells, are all groups isomorphic to G. The identity of such a group is the element .e; ..f; e//1 ; f /. Any subsemigroup of the form feg  G  B (where B  F ) of the Rees product is a right group. This fact is an immediate consequence of Definition 1.2.

Section 1.2 Exercises Exercise 1.11. Let  W S ! T be a homomorphism, where S and T are semigroups. Show that  maps subsemigroups of S to subsemigroups of T , and also idempotents of S to idempotents of T . Show further that if  is onto, then right ideals of S are mapped to right ideals of T , and the identity of S to the identity of T . Exercise 1.12. Let S be a semigroup and T a subsemigroup of S , such that xT D T x for every x 2 S . Prove that , defined by ab , aT D bT , is a congruence. Exercise 1.13. Prove that every equivalence relation on a semigroup of left zeros is a congruence.

10

1 Semigroups

Exercise 1.14. Let S be a semigroup and let  be a congruence on S . Prove that if e is an idempotent, e 2 S , then its equivalence class e= is a subsemigroup of S , and is an idempotent in the quotient S=. Also, prove that if S is finite and x= is an idempotent of S=, then x= contains an idempotent. Exercise 1.15. Prove that the following are equivalent: (i) S is a right group (ii) There exists a right identity element e 2 S such that e 2 aS for every a 2 S . (iii) For each element a 2 S , Sa contains a left identity element of S . Exercise 1.16. Show that if X D f1; 2; 3; 4g and R D f1; 2g, then the semigroup S of all maps from X into X with range R is a right group, but not a group. Find all the eight elements of S and the right group decomposition as the direct product of a group and a right zero semigroup.

1.3 Semigroups with Zero Recall that an element z of a semigroup S is a zero if sz D zs D z for all s 2 S . A zero is unique if it exists. We henceforth adhere to the common convention and denote a zero by 0. The notions of (right, left) simplicity are trivial in the presence of a zero 0. For instance, S is simple if and only if S D f0g: For semigroups with zero, it is therefore useful for many purposes to restrict some of the definitions to nonzero elements of S . It is often practical to have a special notation for these elements; A is the generic notation for the nonzero elements of the set A A  A n f0g  fa 2 Aja ¤ 0g: An ideal I ¤ f0g of a semigroup S is said to be 0-minimal if f0g is the only ideal of S properly contained in I . A 0-minimal left (right) ideal is defined analogously. A semigroup S is called right (left) 0-simple if S 2 ¤ f0g and its only right (left) ideals are 0 and S itself. S is called 0-simple if S 2 ¤ f0g and 0 is the only proper two-sided ideal of S . A semigroup S with the property that all products are 0, S 2 D f0g, is called a null semigroup. Such a semigroup obviously satisfies the second condition of the preceding definitions. Elements (nonzero elements) a and b of a semigroup with 0 are called divisors of zero (proper divisors of zero) if ab D 0. Propositions 1.1 and 1.2 have their counterparts for semigroups with 0, which follow immediately from the original Propositions and Lemma 1.1. Proposition 1.4. A semigroup with 0 is a group with 0 if and only if it is both left and right 0-simple.

1.3 Semigroups with Zero

11

Proposition 1.5. For a semigroup S with 0, the following statements are equivalent: (i) (ii) (iii) (iv)

S  is a right group; For any a 2 S  ; b 2 S the equation ax D b has one and only one solution; S is right 0-simple and contains a nonzero idempotent; S is isomorphic to the Rees quotient of the direct product of a group with zero G 0 and a right zero semigroup E modulo E  f0g.

Lemma 1.1. A right (left) 0-simple semigroup has no proper divisors of zero. Proof. Let S be a right 0-simple semigroup and assume that a and b are proper divisors of zero. For a given a, solutions of the equation ax D 0 form a right ideal of S . Consequently by our assumptions on S , the existence of one nonzero solution b implies that all elements of S are solutions to ax D 0; i.e., aS D f0g. Those as with this property also form a right ideal of S . Again that right ideal has to be all of S , implying S S D f0g, which contradicts the assumption of right 0-simplicity. u t Definition 1.4. A product structure analogous to the Rees product in Sect. 1.2 is the Rees product over a group with zero which is defined as follows: Let G 0 be a group with a zero 0 adjoined. Let E and F be as in the definition in Sect. 1.2.  denotes a map from F  E to G 0 . Form the product E  G 0  F with the same multiplication rule as before .e; g; f /.e 0 ; g 0 ; f 0 / D .e; g.f; e 0 /g 0 ; f 0 /; e; e 0 2 E; g; g 0 2 G 0 ; f; f 0 2 F: This defines a semigroup. The set I  E  f0g  F consisting of triples with zero middle factor is a two-sided ideal of the semigroup. The Rees product over G 0 is then obtained by collapsing all of I into a zero. More precisely, the Rees product over the group with zero is the Rees quotient E  G 0  F mod I: The cells feg  G  ff g are again groups with identity .e; ..f; e//1 ; f / provided the middle term exists, i.e., .f; e/ ¤ 0. These cells are called group cells. Null cells are characterized by .f; e/ D 0. The null cells (with 0 adjoined) are null subsemigroups of the Rees product over G 0 . The sandwich function  is said to be regular if the mappings .f; / and .; e/ are not identically 0 for any f 2 F or e 2 E. In other words, for each f 2 F there is an e 2 E (and for each e 2 E there is an f 2 F ) such that .f; e/ ¤ 0. If  is pictured as a matrix indexed by F  E, then it is regular if and only if there are no zero rows or zero columns. Remark 1.3. The condition that  be regular is necessary and sufficient for the Rees product over G 0 to be 0-simple: Take elements .e; g; f /; .e 0 ; g 0 ; f 0 / 2 E  G  F . The equation .x; y; z/.e; g; f /.x 0 ; y 0 ; z0 / D .e 0 ; g 0 ; f 0 /

12

1 Semigroups

can be solved if and only if .z; e/ and .f; x 0 / can be chosen to be nonzero. In that case, a solution is provided by x D e 0 ; y D ..z; e/g.f; x 0 //1 g; z0 D f 0 ; and y 0 the identity of G. In contrast to the case without 0, sets of the form feg  G 0  B (where B  F ) are not necessarily right 0-simple unless B is all of F . In general semigroup theory, a semigroup S is regular if a 2 aSa for all a 2 S . This notion and the regularity of the sandwich function just discussed are consistent in the following sense: The Rees product over a group with 0 is regular as a semigroup if and only if its sandwich function is regular. To see this, consider the equation .e; g; f / D .e; g; f /.x; y; z/.e; g; f /, which can be solved provided the functions .f; / and .; e/ are not identically 0.

Section 1.3 Exercises Exercise 1.17. Let S be a semigroup with zero 0, and let S ¤ 0. Prove that S is 0-simple if and only if S xS D S for every x ¤ 0 of S . Exercise 1.18. Let S be a 0-simple semigroup, and let for every x ¤ 0 in S , x n ¤ 0 where n D 1; 2 : : :. Show that S contains no proper divisors of zero. Exercise 1.19. Let M be a 0-minimal ideal of a semigroup S with zero. Prove that either M D 0 or M is a 0-simple subsemigroup of S . Exercise 1.20. Define Jk as the set of elements of rank at most k for a transformation semigroup on a finite set X . Let S be a subsemigroup of Tn . Identify the divisors of zero in S=Jk . Prove that Jn =Jn1 does not contain any divisors of zero. If S D Tn , show that Jk =Jk1 has divisors of zero for 2  k < n.

1.4 The Rees–Suschkewitsch Representation Theorem The Rees–Suschkewitsch representation theorem comes in two versions or even four if we take into account the topological considerations to be presented in Sect. 1.5. In Sect. 1.4, we discuss the algebraic case only for semigroups with and without 0. Needless to say, our strategy is dictated by applications where both versions appear naturally. We will begin by studying the case without 0 in some detail. As seen in Chap. 4, this case is more prevalent in the applications. We only outline proofs of results in the latter part of Sect. 1.4, where the semigroups with 0 are treated. However, we point out the main pitfalls in going from one theory to the other. Definition 1.5. An idempotent in S is said to be a primitive idempotent if it is minimal with respect to the partial order  on the set E.S / of idempotents of S defined by e  f ” ef D f e D e

.e; f 2 E.S //:

1.4 The Rees–Suschkewitsch Representation Theorem

13

In TX constant mappings are primitive idempotents. If S is a subsemigroup of TX , X finite, then primitive idempotents must have minimal rank. All idempotents in a Rees product E  G  F are primitive. (If e and f are idempotents such that ef D f e D e then they belong to the same group, which of course has only one idempotent element, its identity.) Definition 1.6. If S is a simple semigroup, i.e., a semigroup without proper two-sided ideals, then it is termed completely simple if it contains a primitive idempotent. As the following example shows, this supplementary condition is by no means vacuous. Let S be the subsemigroup na 0 ˇ o ˇ ˇ a; b > 0 b 1 of the semigroup of 2  2 real matrices (with ordinary matrix multiplication as operation). Clearly, S has no idempotent element, since the identity matrix does not belong to S . Consider the equation 

x 0 y 1

     a 0 u 0 c 0 D ; b 1 v 1 d 1

where a; b; c; d are given positive real numbers. We can solve this equation in S . One solution is given by 

       x 0 3bc=ad 0 u 0 d=3b 0 D ; D : y 1 b=a 1 v 1 d=3 1

Hence S is simple, but not completely simple. Before we proceed to the formulation of the structure theorem for completely simple semigroups, we present some preparatory results, useful in their own right. Proposition 1.6. If S is a simple semigroup, then (i) If the idempotent e 2 S is primitive, then Se and eS are minimal left and right ideals, respectively; the set eSe D Se \ eS is a group. (ii) S is completely simple if and only if it contains at least one minimal left ideal and at least one minimal right ideal. By a minimal (left, right, two-sided) ideal of a semigroup S we mean of course a (right, left, two-sided) ideal that does not properly contain a (right, left, two-sided) ideal of S . Proof. (i) Our first task is to show that xS D eS for any x of the form es. SxS D S , hence txu D e, for some t; u 2 S . In fact, we can choose t from eSe and u from Se, since e is idempotent and x belongs to eS .

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1 Semigroups

Look at xut. We have .xut/  .xut/ D xuet D xut. Hence, xut is idempotent. Furthermore e.xut/ D xut D .xut/e. On the other hand, e was assumed to be primitive. Thus, xut D e, i.e., xS D eS . Hence, eS is a minimal right ideal. The minimality of Se can be established in the same way. Clearly, eSe  Se \ eS . Then e is both right and left identity (thus the unique identity) for Se \ eS , which implies that Se \ eS  eSe. The subsemigroup eSe is right and left simple by the preceding minimality result. Proposition 1.1 tells us that eSe is a group. (ii) The only if-statement follows from (i). Let R.L/ be a minimal right (left) ideal of the simple semigroup S . Then for r; r 0 2 R; rr 0 S D rR D R; a similar statement holds for L. Thus, RLRL D RL and RL is both right and left simple and hence a group by Proposition 1.1. Let e be the identity of the group RL. The e is a primitive idempotent: If f e D ef D f , then f is an idempotent belonging to RL, so it has to coincide with the identity e of the group. Thus S is completely simple. t u We can now formulate the important structure theorem due to Rees and Suschkewitsch. Theorem 1.1. A semigroup S is completely simple if and only if it is isomorphic to a Rees product E  G  F with some sandwich function . Proof. It is quite straightforward and easy to show that a Rees product is completely simple. To prove the converse, let S be a completely simple semigroup. Let e0 be a primitive idempotent. Then Se0 and e0 S are minimal left and right ideals, respectively, by Proposition 1.6, part (i). Their intersection is a group, call it G and e0 is the identity of G. Denote the set of idempotents of Se0 (e0 S ) by E (F ). E is a left zero semigroup, since for e, an idempotent of Se0 , we have Se0 D Se and e is a right identity of Se. Similarly, F is a right zero semigroup: f1 f2 D f2 for f1 ; f2 2 F . Define a Rees product E  G  F with multiplication rule .e; g; f /.e 0 ; g 0 ; f 0 / D .e; gf e 0 g 0 ; f 0 /: This product is well defined, since FE  e0 Se0 D G. The sandwich function simply maps .f; e 0 / to f e 0 . We now claim that the mapping W E  G  F 7! S defined by .e; g; f / D egf is a semigroup isomorphism. is clearly a homomorphism. To see that it is injective, notice that .egf /.e0 egf e0 /1 D .egf /  e0  ..e0 e/  g  .f e0 //1 D .eg/  .f e0 /.f e0 /1  g1  .e0 e/1 D e  ge0 g 1  e0 D ee0 D e: Also .e0 egf e0 /1 .egf / D f and e0 egf e0 D e0 ge0 D g:

1.4 The Rees–Suschkewitsch Representation Theorem

15

(All inverses are taken in the group G  e0 Se0 .) Hence, is injective with inverse s 7! .s.e0 se0 /1 ; e0 se0 ; .e0 se0 /1 s/ on its range. To complete the proof, we have to show that is surjective; i.e., its range is all of S . To this end, let s 2 S and e 2 E. Then se 2 Se0 , which is a left group, by Propositions 1.2 and 1.6 (i); and hence for some idempotent e1 2 Se0 , se D e1 se D .e1 e0 /se D e1 .e0 se/. Since e0 se 2 e0 Se0  G so we can infer that se belongs to the set EG and thus s.egf / 2 EGF for any s 2 S . Similarly, .egf /s 2 EGF: Hence EGF , the range of , is a two-sided ideal of S and consequently equal to S , since S is simple by assumption. t u Definition 1.7. The description of a completely simple semigroup S as a Rees product E  G  F is called its Rees–Suschkewitsch representation, and E; G; and F are called the left, group (or middle), and right factor of S , respectively. Corollary 1.1. A completely simple semigroup S is the union of all its minimal right ideals R˛ as well as the union of all its minimal left ideals Lˇ , where ˛ and ˇ range over some index sets. Furthermore, S is the union of the isomorphic groups R˛ Lˇ . Proof. From Theorem 1.1, it is immediate that S is the union of the minimal right ideals feg  G  F and the union of the minimal left ideals E  G  ff g. (Index sets for ˛ and ˇ can thus be identified with E and F , respectively). S can also be written as the union of the isomorphic groups feg  G  ff g. t u Table 1.1 An egg box picture of the structure of a completely simple semigroup as the union of minimal right ideals R˛ as well as the union of minimal left ideals Lˇ . R˛ Lˇ D R˛ \ Lˇ  G˛ˇ are isomorphic groups; L0 D Se0 ; R0 D e0 S: The idempotents of the first column (row) make up the sets E (F ) L0

Lˇ

L

R0

G00

G0ˇ

G0

R˛ Rı

G˛0 Gı0

G˛ˇ Gıˇ

G˛ Gı

Definition 1.8. A completely simple minimal two-sided ideal of a semigroup is called its kernel. A finite semigroup always has a kernel. This insight goes back to pioneering work by Suschkewitsch, [214]. Proposition 1.7. A semigroup S with at least one minimal left ideal and at least one minimal right ideal contains a completely simple minimal two-sided ideal K. In particular any finite semigroup S has a minimal two-sided ideal K; K is a completely simple subsemigroup of S .

16

1 Semigroups

Proof. Let K consist of those elements of S that generate a minimal left ideal. Thus, K is the union of all the minimal left ideals of S . The set Ssx  Sx is a minimal left ideal of S if x 2 K and s 2 S . The left ideal Sxs is minimal as well: Any txs 2 S xs generates a (minimal) left ideal that contains xs on account of the minimality of Sx. Thus, K is a two-sided ideal. It is a minimal two-sided ideal, since for any k; x 2 K, we have x 2 Skx and hence SkS D K. Let R be a minimal right ideal of S . Then R is a minimal right ideal of K as well. Note that the minimal left ideals S x are minimal left ideals Kx of K too. Thus, we can conclude that the simple semigroup K is completely simple by Proposition 1.6. If S is finite, then we note that x generates a minimal left ideal of S if and only if jS xj, the cardinality of S x, is minimal. Hence, S always admits a minimal left ideal and a minimal right ideal. t u In the important case of a finite transformation semigroup, the kernel has a simple interpretation. Proposition 1.8. Let S be a subsemigroup of TX , where X is a finite set. Then the completely simple minimal ideal K of S consists precisely of those elements that have minimal rank. Proof. The reasoning is based on the discussion of TX (X finite) in Sects. 1.1 and 1.2. The set of mappings K of minimal rank r constitutes a two-sided ideal of S . The ranges of all the elements of K are cross-sections of all the possible partitions generated by the elements of K, otherwise a product of two elements could be formed with lower rank than r. Any k 2 K generates a group of mappings with the same range and the same partition as k. The subset of K consisting of mappings with common range R0 , say, is a right ideal. By Proposition 1.3, it is a right group and hence a minimal right ideal. Similarly, the set of mappings in K with partition 0 constitutes a minimal left ideal. By Proposition 1.6(ii) or Corollary 1.1, K is the completely simple minimal ideal of S . t u The treatment of finite dimensional linear transformations is not very different from the case of finite transformation semigroups. These important semigroups will be discussed more fully in Sect. 1.6. Proposition 1.9. Let S be a semigroup of d  d real matrices under matrix multiplication. Suppose that S admits a completely simple minimal ideal K. Then K consists precisely of the elements of S of minimal rank. This semigroup is no longer necessarily finite. Thus, the existence of a completely simple minimal ideal is not assured in general. As usual, the rank of a matrix (interpreted as a linear transformation on 0: ˇ.n/ D n C 1; n 2 N:

˛.n/ D

Then ˛ˇ D  and ˇ˛ ¤ . Transformations of N generated by ˛ and ˇ can be written ˇ m ˛ m ; m; n 0 in a unique fashion, where  is of course, ˇ 0 ˛ 0 . The bicyclic semigroup is in a sense the example of a semigroup that is simple without being completely simple. Proposition 1.15. If e is any nonzero idempotent of a (0-)simple semigroup S that is not completely (0-)simple, then S contains a bicyclic subsemigroup with identity e. Proof (for the case with 0). The e is by definition of a completely 0-simple a nonprimitive idempotent of S . Then there exists another nonzero idempotent f 2 S such that ef D f e D f . S is 0-simple, so e D xf y for some x; y 2 S   S n f0g. Set p D exf and q D f ye, then pq D eee D e. Direct calculations show that qp is also idempotent and nonzero, but f qp D qpf D qp so qp  f < e in the ordering of idempotents. In particular, qp ¤ e. The preceding relations pq D e; qp ¤ e, and the fact that e is a two-sided inverse of the subsemigroup generated by p and q, guarantee that this subsemigroup is isomorphic to C , [see [43], Sect. 1.12.] t u Proof (of Theorem 1.3, case with 0). The square of each element of a completely 0-simple semigroup lies in a group (Definition 1.4). Conversely assume that some power of each element of S is contained in a subgroup of S . Let a 2 S  . Then a 2 S  aS  , since S is 0-simple. Let a D xay. By iterating this equality, we obtain a D x n ay n . For some n, x n belongs to a subgroup of S , indeed of S  . Let e ¤ 0 be the identity element of this subgroup.

22

1 Semigroups

Suppose S is not completely 0-simple. Then e is not a primitive idempotent, and by Proposition 1.15, e is the identity element of a bicyclic subsemigroup of S with generators p and q, say (pq D e; qp ¤ e). For some n, p n belongs to a group with identity f . Let r be the inverse of p n in that group: rp n D p n r D f . We have f e D fp n q n D p n q n D e and f e D rp n e D rp n D f , so f D e. But then r D p n and p n q n D q n p n D e, which is not possible in the bicyclic semigroup generated by p and q. Hence, S must be completely 0-simple. t u Remark 1.6. Semigroups S with some power of each element in a subgroup of S are termed pseudoinvertible. Thus, a simple semigroup is completely simple if and only it is pseudoinvertible.

Section 1.4 Exercises Exercise 1.21. Prove that a Rees product is completely simple. T Exercise 1.22. Show that if x2S S xS is nonempty, then it is isomorphic to a Rees product E  G  F with some sandwich function . Exercise 1.23. Prove that a Rees product with a sandwich function that maps everything onto a constant is isomorphic to a direct product of E; G; and F . Exercise 1.24. Prove that a subsemigroup U of Tn is completely simple if and only if rank.˛/ D rank.ˇ/ for all ˛; ˇ 2 U . Exercise 1.25. Let C D C .p; q/ be a bicyclic semigroup (as in Definition 1.11). Consider the subsets: S1 D fq 4i C5 p 4j C5 W i; j 0g S2 D fq 4i C7 p 4j C7 W i; j 0g Prove that both S1 and S2 are subsemigroups and that they are isomorphic to C . Prove that their union S D S1 [ S2 is also a subsemigroup. Exercise 1.26. Show by using the properties of minimal ideals that the bicyclic semigroup contains no minimal left or minimal right ideal.

1.5 Topological Semigroups Let S be a semigroup. Suppose S is also endowed with a Hausdorff topology. If the map S  S 3 .s; t/ 7! st 2 S is continuous S is called a topological semigroup. (The topology on S  S is the product topology.) In other words, a semigroup whose multiplication is jointly continuous in the factors is a topological semigroup.

1.5 Topological Semigroups

23

If the multiplication is only separately continuous in the factors, we call S a semitopological semigroup. Equivalently S is semitopological if the right translations a and the left translations a are continuous mappings from S to S for all a 2 S . However, we always assume that our semigroups are topological unless explicitly stated otherwise. Similarly, the topology is always assumed to be Hausdorff. Subsemigroups of a topological semigroup are automatically topological semigroups with respect to the induced topology. By an open (closed, compact, locally compact) subsemigroup (ideal) of S we mean, respectively, an open (closed, compact, locally compact) subset of S that is algebraically a subsemigroup (ideal) of S . We collect some elementary facts about subsets of a topological semigroup in Proposition 1.16. Proposition 1.16. Let S be a topological semigroup. Let F be a closed subset, O an open subset, A an arbitrary subset and C; C1 ; C2 compact subsets of S . (i) For every element s 2 S , sC; C s; and C1 C2 are compact, s 1 F  1 s .F /  ft 2 S jst 2 F g, F s 1  s1 .F /  ft 2 S jts 2 F g are closed, and s 1 O; Os 1 , [ [ s 1 O; OA1  Os 1 A1 O  s2A

s2A

are open subsets of S . If T is a subsemigroup of S , then so is TN , the closure of T . (ii) If in addition S is algebraically a group, then the translations are homeomorphisms from S to S . In particular for any s 2 S , sF; F s; CF are closed, and sO; Os; AO; OA are open subsets of S . In addition, C 1 is closed. Furthermore, sU .Us/ is a neighborhood of s if and only if U is a neighborhood of the identity e. If the net x˛ ! x and x˛1 ! y, then y D x 1 . (iii) The set of idempotent elements of S , E.S /, is closed. If e is an idempotent of a closed subsemigroup T , then eT; T e; eT e are closed. 1 Note that (ii), 1 F is consistent with left s D s 1 , so our notation in (i) for s 1 1 translation of the set F by the element s . (C is of course a notation for the set fs 1 js 2 C g.)

Proof. (i) C1 C2 is the image under multiplication of the compact set C1  C2 in S  S and hence compact. Likewise, sC; C s are continuous images of a compact set. The sets s 1 F; F s 1 are preimages of closed sets, hence closed, and s 1 O, Os 1 are preimages of open sets, hence open. A1 O; OA1 are in turn unions of open sets, hence also open. If t˛ ; u˛ are nets in T converging to t and u, respectively, then t˛ u˛ ! tu, so tu 2 TN . (ii) Since s .s / has the continuous inverse s 1 .s 1 /, most of the assertions follow directly from the results in (i).

24

1 Semigroups

To see that C 1 is closed and to prove the last assertion of (ii), consider a net converging to an element g 2 S . The net c˛ has a subnet, say, cˇ , converging to some c in the compact set C . Then e D cˇ cˇ1 . By the joint continuity of the multiplication, the limit of the right-hand side equals cg, so g D c 1 and hence g 2 C 1 . (iii) The statements are immediate consequences of, respectively, the joint continuity and the continuity of the multiplication. t u

c˛1

One of our basic examples is the semigroup Md of d  d real matrices. If we 2 endow Md with the usual topology of 0. Since M D PM , Mhj D .h; ut /Mut j ; h 2 Ct ; 1  j  d:

(1.8)

Hence, the rank of M restricted to Ct  D is 1. Suppose that Mij > 0. Let i 2 Cs . We claim that there is precisely one Ct with Mij > 0 for i 2 Cs ; j 2 Ct . Furthermore, this correspondence is one-to-one: Mij > 0 and Mi 0 j > 0 for j in some Ct implies that i and i 0 belong to the same Cr .1  r  k/. First of all, let us point out that since all elements of G have the same zero rows and zero columns, X X Pih Nhj D Nij D Nih Phj ; i; j D 1; 2; : : : d (1.9) h…T

h…T

and .NM /ij D

X

Nih Mhj ; i; j D 1; 2; : : : d;

(1.10)

h…T

where M and N are arbitrary matrices in G. Let us now prove our claim. For P the correspondence s $ t is simply the identity mapping by Theorem 1.11: Pij > 0; i 2 Cs ; j 2 Ct if and only if s D t. In particular, Mij > 0 for all i 2 Cs means that the i th row is not a zero row and that j can be chosen from the complement of T ; i.e., j 2 Ct for some t. Similarly, .M 1 /i l > 0, l 2 Cr for some r X Mih .M 1 /hi : 0 < Pii D h…T

Hence, .M 1 /hi > 0 for some h … T . Suppose .M 1 /hi > 0 for an h … Ct . Then 0 < .M 1 /hi Mij  Phj with h and j in different C classes, contradicting Theorem 1.11. Mi l > 0 for l … Ct is impossible too: .M 1 /hi Mi l  Phl D 0: Equation (1.8) shows that Mij > 0 for one i 2 Cs implies Mi 0 j > 0 for all i 0 2 Cs . On the other hand, if Mij 0 D 0 for some j 0 2 Ct then Mi 0 j 0 D 0 for all i 0 2 Cs and hence 0D

X h2Cs

.M 1 /jh Mhj 0 D

X

.M 1 /jh Mhj 0 D Pjj 0 ;

h…T

a contradiction, since j and j 0 both belong to Ct . Hence, all entries in MjCs Ct are positive. This proves the claim.

44

1 Semigroups

Another way of phrasing the claim is that there is a permutation  on D such that Mij > 0; i 2 Cs ; j 2 Ct ” t D .s/: Furthermore, the preceding proof shows that .M 1 /ij > 0; i 2 Cs ; j 2 Ct ” t D  1 .s/; where  1 is the inverse permutation. Let us write M .s/ D t ” MjCs Ct > 0

(1.11)

(in the sense that all the entries in that block are strictly positive).  is an antihomomorphism of G into the group of permutations of D  f1; 2; : : : d g M ı N D NM :

(1.12)

This follows at once from the observation that the expression .NM /ij D

X

Nih Mhj ; i 2 Cs ; j 2 Ct

h…T

is positive if and only if h 2 CN .s/ and j 2 CM .N .s// . In general,  is far from injective, since the group of permutations is a finite group. The matrices M and N have the same image under  if and only if NjCs C.s/ D s MjCs C.s/ ; 1  s  k

(1.13)

for some strictly positive numbers 1 ; 2 ; : : : ; k . (Here, M D N is denoted by just .) Clearly, the condition (1.13) is sufficient, since, by (1.9), the values of both matrices are uniquely determined by their values on T c  T c (and by P ). To prove the converse, let i 2 Cs ; j 2 Ct ; t D .s/. Equation (1.7) says Phj D .h; ut /Put j for h 2 Ct : Premultiplication by M gives Mij D

X

Mih .h; ut /Put j D Put j

h2Ct

X

Mih .h; ut /

h2Ct

Nij D Put j

X

Nih .h; ut /

h2Ct

Equation (1.8) shows that Mih D .i; us /Mus h ; Nih D .i; us /Nus k , so Nih D Mih

Nus h  s .h/Mih Mus h

1.6 Semigroups of Matrices

45

independently of i 2 Cs . Hence, Nij D Mij

P

s .h/Mih .h; ut / P ; h Mih .h; ut /

h

which is independent of j 2 Ct . Let us now collect the results of the preceding discussion of nonnegative matrices in Theorem 1.14. Theorem 1.14. Let G be a group of nonnegative matrices with identity P of rank k. Let the basis of P be fT; C1 ; C2 ; : : : ; Ck g. Then (i) Each M 2 G has rank k, the same zero rows and zero columns as P ; the rank of MjCs D is 1 for all s, 1  s  k; (ii) Each M 2 G is uniquely determined by P and MjT c T c ; (iii) There is an antihomomorphism  from G into a subgroup H of the group of permutations on f1; 2; : : : ; kg such that for M 2 G and 1  s  k MjCs Ct is a zero block if t ¤ .s/ and a strictly positive block of proportional rows if t D .s/; (iv) The constants of proportionality in the strictly positive blocks depend on s only (1  s  k); they are the same for all group elements. Remark 1.17. For any given P , M , and the positive numbers vM .s/ D Mus u.s/ , 1  s  k, determine M uniquely. vM can be any strictly positive vector in 0 8i 0 2 Cs :

48

1 Semigroups

This is used to obtain, for example MjCs Tc D 0 as follows. For i 2 Cs ; j 2 Ct ; t ¤ s, X

0 D .PMP/ij D

Pih Mhl Plj

h2Cs ;l2Ct

C

X

h2Cs ; l2Tc

Pih Mhl Plj C

X

Pih Mhl Plj :

h2Tr ; l2Tc [Ct

Then we can show that Mhl D 0 since the second term is 0. For the details of the proof, we refer to [148].

t u

We now turn to stochastic matrices, i.e., d  d matrices with nonnegative entries P Mij with row sums djD1 Mij D 1. If all column sums are also 1, the matrix is said to be doubly stochastic or bistochastic. If row sums are  1, then the matrix is called substochastic. Our interest in these matrices is of course motivated by their extensive use in probability theory as transition probability matrices for Markov chains. To analyze the evolution of the Markov chain, we must study products of transition probability matrices. The asymptotic behavior of the chain can often be described by idempotent stochastic matrices appearing as limits of products of transition probability matrices. Remark 1.19. Any bistochastic matrix is a convex combination of permutation matrices (i.e., 0–1 matrices with exactly one nonzero element in each row and each column) by a famous theorem of Birkhoff. Any stochastic matrix is a convex combination of stochastic 0–1 matrices, which could then be called transformation matrices. The representation is not unique, but the idempotents of the generated semigroup are (see discussion in [98]). Topological semigroups of (sub)stochastic matrices are compact, since entries are bounded by 1. Then we can use the preceding theory to obtain powerful results. Using notations from Theorems 1.11 and 1.12, we state Theorem 1.16. Theorem 1.16. Let P be a d d stochastic idempotent matrix of rank k. Then there is a basis fTc ; C1 ; : : : ; Ck g for P such that PjCs Cs has identical positive rows of sum 1 and PjCs Ct D 0; s ¤ t: If i 2 Tc , Pij Pih D ; j; h 2 Cs : Pjj Pjh Conversely, any stochastic matrix with these properties is idempotent. Proof. Since P is stochastic, Tr is empty, therefore P is 0 on Cs  .Cs /c . The rows of PjCs Cs are proportional with row sum 1, hence equal. The rest is clear. t u

1.6 Semigroups of Matrices

49

Corollary 1.9. Let P be a d  d bistochastic idempotent matrix of rank k. Then there is a basis fC1 ; : : : ; Ck g for P such that Pij D jC1s j for i; j 2 Cs and Pij D 0 for i 2 Cs ; j 2 Ct ; t ¤ s (1  s; t  k). Proof. This time rows and columns of PjCs Cs are also equal and sum to 1. Hence, the block consist of identical entries. t u Locally compact groups of stochastic matrices are finite by Corollary 1.8. Completely simple semigroups of stochastic matrices need not share the same basis, as Example 1.8 shows. Another example is S D fe; f g, where 0 1 0 1 0 1 0 1 0 0 e D @0 1 0A ; f D @1 0 0A : 0 0 1 0 0 1 If T is empty for all idempotents of S , then Proposition 1.23 says that they all have a common basis. In particular, we have Corollary 1.10. Corollary 1.10. If S is a completely simple semigroup of bistochastic matrices, then S is a finite group. Proof. By Corollary 1.9, there is only one possible idempotent for each basis. Then S consists of just one group. t u Remark 1.20. If (1.15) is satisfied – probabilistically speaking this means that the Markov chain governed by P can jump from a transient state to at least two different recurrent classes – Theorem 1.15 is applicable. Note that Tr is empty.

Section 1.6 Exercises Exercise 1.40. Let G be a compact group of rank one d  d real matrices. Show that G is either a singleton or a two-point group fe; eg. Exercise 1.41. Verify that the minimum number of elements for a nontrivial Rees product is 8. Also show that the set fa; b; c; d; e; f; g; hg constitutes such a Rees product, when         OI O I OM OM aD ;b D ;c D ;d D ; OI OM OM O I         I O I O M O M O eD ;f D ;g D ;h D ; I O M O I O M O and I , M , and O are the 2  2 blocks       1 0 0 1 0 0 ; and : 0 1 1 0 0 0

50

1 Semigroups

y

(x,y)

R(P)

P(x,y)

x N(P)

Exercise 1.42. Find a projection Q, such that PQP D P , when R.P / D f.x; y/jy D xg and N.P / D f.x; y/jy D  13 xg. Also show that such a Q does not exist if N.P / D f.x; y/jy D xg instead. Verify, e.g., by inspection of the multiplication tables, that the semigroup generated by P and Q is isomorphic to the one in Exercise 1.41. Exercise 1.43. Let e be a nonnegative idempotent of rank k with basis fT; C1; C2 ; : : : ; Ck g, where T D Tc [ Tr as in Theorem 1.11. Show that there are column vectors (of appropriate dimensions) i > 0; ˛i > 0; ıi 0; i 0; i D 1; 2; : : : ; k, so that the Ci  Ci blocks are of the form i ˛iT , the Ci  Tr blocks of the form P i iT , the Tc  Ci blocks of the form ıi ˛iT , Tc  Tr block equals kiD1 ıi iT , and all other blocks are all-zero ones. Exercise 1.44. Let S be the multiplicative semigroup generated by the stochastic matrices 1 1 0 0 1 0 0 0 1 A D @1 0 0A ; B D @0 0 1A ; 0 0 1 a 1a 0 0

where 0  a  1. Determine the kernel K of S and its product representation when a D 0 and when a > 0. Exercise 1.45. Let S be the closed multiplicative semigroup generated by a subset A of fcirc.x; x  1; x  1/jx 2 0 such that eih D s .i; j /ejh ; ehi D Ns .i; j /ehj ; h 2 EI

(1.18)

(iii) For s; t 1; s ¤ t; i 2 Cs ; j 2 Ct , eii > 0 and eij D 0:

(1.19)

P Proof. Define T as in (i). eij D k eik ekj , so the products can be taken over the complement T c of T . In other words, the restriction of e to T c  T c is also idemc potent. If i 2 T c and eP ij D 0 for all j 2 T , then the whole row is 0, contradicting the choice of i : eik D j 2T c eij ejk D 0: Similarly, the j th column of ejT c T c is also nonzero. For the moment, we assume that T is empty. We first establish eii > 0 for all i 2 E:

(1.20)

56

1 Semigroups

Assume the contrary: eii D 0 for some i 2 E, i D 1, say. Let A be the set fj 2 Ej e1j > 0g. A is not empty, since we assumed that e had no zero rows or zero columns. Then X 0 D e1j D e1k ekj for j … A k2A

so ekj D 0 for j … A; k 2 A; i.e., e restricted to A  Ac is a zero block. Call the restrictions of e to Ac  Ac , Ac  A, and A  A, P1 ; P2 , and P3 , respectively. Then e 2 has the blocks P12 ; P1 P2 C P2 P3 and P32 , respectively. By idempotency, we obtain equations P1 D P12 ; P2 D P1 P2 C P2 P3 ; P3 D P32 : Algebraic manipulations show that P1 P2 P3 D 0. (P1 P2 D P12 P2 C P1 P2 P3 D P1 P2 C P1 P2 P3 ). Also we obtain a contradiction .P1 P2 P3 /kl .P1 /kj .P2 P3 /js > 0 unless P2 P3 D 0. [.P1 /kj > 0 for some k 2 Ac otherwise there would be a zero column in e.] Finally P2 D 0. This is because P3 cannot have a zero row, so .P3 /kj > 0 for all k and some j (depending on k). Then .P2 P3 /ij .P2 /ik .P3 /kj , which may be >0 unless P2 D 0. We now have a zero row in e: Row 1 is the first row of P2 (1 2 Ac ), and by definition of A, .P1 /1j D 0 for all j 2 Ac . But e has no zero row; hence (1.20) follows. Our next claim is eij D 0 ” eji D 0: (1.21) The proof of this claim rests on a similar decomposition of e into blocks. Suppose e1j D 0 for some j . Define the set B  fkj e1k > 0g. B is nonempty; by (1.20) 1 2 B and j … B. P For k 2 B c , 0 D e1k D l2B e1l elk . Hence, e restricted to B  B c is 0. Call the other B  B, B c  B, B c  B c blocks, respectively, P4 ; P5 , and P6 . Using the relation e 2 D e, we get P4 D P42 ; P5 D P5 P4 C P6 P5 ; P6 D P62 : By a simple argument as in the proof of (1.20), we end up with P5 D 0; hence, ej1 D 0. If Ci D fj j eij > 0g; then ejCi Ci is a strictly positive idempotent matrix: (1.22)

1.7 Semigroups of Infinite Dimensional Matrices

57

P

P

This follows immediately, since .e 2 /ii D eii D k eik eki D k2Ci eik eki , by (1.21). P Likewise .e 2 /ij D eij D k2Ci eik ekj . Finally, A strictly positive idempotent matrix D has rank 1: (1.23) Write Di r D a.i; r; k/Dkr . .D 2 /kr D Dkr Dki Di r : a.i; r; k/ D

1 Di r  : Dkr Dki

Then

1 < 1: Dki

sup a.i; r; k/ D ˇ.i; k/  r

Also for any t, k and r Dtr

Dtk > 0: Dkr We now have X Œˇ.i; k/  a.i; t; k/ Dkt Dtr D ˇ.i; k/Dkr  Di r D Œˇ.i; k/  a.i; r; k/ Dkr : t

Therefore for any t, 0  Œˇ.i; k/  a.i; t; k/ Dkt

Dtr  ˇ.i; k/  a.i; r; k/; Dkr

and 0  Œˇ.i; k/  a.i; t; k/ Dkt Dtk  inf.ˇ.i; k/  a.i; r; k// D 0: r

Hence ˇ.i; k/ D a.i; t; k/ for each t, so Di r D ˇ.i; k/Dkr , whence rank(D) = 1. It is clear from (1.20)–(1.23) that we can now partition E into disjoint classes fC1 ; C2 ; : : :g, where i and j belong to the same class if and only if eij > 0. Also ejCs Cs is a rank 1 idempotent matrix for all s D 1; 2; : : : . Returning to the original matrix e (with nonempty T ), it is easy to verify that (i) If the i th column of e is a zero column then for j and k in the same C -class of T c eik eij D : ejj ejk Consider .e 2 /ij D eij D

X h2T c

and

eik D

X h2T c

X

eih ehj D

eih ehk D

eih ˇ.h; j /ejj

h2Cs

X h2Cs

eih ˇ.h; j /ejk I

58

1 Semigroups

(ii) If the i th row of e is a zero row, then for j and k in the same C -class of T c eji eki D : ejj ekj From (i) and (ii) and (1.23), we conclude that (1.18) is satisfied. This finishes the proof of the Theorem.

t u

Let us now turn to other elements x of the group G with identity e. We find that (i) The zero rows and the zero columns are the same because of the relations ex D xe D x and xx 1 D x 1 x D e; (ii) Since x D xe D ex we can replace e by x in (1.18); (iii) For a given C -class Cs on the basis of e, there exist unique s 0 and s 00 such that the block, Ck  Cs of x is strictly positive or an all zero block, depending on whether or not k D s 0 ; the block Cs  Ck of x is a strictly positive block or an all zero one depending on whether or not k D s 00 . To prove (iii), let i 2 Cs . Since e D xx 1 , we have 0  eii D

X

xiu .x 1 /ui

(1.24)

u2T c

therefore, there is a s 0 such that for some u 2 Cs 00 , xiu > 0; .x 1 /ui > 0. This means that xi u0 > 0 for all u0 2 Cs 00 : xi u0 xiu euu0 > 0. Conversely, xi 0 u0 > 0 for all i 0 2 Cs 0 . On the other hand, xij > 0 ) euj .x 1 /ui xij > 0 ) j 2 Cs 0 : Similar arguments show that xij > 0 for i 2 T c ; j 2 Cs if and only if i 2 Cs 0 for some unique s 0 . Observation (iii) and (1.24) show that there exists a permutation x on the set of C -classes E0 (E0 is the quotient space T c =  where i  j if and only if eij > 0) defined as follows: For s; t 2 E0 , x .s/ D t if and only if the Cs  Ct block of e is all positive. Equation (1.24) shows that x 1 .t/ D s ” x .s/ D t. More generally xy D y ı x , since xy .s/ D t ” for i 2 Cs W .xy/ij > 0 for j 2 Ct and .xy/ij D

X

xik ykj ;

where the summation gives a positive result if and only if k 2 Cx .s/ and j 2 Cy .x .s// .

1.7 Semigroups of Infinite Dimensional Matrices

59

Hence,  is an antihomomorphism from G to the group H of permutations on E0 . As in Sect. 1.6, see especially the proof of (1.13), we see that x D e (the identity permutation) if and only if xjCs Cs D .s/ejCs Cs ; for some positive constants .s/; s 2 E0 . (The finite sums of Sect. 1.6 are replaced by possibly infinite convergent sums.) Let x; y 2 G1  fz 2 Gj z D e g, the kernel of the homomorphism . Then G1 is a group, and the constants of proportionality preserve multiplication: x .s/ y .s/ D xy .s/, see the proof of Proposition 1.22. Suppose that the group G is compact (with respect to a topology where convergence implies pointwise convergence). If x 2 G and x D e , then x .s/ D 1 for n n all s 2 E0 , if not, either xjC or xjC goes to infinity for some s 2 E0 . This s Cs s Cs comment implies that for a compact G,  is one-to-one. G can (algebraically) be mapped isomorphically onto a subgroup of the permutations of E0 . If E0 is finite, then G is finite, see Corollary 1.8. Theorem 1.19. Let G be a multiplicative group of infinite dimensional nonnegative matrices satisfying (1.16). Let e be the identity of G with basis fT; C1 ; C2 ; : : :g. Then there is an antihomomorphism  from G into the group of permutations on the set E0 of C -classes, with the following properties: (i) For each x 2 G, the block Cs  Ct is a strictly positive block or an all-zero block depending on whether or not t D x .s/; (ii) If G is compact with respect to a topology as least as strong as the topology of pointwise convergence and if E0 is finite, then G is finite; indeed if G consists of only strictly positive matrices then G is a singleton; (iii) If G is a group of infinite dimensional stochastic matrices, then G is finite if E0 is finite. Proof. It remains to prove only (iii). (Notice that the infinite dimensional stochastic matrices do not in general form a relatively compact set.) (iii) follows since x n and x n are bounded on Cs  Cs ; hence x .s/ D 1 for all s. Then  is injective, so G has to be finite if E0 is. t u The proof of Proposition 1.23 can be modified to yield the infinite dimensional analog, Proposition 1.25. Proposition 1.25. If S is a completely simple semigroup of infinite bistochastic matrices, then S is a group. Proof. Let the common partition be fC1 ; C2 ; : : :g. Let e be an idempotent of S . On Cs  Cs both the row sum and the column sum are 1. Hence, Cs is a finite set and eij D jC1s j for all i; j 2 Cs (s D 1; 2; : : :). Since there is only one idempotent in S , S is a group. t u

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1 Semigroups

Section 1.7 Exercises Exercise 1.57. Find a matrix semigroup that is isomorphic to the bicyclic semigroup. How can you represent ˛; ˇ; ˇ m ; ˛ n ; ˇ˛ (see p. 21) as matrices? Exercise 1.58. Suppose that S is a completely simple semigroup of infinite dimensional nonnegative matrices satisfying (1.16) and with no zero rows or columns. Show that there is a unique partition fC1 ; C2 ; : : :g of E, the positive integers, such that the following hold: (i) If e is an idempotent in S , then the above partition is the basis of e. (ii) For each x 2 S , the rank of x restricted to the Ck  Ck block is 1, for each Ck . (iii) There is an antihomomorphism p from S into the group of permutations on the set E0 D f1; 2; : : : ; mg, where m  1 is the number of C -classes in the partition above, such that for x 2 S , i 2 Ck , xij > 0 precisely when j 2 Cp.x/.k/ . Exercise 1.59. Let e be an idempotent infinite bistochastic matrix. Show that the partition E0 is infinite.

1.8 Notes and Comments At the previous turn of the century there was a lively discussion on the axioms of group theory. In that connection, some basic semigroup properties were found, but the activity did not give rise, yet, to a theory of semigroup proper. Through the work of Suschkewitsch and Rees, semigroup theory began to take on a life of its own. Suschkewitsch [214] and Rees [192] are considered to be the breakthrough papers. The comprehensive works on the algebraic theory of semigroups, viz., [42, 43, 132] contain a large bibliography and many historical notes, albeit scattered throughout the text. The theory of topological semigroups seems to have been first presented in book form by Paalman-de Miranda [181], whose first edition appeared in 1964. Later monographs are, for example, [9, 10, 29, 30, 91]. Historical notes are to be found in all of these works. The probabilistically oriented monographs by Rosenblatt [199] and Mukherjea and Tserpes [172] provide some necessary background on topological semigroups as well. The exercises are drawn from various sources, most frequently from Paalman-de Miranda [181] and Clifford and Preston [43]. Also [10, 88] have been sources for the exercises.

1.8 Notes and Comments

61

Section 1.1 The basic definitions are given in [43], Chap. 1.4. The semigroup BX of relations on the set X plays an important role in algebraic coding theory, see [11]. Simon [208] is another textbook in theoretical computer science using semigroup theory in an essential way, to develop the theory of automata.

Section 1.5 The general references for this section are [9, 112, 181]. The fundamental Rees– Suschkewitsch theorem was proved by Wallace [229]. The structure theory of locally compact completely 0-simple semigroups (Theorem 1.9) is due to Owen [179, 180], see also [108]. Example 1.2 was studied in [180] and [101].

Section 1.6 As pointed out in the text, Mukherjea [148] studies the completely simple semigroups of matrices in more detail. Clark [41] treats the kernel of a matrix semigroup. H¨ogn¨as [105] studies a semigroup with a closely related structure: the semigroup of analytic functions with a common fixed point (with composition of functions as semigroup operation). In many respects, the essential features of the whole semigroup are expressed by some finite dimensional subsemigroups, which turn out to be semigroups of matrices. A matrix application of a different kind is furnished by Meyer [139] in a study of a finite irreducible Markov transition probability matrix P and the associated invariant probability measure . P belongs to a group of singular matrices. The inverse of P in that group may be used to describe how  varies as a function of P . The striking result on compact groups of nonnegative matrices, Corollary 1.8 was found by Brown [21] and Flor [68]. The proof here, however, follows the one given in [151]. The facts about bistochastic matrices (Theorem 1.16 and its Corollaries) are taken from [133]. Simple examples of nondirect Rees products (such as Exercise 1.41) are given in [110]. The 2  2 stochastic matrices in Exercise 1.49 are treated, e.g., in [172, 199]. We gratefully acknowledge the contribution by Greg Budzban and Phil Feinsilver who compiled the Exercises 1.51–1.54 for our use. They are concerned with the Road Coloring Problem and taken from [24, 25]. For a semigroup approach to this problem see also [27]. The semigroup operation in G= in Exercise 1.52 is the following: The product of [i ] and [j ] is defined to be the equivalence class containing ij , cf. [43], p. 16.

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1 Semigroups

The result in Exercise 1.52(iv) was used by Trahtman in his proof of the Road Coloring Problem, though it was stated using different terminology, see [221]. The definition of solvable transformation used in Exercise 1.54 is a consequence of a result by Radjavi and Rosenthal [189]. The decomposition of the idempotent P in Exercise 1.55 is due to Chakraborty and Mukherjea in [32]. The Exercise 1.56 is also taken from there. There is nowadays a large body of work dedicated to a linear algebra based on the semiring operations (max; C) on 0, there exists a compact subset Kep in B such that  S  Kep < . It follows that for B 2 B, .B/ D inff.V / W V open, V  Bg D supf.K/ W K compact, K  Bg. An infinite nonnegative measure  is called regular when this approximation property still holds and for K compact, .K/ < 1. In a complete separable metric space, all measures in P .S / are regular. (See [182, p. 29].) In what follows, we write: Bx1  fy 2 S j yx 2 BgI x 1 B  fy 2 S j xy 2 Bg; B 2 BI C.S /  the space of all bounded real continuous functions on S I Cc .S /  the space of functions in C.S / with compact support:

2.2 Invariant and Idempotent Probability Measures   For  2 P .S/, the measures x , defined by x .B/ D  Bx1 , and x , defined by  1 B , are both regular. The reason is the following: Given  > 0, if x .B/ D  x   K is a compact subset of Bx1 such that  Bx1  K < , then Kx  fyx j y 2 Kg is a compact subset of B, and moreover      .B  Kx/x 1   Bx1  K < ; since K  .Kx/x 1 . Thus, x is regular relative to compact sets from inside (and, therefore, relative to open sets from outside). The same is the case with x . Proposition open   2.1. Let  2 P .S /, B 2 B, and V be an 1  set in B. Then  V x 1 is a lower semicontinuous function is B-measurable.  1   1  of x and  Bx The same is true for  x V and  x B . R Proof. Notice that for f 2 C.S /, f .sx/.ds/ is a continuous function of x (since  is regular), and moreover    V x 1 D sup

Z

 f .sx/.ds/ W f 2 C.S /, 0  f  1, f D 0 on S  V :

2.2 Invariant and Idempotent Probability Measures

65

This implies the first assertion in Proposition 2.1. This also means that   ˚  F D B 2 B j  Bx1 is a B-measurable function of x contains all open sets and all closed sets. Furthermore, F is a monotone class containing the class F D fV \ W W V is an open set in B, and W is a closed set in Bg: Notice that F0 is closed under finite intersections and the complement of any set in F0 is a finite disjoint union of sets in F0 ; thus it belongs to F . It follows easily that B  F , since F contains the algebra generated by F0 . t u Definition 2.1. Let ;  2 P .S / and f 2 C.S /. Then the iterated integral “ I.f / D

f .xy/.dx/.dy/

is welldefined. By the Riesz representation theorem (see [163, Theorem 5.10]), there is a unique regular probability measure  on B such that for any f 2 Cc .S /, Z f d D I.f /: The measure  is called   , the convolution of  and . Notice that it follows using Fubini’s theorem that “

Z f d.  / D

f .xy/.dx/.dy/ “

D

(2.1) f .xy/.dy/.dx/;

for each f in Cc .S /. Actually, (2.1) leads to a more convenient formula for the convolution. Proposition 2.2. Let ;  2 P .S / and B 2 B. Then Z   .B/ D

   Bx1 .dx/ D

Z

   x 1 B .dx/:

(2.2)

Proof. For B 2 B, define Z .B/ D

   Bx1 .dx/:

(2.3)

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2 Probability Measures on Topological Semigroups

Then  is a probability measure on B. We claim that  is regular. To prove this claim, notice that for any open set G  S , 3 2 Z .G/ D

   Gx 1 .dx/ D

0f 1 f D0 on G c

“ 

7 Z Z 6 7 6 6 sup f .sx/.ds/7 7 .dx/ 6 5 4 f 2Cc .S/

f .sx/.ds/.dx/

sup f 2Cc .S/ 0f 1 f D0 on G c

Z

D sup

f d.  / .sup over the same class as before/

D   .G/; since    is regular. Since S is a metric space, by [182, p. 27],  is regular with respect to open sets from outside and closed sets from inside. It follows that  D   , since .B/ D inff.G/ W G open, B  Gg  inff  .G/ W G open, B  Gg D   .B/.8 Borel set B/:

t u

We remark that if X1 ; X2 ; : : : ; Xn are independent random variables with values in S such that the distribution of Xi is i 2 P .S /, then the product X1 ; X2 ; : : : Xn has distribution 1  2  : : :  n . (The convolution operation is associative.) There is an important interplay between the convolution operation in P .S / and a topology, called the weak topology, in P .S /. We present it next. Definition 2.2. A sequence n in P .S / is said to converge weakly to  in P .S / if for every f in C.S /, Z Z lim f dn D f d: n!1

To denote weak convergence, we often write: w

n !  or n ) : Let us denote by B.S / the set of all (nonnegative) regular measures  on B such that .S /  1. Definition 2.3. A sequence n in B.S / is said to converge vaguely (or in the weak topology) to  in B.S / if for every continuous function f on S with compact support Z Z lim

n!1

f dn D

f d:

2.2 Invariant and Idempotent Probability Measures

67

To denote weak convergence, we write: w

n ! : Let us remark that B.S / as a closed subset of the unit ball in the dual (adjoint) space of Cc .S / is compact in the weak topology (by the Banach–Alaoglu theorem w

of functional analysis, see [164, p. 63]). For a sequence n in P .S /, if n !  and w  is in P .S /, then n ! . P .S / is compact in the weak topology. In fact, P .S / is a compact metric space in its weak topology if and only if S is compact (note that S is already a metric space, see [182, p. 45]). In any case, the weak topology of P .S / is metrizable (in our context). w We remark that n !  in P .S / if and only if 8 closed set F  S; .F /  lim sup n .F / n!1

if and only if 8 open set G  S; .G/  lim inf n .G/: n!1

(This can be verified easily, see [182, p. 40]) Proposition 2.3. Convolution is jointly continuous in P .S / with respect to the weak topology. w

w

Proof. Let n !  and n ! . Let f 2 C.S / and  > 0. There exists a compact set K such that for n  1, n .K/ > 1  ; n .K/ > 1  ; Write: gn .t/ 

R

.K/ > 1   .K/ > 1  :

(2.4)

f .st/n .ds/. Then the sequence fgn W n  1g is a uniformly

K

bounded equicontinuous sequence; moreover, for each t in S , ˇZ ˇ Z ˇ ˇ ˇ lim sup ˇ f .st/n .ds/  f .st/.ds/ˇˇ < 2kf k: n!1 K

(2.5)

K

Calling the second integral in (2.5) g.t/, we notice that if ˇZ ˇ Z ˇ ˇ ˇ gn .t/n .dt/  g.t/n .dt/ˇˇ > 3kf k ˇ K

K

for infinitely many n, then there is a subsequence .ni / such that gni ! g0 uniformly on K

(2.6)

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2 Probability Measures on Topological Semigroups

and

ˇZ ˇ Z ˇ ˇ ˇ g0 dn  ˇ g d n ˇ > 3kf k ˇ K

K

for infinitely many n. But then (2.5) and (2.6) contradict the last inequality. It follows that eventually ˇZ Z ˇ Z Z ˇ ˇ ˇ f .st/n .ds/n .dt/  f .st/.ds/.dt/ˇˇ < 3kf k: ˇ K K

K K

t u

Proposition 2.2 now follows from (2.4).

w

Notice that if n denotes the nth convolution power of ,  2 P .S / and if n ! ,  2 P .S /, then since convolution is jointly continuous (by Proposition 2.3),    D    D :

(2.7)

The next result characterizes all such weak limits when S is abelian. First we need a definition. For  2 P .S /, we define S./, the support of , by S./ D fx 2 S W .V / > 0 whenever V is open and x 2 V g: Note that S./ is the smallest closed subset of S such that .S.// D 1. Note that for ,  in P .S /, S.  / D S./  S./:

(2.8)

This is a basic formula, and it will be used often. It follows that if  is an idempotent (i.e.,  D   ) probability measure in P .S /, then S./ is a closed subsemigroup of S . Theorem 2.1. Let S be abelian and ,  in P .S / such that    D :

(2.9)

  Then  2 P .S / satisfies (2.9) if and only if for ever B 2 B, .B/ D  By 1 for each y 2 S./.   Proof. Let B 2 B and let f .x/ D  Bx1 . Then (2.9) implies that Z f .x/ D

f .xy/.dy/:

(2.10)

Let .Xn / be a sequence of independent S -valued random variables, each with distribution . Let x 2 S . Write: Sn D xX1 X2 Xn .

2.2 Invariant and Idempotent Probability Measures

69

If Fn is the -algebra generated by X1 ; X2 ; : : : ; Xn , then f .Sn / is a martingale with respect to .Fn /. The reason is the following: For A 2 Fn , Z E ŒIA f .SnC1 / D

E ŒIA f .Sn XnC1 / jXnC1 D y .dy/  Z Z D IA f .Sn y/ dP .dy/  Z Z D f .Sn y/ .dy/ IA dP Z D f .Sn / IAdP [using (2.10)]; D E ŒIA f .Sn / ;

which means that E Œf .SnC1 / jFn  D f .Sn /. Thus, .f .Sn // is a bounded nonnegative martingale, and therefore, lim f .Sn / D Z.x/ exists almost surely, and n!1 for n  1, E ŒZ.x/ j Fn  D f .Sn / almost surely: This means that E Œf .Sn / D E ŒZ.x/  a, say: By the Hewitt–Savage zero-one law (see [38, p. 255]), for n  1, f .Sn / D E ŒZ.x/ j Fn  D a, almost surely: Then for n D 1, we have almost surely Z f .xX1 / D a D E Œf .xX1 / D

f .xy/.dy/ D f .x/;

so that f .x/ D f .xy/; for -almost all y. This means that for x 2 S , y 2 S./, and any B 2 B,      Bx1 D  By 1 x 1 ; which implies Z .B/ D



 Bx

1



Z .dx/ D

     By 1 x 1 .dx/ D  By 1 : t u

Our next result considers the same problems in Theorem 2.1 when S is not assumed to be abelian. The method is necessarily different here, and also the result, which is the best possible in a sense, but not so complete as in Theorem 2.1.

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Proposition 2.4. Let ,  2 P .S / such that  D    D   :

(2.11)

Then for B 2 B, x 2 S./, y 2 S./, we have      Bx1 y 1 D  Bx1      y 1 x 1 B D  x 1 B :

(2.12)

Proof. Proposition 2.2 and hypothesis (2.11) imply that for any bounded Borel measurable function f on S , we have “

Z

“

f d D

f .st/.ds/.dt/ D

f .st/.ds/.dt/:

(2.13)

  Let x 2 S./, and let K be any compact subset of S . Let  Kx 1 D a and  > 0. Since x is regular, there exist  open sets G and W and a closed set C such that K  W  C  G such that  Gx 1 < a C . Notice that the set   ˚  A D s 2 S W  C s 1 < a C  is open by Proposition 2.1, and contains x so that .A/ > 0. Define the function g on S by ˚    g.s/ D max  C s 1  a  ; 0 : Then it follows from (2.13) that 

Z Z

g.st/.ds/  g.t/ .dt/ D 0:

(2.14)

Using (2.11), we have for t 2 S ,    C t 1  a   D

Z

 1 1 

 C t s  a   .ds/ 

so that for t 2 S ,

Z g.st/.ds/  0

Z g.t/ 

We have from (2.14) and (2.15) that Z g.t/ D g.st/.ds/;

g.st/.ds/:

(2.15)

./ almost everywhere:

For t 2 A, g.t/ D 0 therefore g.st/ D 0 for almost all s (with respect to ), for almost all t (with respect to ) in A. Since   g.st/ D 0 )  W t 1 s 1  a C ;

2.2 Invariant and Idempotent Probability Measures

71

since W (as well as W t 1 , for each t) is an open set, functions s !  and, 1 W t s 1 and t ! s W t 1 are both lower semicontinuous, it follows by Proposition 2.1 that for all t in A and all s in S./, we must have    W t 1 s 1  a C : Since x 2 A and K  W ,      Kx 1 s 1   Kx 1 for all s 2 S./. Since by (2.11), Z

 

 1    Kx 1 s 1 .ds/ D 0;  Kx

it follows that for almost all s (with respect to ),      Kx 1 D  Kx 1 s 1 :   From the upper semicontinuity of the function s !  Kx 1 s 1 , we then have      Kx 1 D  Kx 1 s 1     whenever x 2 S./ and s 2 S./. Since  is regular,  Bx1 D  Bx1 s 1 for x 2 S./, s 2 S./ and B 2 B. The second result in (2.12) follows similarly. u t Let us remark that the equalities in (2.12) are not sufficient to ensure (2.11) always. If S is a special semigroup such that (i) S contains a zero 0; (ii) xy D 0 for all x; y in S , then, for all x; y in S , By 1 x 1 D By 1 D ;, if 0 … B and

By 1 x 1 D By 1 D S , if 0 2 B:

The same is true for x 1 y 1 B and y 1 B. Thus, (2.12) holds, whereas    D ı0 (the unit mass at 0) 8;  2 P .S /: In Sect. 2.3, we show that the result of Theorem 2.1 continues to hold when S is a group (not necessarily abelian) in the following sense: For ,  in P .S /,  D    if and only if  D y  for each y in theclosedsubgroup generated by S./.  Let us also remark that the result  By 1 D .B/ D  y 1 B for y 2 S./, instead of (2.12), cannot hold in general in Proposition 2.5, unless S is special. This is clear from our results on idempotent probability measures (i.e., those  in

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2 Probability Measures on Topological Semigroups

P .S / for which  D   ), which we present next. Note that when the weak limit of the convolution iterates .n / of a probability measure  in P .S / exists, it is idempotent. There is an important connection between invariant and idempotent probability measures that are exploited in our characterization of idempotent probability measures. Definition 2.4. A (nonnegative) regular measure , not necessarily finite, on the Borel r -invariant (` -invariant) if for B 2 B and x 2 S ,  sets  of S is called   Bx1 D .B/  x 1 B D .B/ . Notice that for a (nonnegative) regular measure , .S  S.// D 0 and S./ is a closed subset of S . If  is r  -invariant on S , then for any x 2 S , x D  so that S./ D S .x / D S./ x. Similarly, if  is ` -invariant, then for any x 2 S , S./ D S .x / D xS./. Lemma 2.1. If  is an r  -invariant (` -invariant) measure on S , then for x 2 S./, xS./.S./ x/ is left-cancellative (right-cancellative). Proof. Let  be a r  -invariant measure on S . Then for any s 2 S./, S./s D S./. Let x, y, z, and w be in S./ such that .xy/.xz/ D .xy/.xw/. Now S./ .xyx/ D S./ so there exist .tn / in S./ such that tn .xyx/ ! x; This means that

n!1

tn .xyx/z ! xz k tn .xyx/w ! xw

so xz D xw. Thus, xS./ is left-cancellative. Similarly, when  is ` -invariant, S./x is right-cancellative. t u Proposition 2.5. Let  be a measure on S such that it is both r  - and ` -invariant. Then  is a unimodular Haar measure on its support, which is a locally compact topological group. Proof. By Lemma 2.1, for x, y 2 S./, x S./ y is bicancellative since xS./y  ŒxS./ \ ŒS./y. We prove Proposition 2.5 by showing that xS./y contains an idempotent element e, because then for any x, y in S./, by the remark preceding Lemma 2.1, S./ D S./e D S./e  S./y  D S./; and S./ D eS./ D eS./  xS./  xS./ D S./: Thus, S./ is both left and right simple, and, therefore, a group. It follows that it is a topological group (see Theorem 1.4, Chap. 1), since it is locally compact.

2.2 Invariant and Idempotent Probability Measures

73

Let us now prove that for x; y 2 S./, xS./y contains an idempotent. Let K be a compact subset of S./ such that .K/ > 0. Let L be a compact subset of S./ so that    x 1 .xKy/ \ L > 0:   Note that x 1 .xKy/y 1  K and thus,  x 1 .xKy/  .K/ > 0. Write A D LC [ L, C  xKy. Define the finite regular measure  by   .B/ D  x 1 B \ A ;

B  S./:

Define the maps  and ˇ by .s; t/ D .s; ts/ ˇ.s; t/ D .t; s/;

s; t 2 S./:

Using Fubini’s theorem, we have Z  ..C C // D

.C s/.ds/ ZC

 ZC  ZC 

   x 1 .xKys/ \ Ls .ds/ 

  x 1 .xKys/ \ Ls s 1 .ds/    x 1 .xKy/ \ L .ds/ > 0:

Now, let E be any compact subset of the set F  D D  .D D/;

D  xS./y:

Then for any s 2 D, Es D ft W .s; t/ 2 Eg  D  Ds     Therefore,  .Es /   x 1 Es D  x 1 Es s 1 y 1 D 0, so that Es s 1 \D 1D ;.1

since S./ \ x Es s y 1 D ;. Thus  .E/ D 0 whenever E is a compact subset of F . Since  .ˇ.C C // > 0 (ˇ being measure preserving) and since ˇ.C C / is contained in D D, it follows that ˇ.C C / \ .D D/ ¤ ;: This means that there exist u, v, w, z in D such that .z; wz/ D .vu; u/ so that z D vu D vwz or vwz D .vw/2 z. Since D is cancellative, vw D .vw/2 is an idempotent element in D. u t

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2 Probability Measures on Topological Semigroups

Our next result is a basic one needed for much of the limit theory in this book. This result characterizes the structure of all idempotent probability measures on S , and its proof here depends on Propositions 2.4 and 2.5. Theorem 2.2. Assume  2 P .S / and  D . Then S./ is a closed completely simple subsemigroup of S . Furthermore, the structure of  can be described as follows: Let e D e 2 2 S./ and let E.S./e/ eS./e E.eS.// be the usual product representation of S./ (see Theorem 1.1, Chap. 1). There exists 1 2 P .X /, 3 2 P .Z/, where X D E.S./e/, Z D E.eS.// such that  D 1  2  3 , where 2 is the Haar probability measure on eS./e, which is a compact group. The measure  can also be regarded as the product measure 1 2 3 . Conversely, if 1 and 3 are in P .S / and 2 is the Haar probability measure on some compact subgroup G  S such that S .3 / S .1 /  G, then 1  2  3 is an idempotent probability measure on S . Here, as usual, E.A/ is the set of all idempotent elements in the set A. Proof. It follows from (2.8) and Proposition 2.4 that S./S./ D S./I

(2.16)

   9   Bx1 y 1  D  Bx1 ; =  y 1 x 1 B D  x 1 B ; ; for x; y 2 S./; B 2 B:

(2.17)

  Consider the probability measure x for x 2 S./, given by x .B/ D  Bx1 . Notice that for any z 2 S./,         x B.zx/1 D  Bx1 z1 x 1 D  Bx1 .xz/1 D  Bx1 D x .B/ by (2.17) whenever B 2 B. Also if zn 2 S./ and zn x ! z as n ! 1, then for any compact subset K  B, using Proposition 2.1, we have:    x Kz1  lim sup x K .zn x/1 D x .K/ n!1

and for any open subset V  B,   x V z1  lim

n!1

Z

 x V .zn x/1 D x .V /:

Thus, it follows easily (using the regularity of ) that for B 2 B,   x Bz1 D x .B/ for all z in S .x / D S./ x.

(2.18)

2.2 Invariant and Idempotent Probability Measures

75

Similarly, for any x 2 S./, x

 1  z B D x .B/

(2.19)

whenever z 2 S .x / D xS./ and B 2 B. Let us now consider for x 2 S./, the regular probability measure  given by   .B/ D  x 1 Bx1 ;

x 2 S./:

Its support is xS./x. It is clear from the preceding considerations that  is both ` and r  -invariant on its support. It follows from Proposition 2.7 that xS./x is a topological group; therefore, it must be compact, since it supports that Haar probability measure . (See [164, Problem 6.7.2].) Now we claim that for any x 2 S./, S./x xS./

is a left group; is a right group:

(2.20)

To prove (2.20), write H D S./x; then by (2.18), for z 2 H , H z D S .xz / D S .x / D H:

(2.21)

Now for z 2 H , H z \ xS./x is a nonempty left ideal of the group xS./x, which implies that xS./x  H z. This means that the identity e of xS./x belongs to H z, so that by (2.21) H  H z  H e D H e D H:

(2.22)

It follows from (2.21) and (2.22) that H is a left group, establishing that S./x is a left group in (2.20). The other part in (2.20) is verified similarly. Because of (2.20), it follows (by Proposition 1.6 and Theorem 1.7 in Chap. 1) that because S./ has a completely simple kernel that is the union of all minimal left ideals; that is, [fS./x W x 2 S./g, which is dense in S./ by (2.16). Since the kernel is known to be closed (see Theorem 1.7), it follows that S./ is a completely simple subsemigroup of S . For the second part of Theorem 2.2, let e D e 2 2 S./. Then, ıe    ıe is the Haar probability measure of the compact group eS./e, by remarks preceding (2.20). Moreover, notice that by (2.17), we have: Z   ıe  .B/ D Z D

    ıe Bx1 .dx/ D

Z

   Bx1 e 1 .dx/

   Bx1 .dx/ D 2 .B/ D .B/;

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2 Probability Measures on Topological Semigroups

so that  D   ıe  :

(2.23)

Also notice that by (2.23),  D .  ıe /  .ıe  /: By (2.18) and (2.19), e D   ıe is r  -invariant on its support S./ e, which is a left group by (2.20), and e  D ıe   is ` -invariant on its support e S./, which is a right group by (2.20). Since S./e is a left group, S./ e is topologically isomorphic to E.S./e/

eS./e via the mapping ˚ W .x; y/ ! x y. We define the probability measure 1 on E.S./e/ by 1 .A/ D e .˚ .A eS1 ./e// and the probability measure 2 on eS./e by 2 .B/ D e .˚.E.S./e/ B//: It is easily verified that 2 is r  invariant on eS./e, since e is r  -invariant; therefore, 2 is the Haar measure (and both r  - and ` -invariant) on the compact group eS./e. Also, it is verified easily that 1  2 .˚.A B// D 1 .A/2 .B/;

(2.24)

where A  E.S./e/, B  eS./e. Now notice that by (2.23), e D e  e : Therefore, if A and B are as in (2.24), Z e .˚.A B// D e  e .˚.A B// D Z

  e x 1 ˚.A B/ e .dx/

   e ˚ E.S./e/ g 1 B e .d˚.f; g//

D ˚.AeS./e/

D 1 .A/2 .B/:

(2.25)

Notice that (2.24) and (2.25) show that both e and 1  2 induce, via the map ˚, the same probability measure 1 2 on E.S./e/ eS./e. It follows that e D 1  2 :

(2.26)

D 2  3 ;

(2.27)

Similarly, e

2.2 Invariant and Idempotent Probability Measures

77

where 3 is the probability measure on E.eS.// defined by 3 .C / D e .˚.eS./e C //: Theorem 2.2 now follows from (2.26) and (2.27), since  D e  e  D .1  2 /  .2  3 / D 1  2  3 :

t u

Theorem 2.3. Suppose that  2 P .S / and  is r  -invariant. Then S./ is a left group and consequently topologically isomorphic to E.S.// eS./, under the map ˚.x; y/ D xy, where e is a fixed idempotent in S./ and eS./ is a compact group. Moreover if 1 is the probability measure on E.S.// defined by 1 .A/ D .˚.A eS.///; then  D 1 2 , where 2 is the Haar probability measure on eS./. Proof. Notice that any r  -invariant probability measure on S is idempotent. Since for x 2 S./, by (2.20), S./ x is a left group, then for an idempotent e in S./x, we have, by comments preceding Lemma 2.1, S./ D S./e D S./e  S./x  S./x D S./ So that S./ is a left group. The rest of Theorem 2.3 follows from Theorem 2.2. u t Note that whenever  is r  -invariant and consequently, S./ is a left group by Theorem 2.3, then since S./ is right-cancellative, for B  S./ and x 2 S./, Bxx 1 \ S./ D B so that whenever B is a compact subset of S./,     .Bx/ D  Bxx 1 D  Bxx 1 \ S./ D .B/: This property of  is called right-invariance. Thus, every r  -invariant probability measure on S is right-invariant when restricted to its support. Conversely, if  is a right-invariant probability measure in P .S /, then  is r  -invariant. The reason is 1 the following: for B 2 B, x 2 S , any  compact set A  Bx , we  have  .A/ D 1 1 .Ax/  .B/, since Ax  Bx  x  B. This means that  Bx  .B/   for any B 2 B, and therefore 1   Bx1 D  B c x 1   .B c / D 1  .B/, so that  is r  -invariant. This right-invariance property helps us characterize all r  -invariant (possibly infinite) measures on S . Theorem 2.4. Let  be a regular r  -invariant measure (not necessarily finite) on S . Then S./ is a left group if and only if  is right-invariant on its support. Furthermore, if S./ is a left group and S./ has the usual left group representation

78

2 Probability Measures on Topological Semigroups

E G, where E  E.S.// and G  eS./, G a group, then  is of the form 1 2 where 1 2 P .E/ and 2 is a right Haar measure on G. Conversely, any measure of the form 1 2 (as here) on a left group E G is both r  - and right-invariant. Proof. Suppose  is an r  -invariant, right-invariant regular measure on S . Note that since  is r  -invariant, S./x D S./ for x 2 S . Consider the map from S./ to L, the space of bounded linear operators on L2 .S./; / of norm 1, defined by .x/Œf .s/ D f .sx/: Note that kf k2 D k .x/Œf k2 since  is r  -invariant. Thus, .x/ is an isometric linear operator on L2 . We show that it is unitary. It suffices to show that .x/ is surjective. (See [164, Proposition 7.17, p. 153].) To this end, let f 2 L2 . Then there exist simple functions fn , where each fn is a linear combination of indicator functions of compact subsets of S such that kfn  f k2
0g is an open set with positive  measure. Then .xs1 / Œf  ¤ .xs2 / Œf , which is a contradiction.

2.2 Invariant and Idempotent Probability Measures

79

Thus, for each x 2 S./, since xS./ is anti-isomorphic to a subsemigroup of a group, it is bicancellative. Therefore, as in the proof of Proposition 2.5, we can also show that for any x, y 2 S./, xS./y contains an idempotent. (In this part of the proof of Proposition 2.5, we did not use the ` -invariance of .) Thus, if e D e 2 2 xS./y  S./y, S./ D S./ e D S./e  S./y  S./y D S./; so that for any y 2 S./, S./y D S./: This means that S./ is a left group. For the last part of Theorem 2.4, let us assume that S./ is a left group with its usual product representation E G, so that ˚ W E G ! S./; where ˚.x; y/ D xy is a topological isomorphism and E; G inherit the topology from S./. Here, G D e0 S./, a group, and E  E.S.//, where e0 is a fixed idempotent in S./. Let h be a continuous function on E with compact support such that h  0. For f , a continuous function with compact support on G, we define Z I.f / D

h.e/f .g/0 .d.e; g//;

where 0 .A/ D .˚.A//, A  E G. Notice that if fy .g/ D f .gy/ for g and y in G, then Z   I fy D h.e/fy .g/0 .d.e; g// D I.f /; since 0 is r  -invariant and h.e/fy .g/ is the right translate of the function .e; g/ ! h.e/f .g/ by the element .e; y/, e 2 E. Since I is a right-invariant positive linear functional on Cc .G/, there exists a right Haar measure h on G such that for f2 Cc .G/, Z Z f .g/h .dg/ D

h.e/f .g/.d.e; g//:

Thus, given a right Haar measure 2 on G, there exists a positive number ˛.h/ such that Z Z ˛.h/ f .g/2 .dg/ D h.e/f .g/.d.e; g//:   Clearly, ˛ (with ˛.h/ D ˛ hC  ˛ .h /) is a positive linear functional on Cc .E/ so that there is a nonnegative regular measure 1 on E such that for h 2 Cc .E/, Z ˛.h/ D

h.e/1 .de/

80

2 Probability Measures on Topological Semigroups

so that for h 2 Cc .E/, f 2 Cc .G/, Z “ h.e/f .g/.d.e; g// D h.e/f .g/1 2 .d.e; g//: The rest of Theorem 2.4 follows easily by an application of the Stone–Weierstrass theorem (see [163, p. 78]). t u Before we close Sect. 2.2, we discuss two related questions, one on (infinite) regular measures  satisfying Z   .B/ D  Bx1 .dx/ (2.28) for all Borel sets B (for which the integral on the right makes sense), and the other on the existence of right-invariant (i.e., .Kx/ D .K/ for compact K  S , x 2 S ) measures on S . Notice that (2.28) implies that .S / D .S / .S /

(2.29)

since for x 2 S , S x 1 D S , so that a measure satisfying (2.28) must be, if not a zero measure, either an idempotent probability measure or an infinite measure. When S is a group, it has been proven in [144] that there cannot exist an infinite regular measure  satisfying (2.28). The proof is quite involved (and works in contexts much more general than groups), and thus is not presented here. However, we point out that the proof is based on the following observation: If  is a solution of (2.28), then for every x in S./, there are continuous homomorphisms ˇx and x from S./ into the positive reals such that      Bx1 y 1 D ˇx .y/ Bx1 and

     y 1 x 1 B D x .y/ x 1 B ;

for every Borel set B 2 B and x; y in S./. Furthermore, there cannot exist an infinite regular measure  on S when S is a group such that    By 1 D ˇ.y/.B/; for B 2 B and y 2 S./, R where ˇ is a continuous homomorphism from S into the positive reals such that ˇd D 1. Finally, we consider the question of the existence of right-invariant measures on S . Recall that  is right-invariant if  is regular, and for compact K  S and x 2 S , .Kx/ D .K/. If  is finite and right-invariant, then  is r  -invariant (as was noted earlier); therefore, finite right-invariant measures are completely known and fully characterized earlier. The problem is then with those that are infinite. How do they arise?

2.2 Invariant and Idempotent Probability Measures

81

Suppose that right translations are open in S ; that is, the maps x ! x y are open maps for each y in S . Then if  is a right-invariant measure on S and there are x, y, z in S./ such that xy D zy; x ¤ z; then there are disjoint open sets V .x/, V .z/ containing x and z, respectively, such that .V .x/ [ V .z// D .V .x/y [ V .z/y/ D .V .x/y/ C .V .z/y/  .V .x/y \ V .z/y/ < .V .x// C .V .z// since  is right-invariant (so that S./ is a subsemigroup of S ), V .x/y \ V .z/y is nonempty open and it intersects S./. This means that right-invariance and open right translations imply that S./ is right-cancellative. Theorem 2.5 is given in [170]. Theorem 2.5. Let S be right reversible (that is, S x \ Sy ¤ ; for every x; y 2 S ). (This property holds for example when S is abelian.) Suppose S has translations open. Then S can support a left- and right-invariant measure  if and only if S can be topologically embedded (via a topological isomorphism ˛ W S ! ˛.S /  G) as an open subspace in a locally compact Hausdorff topological group G and  on S is the restriction of a unimodular Haar measure on G. Note that in Theorem 2.5, to embed S topologically as an open subspace in a topological group, it is necessary for translation maps in S to be open. It is clear that Theorem 2.5 gives only a partial solution to the problem of characterizing all right-invariant measures on S . The reader can consult [3] for other (partial) results in this context. The problem in the case when S is abelian has a more satisfactory solution (see Theorem 2.6) given in [183]. Theorem 2.6. Let S be abelian. If  is right-invariant on S , then there is a locally compact abelian group G and a continuous homomorphism q W S ! G such that q.S / has a nonempty interior and that there is a Haar measure m on G such that for compact K  S , .K/ D m.q.K//. The map q is injective if and only if S is cancellative; it is a homeomorphism onto q.S / if and only if S has the following property: sn ! s; sn tn ! st ) tn ! t:

Section 2.2 Exercises Below, S is always a locally compact Hausdorff second countable topological semigroup (unless otherwise mentioned).

82

2 Probability Measures on Topological Semigroups

Exercise 2.1. Let  2 P .S / and F D fB  S W B is a Borel set such that given  > 0, there is an open set u and a closed set V such that V  B  U and .U  V / < g. Show that F is the class of all Borel subsets of S . (Hint: F is a -algebra and it contains all closed subsets, since in a metric space, a closed subset is the countable intersection of decreasing open subsets.) Exercise 2.2. Show that for  2 P .S /, given any Borel subset B  S and any  > 0, there exist an open set U and a compact subset K such that K  B  U and .U  K/ < . Exercise 2.3. Show that for  2 P .S /, the set S./, defined by S./ D fx 2 S W .v/ > 0 for any open set V containing xg; is a nonempty closed subset of S and .S.// D 1. This set S./ is the support of . Exercise 2.4. Let 1 and 2 be in P .S / such that 1 .B/  2 .B/ for any Borel set B  S . Show that 1 D 2 . Exercise 2.5. Show that in P .S /, n converges to  weakly iff for any open subset G  S , .G/  lim inf n .G/. n!1

Exercise 2.6. Let n , n  1, be in P .S / and  2 B.S / such that n converges to  in the weak -topology. Show that n converges to  weakly iff  2 P .S /. Exercise 2.7. Explain how the inequalities in (2.4) follow under the assumptions made there. Exercise 2.8. Prove the formula (2.8). Exercise 2.9. Prove (2.13) assuming (2.11). Exercise 2.10. Let  be an infinite nonnegative regular measure (definition given at the end of Sect. 2.1) on the Borel subsets of S . Show that .S  S.// D 0, when S./ is as defined in Exercise 2.3. Exercise 2.11.  Let  2 P .S / or  be as given in Exercise 2.10. For x 2 S , define x .B/ D  Bx1 . Show that x is a regular measure and S .x / D S./ x. Exercise 2.12. Let  2 P .S / and V be an open subset of S . Show that if xn 2 S and xn ! x, then      V x 1  lim inf  V xn1 : Give the corresponding inequality when V is a closed subset of S . Exercise 2.13. Let X G Y , X [ G [ Y  S , be the usual product representation of a completely simple semigroup S . Here, G is a compact group and Y:X  G. Let 1 2 P .X /, 3 2 P .Y / and 2 be the Haar probability measure on G. Show that 1  2  3 is an idempotent probability on S .

2.3 Weak Convergence of Convolution Products of Probability Measures

83

Exercise 2.14. Let  be an r  -invariant probability measure on the Borel subsets of S . Show that  is right invariant (that is, for compact K  S./ and x 2 S./, .Kx/ D .K/). Exercise 2.15. Let  2 P .S / be such that for any compact K  S and x 2 S , .Kx/  .K/. Show that S./ is a left group. (Hint: Show that  is r  -invariant.)

2.3 Weak Convergence of Convolution Products of Probability Measures As before, S is a locally compact Hausdorff second countable topological semigroup and Cc .S / is the vector space of real continuous functions on S with compact support. It is then wellknown (from Banach–Alaoglu’s theorem in functional analysis) that the unit ball in the dual of Cc .S / is weak compact. Thus, the set B.S /  f j  is a nonnegative regular Borel measure on S with .S /  1g  is compact in the weak topology. Recall: A net . R ˛ / in B.S /,R w converges to  in B.S / if and only if for every f in Cc .S /, f d˛ ! f d. However, P .S /  f 2 B.S / W .S / D 1g need not be weak compact, unless S is compact. Note that in P .S /, weak compactness is equivalent to weak compactness, and thus P .S / is weak compact if and only if S is compact. (See [182, Theorem 6.4, p. 45].) For a subset   P .S /, the weak closure of  in P .S / is weak compact, if  is tight; that is, given  > 0, there is a compact subset K  S such that

 2  )  .K / > 1  :

(2.30)

The reason for this is obvious since  2 w -closure of  and  is tight only if  2 P .S /, and since B.S / is w -compact. In Sect. 2.3, we present various results on the weak (and weak ) convergence of convolution products of probability measures of the form k;n  kC1  kC2   n

.n > k/:

Let us consider a few simple examples. Example 2.1. Suppose there is an ideal I 2 B of S such that for a sequence n in P .S /, we have: 1 X ki ;ki Cri .I / D 1; (2.31) i D1

for sequences .ki / and .ri / of positive integers with the property that for i  1, ki C ri < ki C1 ;

1  ri  ri C1 :

Here, we show that for s  1, lim s;t .I / D 1. t !1

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2 Probability Measures on Topological Semigroups

To this end, if .Xi / is a sequence of independent S -valued random variables such that the distribution of Xi is i , it follows from (2.31) that 1 X

  Pr Xki C1 Xki C2 Xki Cri 2 I D 1:

i D1

By the Borel–Cantelli lemma, we then have ! 1 [ 1 \   Xki C1 Xki C2 Xki Cri 2 I D 1:

Pr

mD1 i Dm

Hence, given  > 0, for every m  1, there exists n.m/ such that 0 Pr @

n.m/ [



Xki C1 Xki C2 Xki Cri

1  2 I A > 1  :

i Dm

But notice that the set inside the parentheses is contained in the set fXs XsC1 Xt 2 I g ; whenever s  km C 1 and t  ki C ri for i D n.m/. This is true because I is an ideal. This means that for s  1, there exists t.s/, depending on , such that t  t.s/ ) s;t .I / > 1  : It is clear that if I is a singleton say, I D f0g, then for each s  1, the sequence s;n converges weakly to the unit mass at 0. Example 2.2. Let S be the (multiplicative) topological semigroup of nonnegative reals with usual topology. Let  be the normalized Lebesgue measure on the interval Œ0; e, where e D exp.1/. We show that the sequence .n / converges vaguely (that is, in the weak sense) to the measure  given by .f0g/ D 1=2 and .0; 1/ D 0. To show this, we use the Central Limit Theorem. Let ˇ be the probability measure on the reals .1; 1/, under addition, induced by  and the map W x ! e x , so that ˇ.B/ D . .B//: Let .Xn / be an independent and identically distributed (i.i.d.) sequence of real random variables with distribution ˇ. Then it follows that m D E .Xi / D

1 e

Z

e

ln x dx D 0 0

2.3 Weak Convergence of Convolution Products of Probability Measures

and

85

   2  Var .Xi / D E Xi2  m2 > 0:

Notice that for 0 < k < 1, n .Œ0; k/ D ˇ n ..1; ln k/: n P Xi . Then, for any k > 0, ˇ n ..1; ln k/ D Pr .Yn 2 .1; ln k/ D Let Yn D i D1 i   p and this converges by the Central Limit Pr ŒYn  nm =s n 2 1; slnpkn Theorem to Z 0 1 1 2 p e .1=2/x dx D : 2 2 1

Example 2.3. In Example 2.2, if we had considered the interval Œ0; a, a > 0, (instead of the interval Œ0; e), but defined the probability measure  in the same manner, then by using the Central Limit Theorem, we would easily obtain a weak convergence result for .n / that is strikingly different. If a > e, the sequence n converges vaguely to the zero measure, which would mean that for any ˛ > 0, lim n .Œ0; ˛/ D 0:

n!1

If a < e, the behavior is just the opposite; in this case, for any ˛ > 0, lim n .Œ0; ˛/ D 1;

n!1

that is, the sequence n converges weakly to the unit mass at 0. Example 2.4. Let S D Œ0; 1, the real numbers x, 0  x  1, with the usual topology and multiplication. Then S is a compact topological semigroup. Consider any sequence of probability measures n in P .S /. Let .Xn / be a sequence of independent S -valued random variables such that Xn has distribution n . Write Xk;n  XkC1 XkC2 Xn ;

n > k:

Notice that Yk D lim Xk;n exists pointwise. Let Ak  fYk > 0g. Then Ak  n!1 AkC1 . Notice that on the set E

1 [ 1 \

Ak ;

nD1 kDn

lim Ym D 1. Since E is a tail event (Ak  AkC1 ), it follows by the Kolmogorov

m!1

zero-one law (see [38, p. 254]) that Pr.E/ D 0 or 1. Also, on S  E, lim Ym D 0. m!1 This means that the limit Y D lim Ym m!1

always exists, and either Y D 0 almost surely or Y D 1 almost surely.

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2 Probability Measures on Topological Semigroups

It is now clear from the preceding discussion that for each k  1, the sequence k;n ( kC1   n ) as n ! 1 converges weakly to k 2 P .S /, where k is the distribution of Yk , and that as k ! 1, k converges weakly to either the unit mass at 0 or the unit mass at 1. Notice that for any bounded continuous function f on S , Z Z Z Z   f dk;n D f Xk;n dP ! f .Yk / dP D f dk ; by the standard bounded convergence theorem of integration theory. Example 2.5. Let S be the circle group fz is a complex number such that jzj D 1g with usual topology and multiplication. Let n 2 P .S / be defined by  



1 2 i 2 i D n exp  p D : n exp p 2 n n Then notice that the rth Fourier coefficient of n is given by

Z fr .n / D

t r n .dt/ D cos

2 r p n

 :

Let r be any integer. Then we have (i) There exists m such that ˇ ˇ ˇ ˇ ˇ r ˇˇ 1 ˇˇ r ˇˇ ˇ n  m ) ˇsin p ˇ > ˇ p ˇ : 2 n n (ii) For each r ¤ 0,

1 Y

2 r cos p n nDm since

1  X nDm

2 r 1  cos p n



 D 0;

1 1 2 2X 1 D 1;  r 2 n nDm

where m is as in (i). This means that for each nonzero integer r,   lim fr k;n D 0:

n!1

Notice that if  is the Haar probability measure on the circle group, then for any integer r, fr./ D 0, so that   lim fr k;n D fr./:

n!1

Then it follows (see [182, Theorem 3.3, p. 76]) that k;n converges weakly to the Haar probability measure on S as n ! 1.

2.3 Weak Convergence of Convolution Products of Probability Measures

87

Theorem 2.7 (below) is a general result on the weak convergence of the sequence of averages of convolution powers of a probability measure. It is not essential in Theorem 2.7 for S to be locally compact; for example, Theorem 2.7 also holds when S is a complete metric separable topological semigroup. Theorem 2.7. Let  2 P .S / and suppose that the sequence .n / is tight; that is, fn W n  1g satisfies (2.30). Suppose also that " S D cl

1 [

# S./n :

nD1

Let K D f 2 P .S / W  is a weak limit point of the sequence .n /g. Also, let us define: S0 D [ fS./ W  2 K g and S1 D S 0 : Then the following assertions hold: (i) The sequence 1=n

n P

k converges weakly to a probability measure  such

kD1

that  D    D    D   , and S./ is the closed completely simple kernel of S with a compact group factor. (ii) The set K is a group with respect to convolution as multiplication. The support S./ of  coincides with the set S1 . If is the identity of K , then S. /  S1 , and they have the following product representations: S1 D X H Y;

S. / D X H1 Y

(the usual product representation for completely simple semigroups), where H1 is a closed normal subgroup of the compact group H , and X; Y have their usual meanings. Also if  2 K and  ¤ , then S./ D X gH1 Y;

g 2 H  H1 ;

if the product representation of the idempotent probability measure  (see The orem 2.2) is 1  2  3 , then the product representation of  is 1  2  ıg  3 , where 1 2 P .X /, 3 2 P .Y /, and 2 is the Haar probability measure on H1 . (Here, equality for sets is used in the sense of a topological isomorphism.) Furthermore, the set S0 is also a completely simple subsemigroup of S , and K D f g if and only if   D . (iii) The sequence n converges weakly if and only if there does not exist a closed normal subgroup H0 of H such that S./ .X H0 Y /  X gH0 Y

88

2 Probability Measures on Topological Semigroups

for some g 2 H H0 and YX  H0 if and only if there does not exist a closed normal subgroup H0 of H such that YX  H0 and eS./e  gH0 for some g 2 H  H0 , where e is the identity of the group H . (iv) The sequence n converges weakly if and only if lim inf .S./n / is nonempty, n

where lim inf .S./n / is defined as fx 2 S W for every open set V .x/ containn

ing x, there exists a positive integer N such that n  N implies that V .x/\ S./n is nonemptyg. Proof. (i) Write: n  1=n

n X

k :

kD1

Then, for k  1, k  n D n  k and     lim n  k  n  D 0:

n!1

(2.32)

Since the sequence n is tight, the same is true for the sequence n so that fn W n  1g is weakly relatively compact. Let 1 and 2 be two weak limit points of .n /. By (2.32), for every k  1, k  1 D 1  k D 1 ; k  2 D 2  k D 2 :

(2.33)

It follows that for every n  1, n  1 D 1  n D 1 n  2 D 2  n D 2 ;

(2.34)

which then implies that 1 D 2 ( , say), and  D  D  D . It follows easily that S./ is the completely simple semigroup with a compact group factor (since it is the support of an idempotent probability measure; see Theorem 2.2), which is also the kernel of S (since clŒS./S./ D clŒS./S./ D S./). (ii) Notice that if T is a compact Hausdorff (second countable) topological semigroup, then for any element x 2 T , the subsemigroup A.x/ D cl fx n W n  1g is a compact abelian subsemigroup. Therefore, it has a closed kernel (see Corollary 1.3) K. If B denotes the set fy j y is a limit point of the sequence x n g, then it is easy to verify that (i) K  B; (ii) Bz D zB D B for every z 2 B.

2.3 Weak Convergence of Convolution Products of Probability Measures

89

This means that B is a group containing the ideal K. For k 2 K, k 1 2 B, so that the identity element of B belongs to K, which means B D K. From the preceding arguments, it follows that the set K is a group with respect to convolution. Let us now prove that the set S1 given in the theorem is the closed, completely simple kernel of S . To this end, it suffices to show that S1 D S./, where  is as in Assertion (i). Let U be an open set containing S./. Then we claim that lim n .U / D 1:

(2.35)

n!1

Let  > 0. Since .n / is tight, there exists a compact subset K1  S such that n .K1 / > 1  ;

n  1:

(2.36)

Let K2 be a compact subset of S./ such that  .K2 / > 1  :

(2.37)

Since S./ is an ideal of S , the set K1 K2  S./  U , and therefore, there exists an open set V  K2 such that K1 V  U: (2.38) Now n !  weakly; therefore, 1   < .V /  lim inf n .V /; n!1

which means that there exists a positive integer m such that m .V / > 1  :

(2.39)

By (2.36), (2.38) and (2.39), for all n  1, nCm .U /  n .K1 / m .V / > .1  /2 : This proves (2.35). Let us now write S./ D

1 \

Un ;

Un open;

Un  UnC1 :

nD1

Let  2 K and x 2 S./. Suppose that x … Uk for some k > 1. Then there is an open set V .x/ containing x such that V .x/ \ Uk D ;. Since .V .x// > 0, there exists infinitely many n such that for some ı > 0, n .V .x// > ı;

90

2 Probability Measures on Topological Semigroups

which means that lim inf n .Uk /  1  ı:

n!1

This contradicts (2.35). This means that for each  2 K , 1 \

S./ 

Uk D S./;

kD1

which implies that S1  S./: Since for  2 K ,    as well as    is in K , it is clear that S1 is an ideal of S . Since S./ is simple, it follows that S1 D S./. Let D  be the identity of the group K . Then S. / ( S1 ) as well as S1 is a closed, completely simple subsemigroup of S with a compact group factor. Let e D e 2 be an idempotent element in S. /, and let us define X  E .S1 e/ ; Y  E .eS1 / ;

X1  E.S. /e/; Y1  E.eS. //;

H  eS1 e;

H1  eS. /e:

(Here, E.D/ denotes the set of idempotents in the set D.) Then it follows from Theorem 1.7 that S1 D X H Y;

S. / D X1 H1 Y1 ;

where equality means topologically isomorphic. Let us now show that X D X1 and Y D Y1 . Let .x; h; y/ 2 S1 . It follows from the definition of S1 that there exist n 2 K , elements .xn ; hn ; yn / in S .n / such that .xn ; hn ; yn / ! .x; h; y/ as n ! 1. By the group property of K , there exist ˇn 2 K such that ˇn  n D n  ˇn D . This means that xn 2 X1 and yn 2 Y1 . It is easily verified that X1 and Y1 are both closed in S so that x 2 X1 and y 2 Y1 . Thus, X D X1 and Y D Y1 . Consequently, YX  H1 . Now we prove the last assertions (including the normality of H1 ) in Assertion (ii). Let  2 K . Since   D , it is clear that if .x; g; y/ 2 S./, then X H1 g fyg  .X H1 Y / .x; g; y/  S./: Since   D  and YX  H1 , then    ˚ X H1 g Y  .X H1 g fyg/ fxg .yx/1 Y  S./: Similarly, X gH1 Y  S./. Thus, we must have S./  X U Y;

2.3 Weak Convergence of Convolution Products of Probability Measures

91

where the set U is a union of right cosets of H1 , and similarly, it is also a union of left cosets of H1 . Since there exists ˇ 2 K such that   ˇ D , it is not difficult to verify that U consists only of a single right coset of H1 . It follows similarly that S./ . X H1 g Y / D X gH1 Y: Thus, whenever .x; g; y/ 2 S./ for some  2 K , we have H1 g D gH1 . If g 2 H , then there exist n 2 K and gn 2 H such that S .n / D X H1 gn Y; H1 gn D gn H1 , and gn ! g. If h 2 H1 , then there exist hn 2 H1 such that hgn D gn hn : Since H1 is compact, it follows that hg 2 gH1 proving H1 is a normal subgroup of H . Now we must show that if S./ D X gH1 Y , then g … H1 whenever  ¤ . To prove this, notice that if g 2 H1 , then S./ D S. /. Since   D   D , it follows from Proposition 2.5 that for any Borel subset B  S./ and u, v in S./,      Bu1 v1 D  Bu1 :

(2.40)

It follows from (2.40) that for u 2 S./,   2 Bu1 D

Z

     Bu1 v1 .d v/ D  Bu1 :

This means that since 2  D 2 and   D  and S. / D S./, Z  .B/ D 2

   Bu1 .d u/ D 2

Z

   Bu1 .d u/ D .B/;

for all Borel subsets B  S./. This implies that  D 2 , so that  D , which is a contradiction. Now about the product representation of : First notice that  D    . Following Theorem 2.2, consider the product representation D 1  2  3 , where 1 2 P .X measure on  Haar probability

 H1 . Then ˚ /, 3 2 P .Y /, and 2 is the   D 1  2  . 3    1 /  2  ıg1  ıg  3 D 1  2  ıg  3 , since the support of 3    1 is gH1 , so that 3    1  2  ıg1 2 P .H1 / and consequently the product inside f g is simply 2 . The proof of Assertion (ii) will be complete if we show that the set S0 is also a completely simple semigroup. Notice that if J is an ideal of S0 , then for u 2 J , uS0 \ S./ ¤ ; for some  2 K . Therefore, since K is a group, uS0 \ S. / ¤ ;. This means that S. /uS0 \ S. / ¤ ;:

92

2 Probability Measures on Topological Semigroups

Since S. /uS0 \ S. / is an ideal of S. /, and S. / is simple, S. /  S. /uS0  J: For  2 K ,   D  and therefore, S./ D S./S. /. Consequently, S./  J . Thus, J D S0 , and S0 is simple. Since S0  S1 and S1 is completely simple, it follows that S0 is a completely simple subsemigroup of S . (iii) The if part in the first if-and-only-if statement follows from Assertion (ii) since if n does not converge weakly, then   ¤ , and the support of   must be of the form X gH1 Y; g 2 H  H1 ; where H1 is a normal subgroup of H . This means that S./ .X H1 Y /  S.  / D X gH1 Y: Now suppose there exists a closed normal subgroup H0 of H such that YX  H0 and S./ .X H0 Y /  X gH0 Y; g 2 H  H0 : (2.41) Then this implies that for each positive integer n, S./n .X H0 Y /  X g n H0 Y: "

Since S D cl

1 [

(2.42)

# S./n

nD1

and S S1 D S1 , it is clear that " H D cl

1 [

# n

g H0 :

nD1

Suppose there is a (smallest) positive integer n such that g n 2 H0 . In that case, H0 and gH0 are both open subsets of H . If ˇ is a probability measure with its support contained in X H0 Y , then S .n  ˇ/  X H0 Y

(2.43)

for infinitely many n and also S .n  ˇ/  X gH0 Y

(2.44)

for infinitely many other values of n. Since the sets on the right of (2.43) and (2.44) are both closed and open in X H Y , it is then clear that the sequence n  ˇ (and therefore the sequence n ) cannot converge weakly. Now let us consider the other

2.3 Weak Convergence of Convolution Products of Probability Measures

93

possibility when g n … H0 for all positive integers n. In this case, we consider a limit point h of the infinite sequence gn . Notice that hH0 \ hgH0 D ;, and therefore there are open subsets V1 and V2 with disjoint closures such that hH0  V1 and ghH0  V2 . Since g n H0  V1 for infinitely many n and again g n H0  V2 for infinitely many other values of n, we see that S .n  ˇ/  X V1 Y

(2.45)

for infinitely many n and also S .n  ˇ/  X V2 Y

(2.46)

for infinitely many other values of n, where ˇ 2 P .S / and S.ˇ/  X H0 Y . It follows from (2.45) and (2.46) that the sequence n  ˇ (and therefore, the sequence n ) cannot converge weakly. This completes the proof of the first if-and-only-if statement in Assertion (iii). To complete the proof of Assertion (iii), it is enough to show that (2.41) is equivalent to eS./e  gH0 , where e is the identity of H . To this end, suppose that eS./e  gH0 :

(2.47)

Let x 2 S./ and y D .e1 ; h; f1 /, where h 2 H0 , e1 2 X and f1 2 Y . Notice that ex D e.ex/ 2 eS1 so we can write   ex D e; h0 ; f ; h0 2 H; f 2 Y: Also, exe D .e; h0 ; f / .e; e; e/ D .e; h0 ; e/, since f e D e, where Y is a right zero semigroup. By (2.47) h0 D exe 2 gH0 . Now exy D .e; h0 ; f / .e1 ; h; f1 / D .e; h0 f e1 h; f1 / so that exy 2 X gH0 Y , since h0 2 gH0 and YX  H0 . Let us now write xy D .e2 ; h2 ; f2 / ; h2 2 H: Then we have exy D .e; e; e/ .e2 ; h2 ; f2 / D .e; h2 ; f2 / since ee2 D e, and this means that h2 2 gH0 ; thus (2.41) holds. Conversely, (2.41) implies that S./ .XH0 Y /  X gH0 Y so that S./e  X gH0 Y; and therefore, eS./e  .eX / .gH0 / .Y e/ D gH0 ; which implies (2.47). w (iv) If n ! , then for any x in S./ and any open set V containing x, 0 < .V /  lim inf n .V /; n!1

94

2 Probability Measures on Topological Semigroups

which means S./  lim inf .S./n / : n!1

To prove the converse, let us assume that lim inf .S./n / ¤ ;:

n!1

Then it is easily verified that lim inf .S./n /

n!1

is an ideal of S . Suppose that the sequence n does not converge weakly. Then, by Assertion (iii) it follows that there exists a closed normal subgroup H0 of H such that for each positive integer n and for some g 2 H  H0 , eS./n e  g n H0 ;

e D e 2 2 H:

(2.48)

This follows using the same argument used preceding (iv) above. Let x 2 H . Since H is a topological group, given any open set U.x/ containing x, there exist open sets U.y/ containing y 2 H and U.z/ containing z 2 H such that U.x/ \ H  ŒeU.y/eŒeU.z/e1 :

(2.49)

Now S1 , as the kernel of S , is a subset of lim inf .S./n / ;

n!1

and this means there exists a positive integer N such that U.y/ \ S./N ¤ ;;

U.z/ \ S./N ¤ ;:

It follows from (2.48) that eU.y/e \ gN H0 ¤ ;;

eU.z/e \ g N H0 ¤ ;:

It follows from (2.49) that U.x/ \ H \ H0 ¤ ;: This means that x 2 H0 . Thus, H0 D H , and this is a contradiction.

t u

Noting that a completely simple subsemigroup (with a compact group factor) in a group is a compact subgroup, one can now easily obtain the following corollary. Corollary 2.1. Let S be a compact group and  2 P .S /. Let D 2 be the identity of the group K D f 2 P .S / W  is a weak limit point of the sequence n g ;

2.3 Weak Convergence of Convolution Products of Probability Measures

95

where convolution is the multiplication. Assume that S D cl

1 [

! S./

n

:

nD1

(Recall that a compact cancellative semigroup is a compact topological group so that the smallest closed subsemigroup containing S./ in a compact group is also the smallest closed subgroup containing S./). Then the following assertions are equivalent: (i) The sequence .n / is weakly convergent; (ii) The set lim inf .S./n / is nonempty; n!1

(iii) The sets lim inf .S./n / and lim sup .S./n /

n!1

n!1

are equal; if  2 K , then S./  lim sup .S./n / : n!1

Note that lim sup An  fx 2 S j if V is open and x 2 V , then n!1

V \ An ¤ ; for infinitely many ng. (iv) S is the smallest closed subgroup containing the set [ fS./n S./n W n  1g I (v) S./ is not contained in a proper coset of any closed normal subgroup of S ; (vi) S./  is not contained   in any proper coset of S. /; (vii) Bx1 D x 1 B D .B/ for x 2 S , B  S . Proof. The proof is left to the reader.

t u

In Sect. 2.2, when S is abelian, we determined in Theorem 2.1 all the -invariant ( 2 P .S /) probability measures . Corollary 2.2 and 2.3 address the same problem when S is a group. Corollary 2.2. Suppose that S is a group and ;  are in P .S /. Then  D    if and only if  D x for each x in the smallest closed subgroup of S containing S./. Proof. The if part is trivial. For the only-if part, let  D   . Then we have for n  1,  D    n: (2.50)

96

2 Probability Measures on Topological Semigroups

Given  > 0, let K be a compact subset such that .K/ > 1  . Then Z

   n x 1 K .dx/

1   < .K/ D Z

   n x 1 K .dx/ C 

 K

    n K 1 K C ; ˚  where K 1 K D [ x 1 K W x 2 K . Since the set K 1 K is a subset of S (which is a topological group), it is compact and, consequently, the sequence  n is tight. We can now use Theorem 2.7 so that n 1X k  n kD1

converges weakly to some idempotent probability measure ˇ. Also we have from n P (2.50),  D   1=n  k . Since, by Proposition 2.3, convolution is continuous kD1

with respect to weak topology, we have  D   ˇ;

(2.51)

where ˇ D ˇ  ˇ and, consequently, by Theorem 2.2, S.ˇ/ is a compact subgroup of S , S.ˇ/  S./, since  ˇ D ˇ  D ˇ, (and ˇ is the Haar probability measure on S.ˇ/).   Now let K be any compact subset of S . Since the map x !  Kx 1 is upper-semicontinuous (by Proposition 2.1), there exists z 2 S.ˇ/, which is compact, such that   ˚     Kz1 D sup  Kx 1 W x 2 S.ˇ/ : It follows from (2.51) that 

1

 Kz so that



Z D

   Kz1 x 1 ˇ.dx/

     Kz1 D  Kz1 x 1

(2.52)  for ˇ-almost all x in S.ˇ/. Suppose there exist x0 in S.ˇ/ such that  Kz1 x01 <    Kz1 . Then since x !  Kz1 x 1 is upper-semicontinuous, 

˚

    x 2 S j  Kz1 x 1 <  Kz1

2.3 Weak Convergence of Convolution Products of Probability Measures

97

is an open set containing x0 in S.ˇ/ (and, therefore, has positive ˇ-measure) and this contradicts that (2.52) holds almost everywhere .ˇ/. Thus, we have:      Kz1 D  Kz1 x 1   for all x in S.ˇ/. Since S.ˇ/ is a group, it is clear that .K/ D  Kx 1 for all x in S.ˇ/. t u As far as we know, it is still an unsolved problem to determine all the -invariant probability measures , where  and  are in P .S / and S is nonabelian or not a group. However, when S is compact, or more generally, when the sequence  n is tight, we have Corollary 2.3. Corollary 2.3. Let  2 P .S / such that the sequence . n / is tight and ! 1 [ n S./ : S D cl nD1

Let  2 P .S / such that    D :

(2.53)

Then the following assertions are valid: (i) S has a completely simple kernel K D S .0 /, where 0 is the weak limit of n 1X k  n kD1

and   0 D 0   D 0 . (ii) S./  K and  D   . (iii) If e D e 2 is an idempotent element in S./, which is a completely simple subsemigroup of K, then the product representations of S./ and K take the form K DX G Y (2.54) S./ D A G Y , A  X; where X  E.Ke/, Y  E.eK/, G  eKe; moreover,  and 0 have the following representation: 0 D 1  2  3 ;  D 1  2  3 ;

(2.55)

where 1 2 P .X /, 1 2 P .A/, 3 2 P .Y /, and 2 is the Haar probability measure on G. Conversely, if 0 is the weak limit of n 1X k  n kD1

and takes the form of (2.55), then any  2 P .S / written as in (2.55) satisfies (2.53).

98

2 Probability Measures on Topological Semigroups

Proof. Assertion (i) follows from Theorem 2.7. Suppose now that (2.53) holds for some  2 P .S /. Then ! n 1X k D ; n  1I   n kD1

therefore,   0 D  and S./ D S./S .0 /  S .0 / D K. Now let x 2 S./ and B be any Borel set. Then   0 D  implies that Z  1       x B D 0 y 1 x 1 B .dy/ D 0 x 1 B ; (2.56) by Proposition 2.5, since 0 D 0  0 . This means that Z     .B/ D  x 1 B .dx/ Z   D 0 x 1 B .dx/ D   0 .B/ D .B/ so that  D   . By Theorem 2.2, S./ is a completely simple subsemigroup of K D S .0 /. Since S .0 / D cl ŒS./S .0 /, the representations in (2.54) follow immediately. By Theorem 2.2, 0 takes the form of (2.55), where 2  3 D ıe  0 (see (2.27)). By (2.56), ıe   D ıe  0 ; therefore, since  D   ,  must take the form given in (2.55). For the converse part, if  and 0 take the form of (2.55) and 0   D 0 , then since S .3  1 /  YX  G, 2  .3  1 / D 2 D 2  2 : Therefore, it follows that   0 D  and, consequently,    D .  0 /   D   .0  / D   0 D : t u Notice that in Corollary 2.3, if (2.53) is replaced by the stronger condition  DD then  has to coincide with 0 there. However, what happens when this equation holds, but the sequence  n is not tight? Look at Example 2.2. There, with  as the normalized Lebesgue measure on Œ0; e, where e D exp.1/, n 1X k  ; n kD1

2.3 Weak Convergence of Convolution Products of Probability Measures

99

converges vaguely to the measure 1=2 ıf0g, and with  D ıf0g ,    D    D . Corollary 2.4 shows that this type of behavior cannot occur in the presence of a mild compactness condition. Corollary 2.4. Suppose that all the assumptions in Corollary 2.3 (except the one on the tightness of  n ) hold. Suppose also that S satisfies the following compactness condition: K compact, x 2 S ) x 1 K is compact: (2.57) Then (2.53) implies that

1 n

n P

 k converges weakly to 0 in P .S / and, conse-

kD1

quently, all the results in Corollary 2.3 remain valid. Proof. Let  be a weak limit point of the sequence n 

n 1X k  : n kD1

If all such weak limit points are probability measures, then it follows from n P Theorem 2.7 that the sequence n1  k converges weakly to some 0 in P .S /, and kD1

the rest of Corollary 2.4 then follows exactly as in Corollary 2.3. Thus it suffices to show that  2 P .S /. To this end, let f 2 Cc .S / and x 2 S . Define fx .y/  f .xy/. Then (2.57) implies that fx 2 Cc .S /. Let .nk / be the subsequence such that nk weak converges to . Then let us define the functions gk and g by Z gk .x/ D f .xy/nk .dy/; Z g.x/ D f .xy/.dy/: It is clear that for each x 2 S , gk .x/ converges, as k ! 1, to g.x/; therefore, by the bounded convergence theorem, for f 2 Cc .S /, we have: Z Z f .x/.dx/ D f .x/  nk .dx/ “ D f .xy/.dx/nk .dy/ Z D gk .x/.dx/ Z ! g.x/.dx/ “ D f .xy/.dx/.dy/ Z D f .x/  .dx/: This means that  D    and, consequently,  2 P .S /.

t u

100

2 Probability Measures on Topological Semigroups

In Examples 2.2 and 2.3, we found that even though f0g was an ideal of the multiplicative semigroup Œ0; 1/ (with the usual real-line topology), lim sup n .Œ0; a/ < 1

n!1

(2.58)

for a > 0, when  is the normalized Lebesgue measure on Œ0; b, b  e, where e D exp.1/. However, if we consider the Lebesgue measure ˇ on the multiplicative compact semigroup Œ0; 1 with usual topology, then ˇ n converges weakly to ıf0g , the unit mass at 0. Proposition 2.6 explains this result. Proposition 2.6. Suppose S is compact,  2 P .S /, and S D cl

!

1 [

S./

n

:

nD1

Then for any open set G  K, where K is the kernel of S , lim n .G/ D 1:

n!1

In fact, there exist 0 < ı < 1 and a positive integer m such that for n  m, n .S X G/  ı Œn=m ; where Œn=m is the greatest integer  metrically fast.

n m.

In other words, the convergence is geo-

Proof. Let K  G, G open. Notice that S is compact and KS [ SK [ SKS  K  G. Thus, there exist open sets V and W such that K  V  W  G such that V S [ S V [ S V S  W;

W S  G:

(2.59)

There exists a positive integer m such that m .V / D 1  ı > 0. Let .Xn / be an i.i.d. sequence of S -valued random variables, each with distribution m . Then, for 1  k < m, using (2.59), we have: nmCk .G/  nm .W /k .S / D nm .W / D P .X1 X2 : : : Xn 2 W / ! n [ P fXi 2 V g i D1 nm

D 1ı The proof is now clear.

: t u

2.3 Weak Convergence of Convolution Products of Probability Measures

101

We obtain a similar result with an assumption different from compactness (see Proposition 2.7). Proposition 2.7. Let I be a Borel set that is an ideal of S . Suppose that for some positive integer m; m .I / > 0 for some  2 P .S /. Then the sequence n .I / monotonically increases to 1. Proof. Notice that I  I S so that nC1 .I /  n .I /.S / D n .I /, for all positive integers n. Also, notice that Z nC1



  n I c x 1 .dx/

.I / D c

 n .I c /  .I c /  Œ .I c /nC1 : Thus, .m /nC1 .I c /  Œm .I c /nC1 . Now the proof follows the same lines as in Proposition 2.6 since SIS  I . u t An important question in the context of weak convergence is what makes the sequence n for  2 P .S / a tight sequence when n does not converge vaguely (that is, in the weak sense) to the zero measure. Note that in Example 2.2, the sequence n is not tight nor does it converge vaguely to the zero measure. Although this problem is not completely solved (to the best of our knowledge), we present a number of satisfactory results in this context. First, we prove Lemma 2.2. Lemma 2.2. Let n be a sequence in P .S / such that the subsequence 0;nt , where k;n D kC1   n , has at least one weak limit point in P .S /. Suppose that S has the property such that convolution as a map from P .S / B.S / ! B.S / is continuous in the weak sense. Then there is a subsequence .pt /  .nt / such that for each positive integer k, w

k;pt ! k 2 P .S /; w

pt !  D    2 P .S /; k   D k : w

Proof. Suppose 0;nt ! 0 2 P .S /. Note that w -convergence is weak convergence when the limit is in P .S /. Now for each positive integer t, ynt  .0;nt ; 1;nt ; : : : ; nt 1 ; 0; 0; 0; : : :/ are elements in the product space Y D

1 Y

Xi ;

Xi  B.S /, for each i;

i D1

where each Xi has weak topology, Y has the product topology so that Y is compact, since B.S / is w -compact. Since Y is compact (and also first countable), there

102

2 Probability Measures on Topological Semigroups

is a subsequence .mt /  .nt / such that ymt ! y 2 Y , in the topology of Y . This means that for each k  0, there exists k 2 B.S / such that k;mt ! k : Since convolution is continuous as a map from P .S / B.S / ! B.S /, it follows that for each k  1, 0;mt D 0;k  k;mt ! 0;k  k in the w -sense, and this means that 0;k  k D 0 ;

k  1:

However, since 0 2 P .S /, this implies that k 2 P .S / for each k  1. Let .pt /  .mt / be a subsequence such that pt !  2 B.S / in the weak sense. Now for fixed integer s and t > s such that ps > k, we have k;ps  ps ;pt D k;pt : Again by the continuity of convolution, it follows that given k  0, for each s such that ps > k, k;ps  ps D k which in turn implies that k   D  k ;

k  1:

Since k 2 P .S /,  2 P .S /. The last equation implies that    D .

t u

Lemma 2.3 shows when the convolution operation as a map from P .S / B.S / into B.S / is continuous. The compactness conditions in the next two lemmas hold when S is either compact or a group. Lemma 2.3. Suppose S satisfies the following compactness condition: K compact and x 2 S ) x 1 K is compact: This condition holds when S is either compact or a group. If n !  weakly in P .S / and n !  2 B.S / in the weak sense with each n 2 P .S /, then n  n !    in the weak topology. Proof. Let f 2 Cc .S /. Then for each s 2 S , t ! f .st/ is in Cc .S /. Hence, if Z gn .s/ 

Z f .st/n .dt/;

g.s/ 

f .st/.dt/

2.3 Weak Convergence of Convolution Products of Probability Measures

103

then lim gn .s/ D g.s/, s 2 S . Since  is a regular measure, it is easily seen that n!1 g is a bounded continuous function on S . Also, by Egoroff’s theorem in analysis, given  > 0, there exists a compact set K such that .K/ <  and on S  K, gn ! g uniformly. Since n !  weakly, lim sup n .K/  .K/ < :

n!1

Then we have ˇ Z ˇZ ˇ ˇZ Z Z ˇ ˇ ˇ ˇ ˇ gn dn  gdˇ  ˇ gn dn  gdn ˇˇ C jgn  gj dn ˇ ˇ ˇ K Kc ˇZK ˇ Z ˇ ˇ C ˇˇ gdn  gdˇˇ ; Z

which shows that lim

n!1

Z gn dn D

gd:

This means that “

Z

f dn  n D f .st/n .ds/n .dt/  Z Z D f .st/n .dt/ n ds Z Z Z D gn .s/n .ds/ ! gd D f d  : t u Lemma 2.4. Let S satisfy the following compactness condition: A1 B is compact whenever A and B are compact: Let n 2 P .S / and 0;n  1  2   n . Then     sup lim sup 0;n Kx 1 W K compact  b n!1 x2S

is either 0 or 1. Proof. First note that   0;nC1 Kx 1 D

Z

    0;n K.yx/1 nC1 .dy/  sup 0;n Kx 1 ; x2S

which shows that

  lim sup 0;n Kx 1

n!1 x2S

exists.

104

2 Probability Measures on Topological Semigroups

Suppose that 0 < b < 1. Choose c DbC

b.1  b/ : 4

Then b < c < 1 and c.1 C c/ < 2b. Let K be any compact set. Then we have for some k  1 and q > 0,   sup 0;k Kx 1 < c  q:

(2.60)

x2S

Notice that there exists a compact set K1 (depending only on k) such that 0;k .K1 / >

bCc : 2c

Then given any compact set A, for n sufficiently large, we must have:   k;n Ax 1 < c;

x 2 S:

(2.61)

  Otherwise, since .K1 A/ x 1  K1 Ax 1 ,     bCc bCc c D >b 0;n .K1 A/ x 1  0;k .K1 / k;n Ax 1  2c 2 for infinitely many n, contradicting the definition of b. Now let D be a compact set such that 0;k .S  D/
0, a compact set K1 and a subsequence .ni / of positive integers such that 0;ni .K1 / > ;

i  1:

(2.63)

By Lemma 2.4, for 0 < ı < , there exist elements xni 2 S and a compact set K2 such that for i  1,   0;ni K2 xn1 > 1  ı: (2.64) i It follows from (2.63) and (2.64) that ¤ ;; K1 \ K2 xn1 i which means that

xni 2 K11 K2 ;

i  1; i  1:

 1 Then from (2.64) we have for the compact set Aı  K2 K11 K2 , 0;ni .Aı / > 1  ı:

Thus, the subsequence 0;ni is tight.

(2.65)

106

2 Probability Measures on Topological Semigroups

By Lemmas 2.2 and 2.3, there exists a further subsequence .pi /  .ni / such that for each k  1, w k;pi ! k 2 P .S / w

 pi !  D    2 k   D k :

(2.66)

It follows from Theorem 2.2 that S./ is a completely simple semigroup with a compact group factor. The compactness condition in this theorem then implies that S./ is also compact. Let us now write: S./ D

1 \

Gn ;

nD1

where Gn is a sequence of open sets with compact closure. Now there exists a subsequence .si /  .pi / such that w

si ;si C1 !  and for each i  1, j  1, 1 si ;si Cj .Gi / > 1  : i Then for any m such that si < m  si C1 , there exists zm in S such  positive  integer that si ;m Gi z1 > 1  1i , since m Z   1 si ;m Gi y 1 m;si C2 .dy/ D si ;si C2 .Gi / > 1  : i Let z 2 S./. We claim that for all nonnegative integers k, the sequence k;n  ızn z converges weakly to k  ız .   To prove this let k be a weak limit  claim,   point of k;n  ızn z . Then given tj , there is a subsequence mj such that for some subsequence any sequence  si.j / of .si / such that si.j / < mj  si.j /C1 and k;mj  ızmj z ! k (vaguely):

(2.67)

The sequence si.j / ;mj ızmj is tight (because of the way we constructed the sequence zm ), and it follows that any weak limit point ˇ of this sequence must satisfy S.ˇ/  S./. Also note that for any Borel set B, Z

   Bz1 y 1 ˇ.dy/   D  Bz1

  ˇ  ız .B/ D

D   ız .B/

2.3 Weak Convergence of Convolution Products of Probability Measures

107

by Proposition 2.4 (since  D  2 ). Since we have k;mj  ızmj z D k;si.j /  si.j / ;mj  ızmj z ; it is clear that k D  k  ˇ  ı z D .k  /  ˇ  ız D k  .  ˇ  ız / D k  .  ız / D k  ı z : This means that for each k  1, w

k;n  ızn z ! k  ız : t u Let us remark that with a little extra work, Assertion (i) of Theorem 2.8 can be strengthened when S is a group to the assertion that   lim sup 0;n Kx 1 D 0 n!1 x2S

for any compact subset K  S . Theorem 2.8 leads to the following important result. Theorem 2.9. Let S be a group and  2 P .S /. Suppose S is noncompact and S has no proper closed subgroup containing S./. Then the sequence n converges vaguely to the zero measure. Proof. Suppose S0 D cl

1 [

! S./

n

nD1

and that n does not converge vaguely to the zero measure; then there exists a compact subset K  S0 such that for some ı > 0, n .K/ > ı

(2.68)

for infinitely many n. This means that there exists an element x in K such that for any open set N.x/ containing x lim sup n .N.x// > 0:

n!1

(2.69)

In other words, if .Xn / is a sequence of independent S -valued random variables, each with distribution , and if Zn is defined by Zn D Xn Xn1 X1 , then Zn has distribution n and (2.69) implies that Pr .Zn 2 N.x/ infinitely often/ > 0:

108

2 Probability Measures on Topological Semigroups

We prove in Chap. 3 that this infinitely often probability is either 0 or 1 when S is a group, and therefore, Pr .Zn 2 N.x/ infinitely often/ D 1

(2.70)

for any open N.x/ containing x. Now let e be the identity of S and y 2 S0 . Let N be a given open set such that e 2 N . There exist open sets N1 and N2 containing e such that N1 N11  N; N2 xy 1  xy 1 N1 : Now there exists a positive integer k such that k .N1 y/ > 0, since y 2 S0 . Then we have fZkCn 2 N2 x infinitely ofteng ˚   fZk … N1 yg [ ZkCn Zk1 2 xy 1 N infinitely often ; which means that   Pr Zn 2 xy 1 N infinitely often > 0 so that xy 1 2 S0 , and it is a recurrent state for the random walk .Zn / whenever y 2 S0 . Thus, e 2 S0 and it is recurrent. Then for any y 2 S0 , repeating the previous argument, e y 1 2 S0 . It is also clear that S0 is a closed semigroup, so that S0 D S . Now by Theorem 2.8, since n does not converge vaguely to the zero measure, there exist elements xn in S such that for each k  0, nk  ıxn ! k 2 P .S /: Then there exists n0 such that   n  n0 ) n K1 xn1 > 1  ı; for some compact set K1 ; it follows from (2.68) that K1 xn1 \ K ¤ ;; for infinitely many n. This means that xn 2 K 1 K1 , which is compact, for infinitely many n. Let z be a limit point of the sequence xn , so that 0  ız1 is a weak limit point of .n /. By Lemma 2.2, there is a subsequence nt such that for k  0, w

nt k ! k 2 P .S /; w

nt !  D    2 P .S /:

(2.71)

2.3 Weak Convergence of Convolution Products of Probability Measures

109

It follows from (2.71) (since S is second countable, the weak topology in P .S / is metrizable) that there is a subsequence .mt /  .nt / such that w

mt C1 mt ! : For any weak limit point k of mt C1 mt k , we have by Lemma 2.3, k  k D ;

k  1:

(2.72)

Since S./  H is a compact subgroup by Theorem 2.2, there exists (by (2.72)) xk 2 S .k / such that S./k  H xk1 I also, for k  1, there exists hk in H such that x k D hk xk1 ;

S./k  H x k ;

x 2 S./:

(2.73)

˚  By [88, p. 85], either the set x k W k  1 has compact closure or the sequence x k ! 1 as k ! 1. It follows from (2.73) that since S0 is noncompact, x k ! 1. But then given any compact set A, A \ H xk D ; for all but finitely many k, since otherwise x k 2 H 1 A for infinitely many k. Thus, k .A/ D 0 for all but finitely many k. t u Theorem 2.10 further generalizes Theorem 2.9. Theorem 2.10. Let S be a completely simple semigroup with (usual) product representation X G Y . Suppose  2 P .S / and S D cl

1 [

! S./

n

:

nD1

Then n converges to the zero measure vaguely if and only if G is a noncompact group. Proof. The proof consists of several steps consisting of noting that observations made earlier are still valid in this more general situation. Step 1. Given  > 0, there still exist compact subsets K1  X , K2  Y such that  .K1 G K2 / >

p

1  :

(2.74)

110

2 Probability Measures on Topological Semigroups

Notice that for n  1, S S .K1 G K2 / „ ƒ‚ S … .K1 G K2 / n times

D K1 G K2 ; which means that for n  1, nC2 .K1 G K2 / > 1  :

(2.75)

Thus, if G is compact, the sequence n is tight. This proves the only if part of Theorem 2.10. Step 2. Let K be a compact subset of X . Then it easily verified that h

i .A1 A2 A3 /1 .B1 B2 B3 / \ ŒK G Y  h i  K .A3 K/1 A1 B 2 B3 ; 2

(2.76)

which means that if A, B are compact subsets of S , then A1 B \ K G Y is closed in S and therefore compact (from (2.76)). Step 3. Using Step 2 and the same proof as in Lemma 2.4 along with the regularity property in (2.75), it is easily verified that  sup

   lim sup n Kx 1 W K compact

n!1 x2S

is either 0 or 1. Step 4. Suppose from now on that G is noncompact and (if possible) for some subsequence ni and some compact subset A  S , ni .A/ >  > 0;

i  1:

(2.77)

In Step 4, we show that .ni / is tight. Notice that Z  .A/ D ni

    ni 1 Ax 1 .dx/  sup ni 1 Ax 1 x2S

and therefore, from Step 3, given 0 < 2ı < , there is a compact set B and elements xni 2 S such that   ni C1 Bx1 i  1: (2.78) ni > 1  ı;

2.3 Weak Convergence of Convolution Products of Probability Measures

111

Choose the compact set K1  X such that  .K1 G Y / > 1  ı: Then we have   ni C1 Bx1 ni D

Z

  1 ni Bx1 .dy/  ni y

Z Cı; K1 GY

so that there exists y 2 K1 G Y such that for zni D yxni (2 K1 G Y ),   ni Bz1 ni > 1  2ı:

(2.79)

It follows from (2.77) and (2.79) that A \ Bz1 ni ¤ ; so that

zni 2 A1 B \ K1 G Y  C;

say, for i  1. By Step 2, C is compact. Then it follows from (2.79) and (2.75), with  replaced by ı, that for i  1,   ni BC 1 \ .K1 G K2 / > 1  3ı; where K1  X , K2  Y are compact subsets. It is easily verified that the set BC 1 \ ŒK1 G K2  is compact. Thus, the subsequence .ni / is tight. Step 5. Using the same argument as in the proof of Lemma 2.2, there is a subsequence .mi /  .ni / such that for each k  0, mi k ! k

(2.80)

vaguely as i ! 1. Let us show that k 2 P .S /, k  0. Let k be a fixed positive integer. Since .mi / is tight, using Step 4 and (2.75), given  > 0, we can find compact subsets K1  X , L  G, K2  Y such that mi .K1 L K2 / > 1  ;

i  1;

 .K1 L K2 / > 1  ; k

nC2



Z

Thus, 1 < or

.K1 G K2 / > 1  ;

(2.81) n  1:

  mi k .K1 L K2 / y 1 k .dy/

 1  2  mi k .K1 L K2 / .K1 L K2 /1 :

112

2 Probability Measures on Topological Semigroups

It follows from (2.81) that for i  1,  mi k .K1 L K2 / .K1 L K2 /1 \ ŒK1 G K2  > 1  3: The inside the parentheses is easily verified to be compact. This proves that  m set  i k is tight, so that k 2 P .S /. Notice that it follows from (2.81) that given  > 0, we can obtain the same compact set A  .K1 L K2 / .K1 L K2 /1 \ .K1 G K2 / such that for i > j  1,

mi mj .A / > 1  3:

This means that the sequence mj is tight. Let .si /  .mi / be such that w

si !  2 P .S /:

(2.82)

For k < j , (2.80) gives w

j k  mi j ! k : Then

j  j k D j k  j D k ;

k < j:

(2.83)

Taking j D si in (2.83), we have:   k D  k   D  k ;

k  1:

(2.84)

Therefore,    D : (2.85)   Let us now show that for each k  1, si Ck is tight. To this end, as in (2.81), given  > 0, we can find compact sets K1 , L0 , K2 such that   si Ck K1 L0 K2 > 1  ;

i  1;

and any given j  1, Z 1
sj C k, we have   si .sj Ck / A0 > 1  3:

(2.86)

  It follows from (2.86) and (2.80) that the subsequence sj Ck is tight for each k  1. From (2.83), rCk  r D r  rCk D k

k  1;

r  1;

so that rCsi  r D r  rCsi D si ; i  1:   If r is a weak limit point of rCsi , then r 2 P .S / and r  r D r  r D 

r  1:

(2.87)

  r D  r   D  r ;

r  1:

(2.88)

From (2.84), "

Since S D cl

1 [

# S./

k

;

kD1

it follows from (2.87) that S./, which is a closed, completely simple subsemigroup with a compact group factor by Theorem 2.2 ( is idempotent), must take the form S./ D X H Y; where H is a compact subgroup of G and YX  H . We claim that S./  X gH Y; g 2 G  H;

(2.89)

(2.90)

and H is a normal subgroup of G. To prove (2.90), let .x1 ; g1 ; y1 / be an element of S./ and let .x; g; y/ 2 S .1 /. Then it follows from (2.86) and (2.87) that .x1 ; g1 ; y1 / .x; g; y/ 2 X H Y; which, since YX  H , implies that g 2 Hg11 H:

(2.91)

114

2 Probability Measures on Topological Semigroups

Using (2.91), we have S./  X Hg1 H Y S .1 /  X

Hg11 H

(2.92)

Y:

(2.93)

From (2.88), we have .X H Y / S .1 /  S .1 / :

(2.94)

  Let x; h1 g11 h2 ; y 2 S .1 /, where h1 and h2 are in H . Then by (2.94),   .X H Y / x; h1 g11 h2 ; y D X Hg11 h2 fyg  S .1 / :

(2.95)

By (2.92), a typical element in S./ is .x1 ; h3 g1 h4 ; y1 /, where h3 and h4 are in H . Then from (2.87) and (2.95),   .x1 ; h3 g1 h4 ; y1 / X Hg11 h2 fyg  X H Y; which means that

g1 Hg11  H

or g1 H  Hg1 :

From (2.87), 1   D , we obtain similarly g11 Hg1  H

or Hg1  g1 H:

It follows from (2.92) that S./  X g1 H Y;

(2.96)

S./k  X g1k H Y:

(2.97)

Therefore for k  1, "

Since S D X G Y D cl

1 [

# S./

k

;

kD1

it follows that G D cl

1 [

! g1k H :

(2.98)

kD1

Since g1 H D Hg1 , (2.98) ˚ implies that  H is a normal (and compact) subgroup of G. Since G is noncompact, g1k W k  1 cannot have compact closure. Therefore, g1k must converge to infinity as k ! 1, see [88, p. 85]. This means that given any compact subset M  G, M \ g1k H ¤ ;; (2.99)

2.3 Weak Convergence of Convolution Products of Probability Measures

115

for at most finitely many k, since (2.99) implies that g1k 2 MH 1 , a compact subset of G. This contradicts (2.77) and the proof of Theorem 2.10 is complete. u t Proposition 2.8. Suppose S satisfies the following compactness condition: Kx 1 and x 1 K are compact for compact K and x 2 S . Let  2 P .S / and " S D cl

1 [

# S./n :

nD1

Then the sequence n 

n 1X k  n kD1

converges to the zero measure vaguely if and only if S is noncompact. (Note that in Proposition 2.8, the compactness conditions mentioned are necessary. The completely simple semigroup X G Y has the compactness condition “Kx 1 is compact for compact K and x 2 S ” when Y is compact, and it has the other compactness condition when X is compact. Yet, this semigroup can support an idempotent probability measure when G is a compact group even when X and Y are not compact (as long as X and Y can support probability measures). See Theorem 2.2.) Proof. Because of Theorem 2.7, we need to prove only the if part. Notice that if K is a compact subset of S , then for S , Kx 1 and x 1 K are both compact, and  c x x 1 K  K c ;  1 c Kx x  Kc:

(2.100)

Now consider the one-point compactification S  D S [ f1g of S . For x 2 S , let us define, x 1 D 1 x D 1;

1 1 D 1:

It is then clear that S  is a compact separately continuous (that is, the maps x ! x y and y ! x y are continuous) semigroup. It is easily verified that P .S / is also a separately continuous semigroup with respect to convolution and weak topology. We now consider  as a probability measure on S  , so it is clear that S  is the closed semigroup generated by S./. Then using the same procedure as in Theorem 2.7, we observe that there is an idempotent probability measure  such that n !  weakly as n ! 1. This means    D    D ;

(2.101)

which implies that S./ is an ideal of S  , thus 1 2 S./. Also, it can be proven (following the proof of Theorem 2.2) that even though S  is only separately

116

2 Probability Measures on Topological Semigroups

continuous, S./ contains a kernel of S  , which is dense in S./. This means that S./ D f1g. Thus, for compact K  S , lim sup n .K/  .K/ D 0:

n!1

t u A stronger version of the preceding result holds under a stronger compactness assumption. Theorem 2.11. Let  2 P .S / and " S D cl

1 [

# S./

n

:

nD1

Suppose the following compactness condition holds in S : For compact subsets A and B, A1 B and AB 1 are compact. Then n converges vaguely to the zero measure if and only if S is noncompact. (Note that when S is the multiplicative semigroup of d d , d finite, nonnegative matrices (with no zero rows nor zero columns) with usual topolgy, then S has the compactness conditions assumed in Theorem 2.11.) Proof. Suppose there is a compact set K  S , a subsequence .nk /, and  > 0 such that for k  1, nk .K/ > : (2.102) Let us define the set H  S by ) ( 1 X n  .N.x// D 1 for any open set N.x/ containing x : H D x2SW nD1

Then (2.102) implies that H is nonempty. It is clear that H is an ideal of S , since if x 2 H and y 2 S , then given any open set N containing xy, there are open sets N.x/, N.y/ such that x 2 N.x/, y 2 N.y/, and clŒN.x/N.y/  N ; therefore, if k .N.y// > 0 (such a positive integer k always exists), then 1 X

nCk .N / D

nD1

Z X 1 Z



  n N z1 k .d z/

nD1 1 X

N.y/

Z

 N.y/

  n N z1 k .d z/

nD1 1 X

n .N.x//k .d z/ D 1:

nD1

Thus, HS  H ; similarly, SH  H . We claim that H is a closed completely simple kernel of S .

2.3 Weak Convergence of Convolution Products of Probability Measures

117

To prove this claim, we use an argument from Chap. 3. Let X1 ; X2 ; : : : be a sequence of independent S -valued random variables, each with distribution , and let Zn D X1 X2 Xn . Then given any positive integer k, we can find a positive integer m with 1  m  k such that 1 X

Pr .ZmCi k 2 N.x// D 1;

(2.103)

i D0

where N.x/ is any open set containing x, an element in H . Also, we have 1  Pr .Zn 2 N.x/ finitely often/  

1 X i D0 1 X

Pr .ZmCi k 2 N.x/, Zn … N.x/ for n  m C .i C 1/k/  Pr ZmCi k 2 N.x/, XmCi kC1 Xn … N.x/1 N.x/

i D0

for all n  m C .i C 1/k/ 1  X  Pr .ZmCi k 2 N.x// : D Pr Zn … N.x/1 N.x/ for all n  k i D0

It follows from (2.103) that for each positive integer k,   Pr Zn … N.x/1 N.x/ for all n  k D 0:

(2.104)

This means that for any open set N.x/ (with compact closure), the set N.x/1 N.x/ ¤ ;. Thus, there exist   zn 2 N.x/1 N.x/  cl N.x/1 N.x/ ; which is compact. This means that there is a sequence xn such that xn ! x;

xn zn ! x:

It follows that x 2 xS . Let y 2 xS . Then we claim that x 2 yS and this will prove that xS is a minimal right ideal of S . If x … yS , then by our compactness assumptions in Theorem 2.11, there must exist open sets N.x/ containing x and N.y/ containing y such that N.y/1 N.x/ is empty. Write y D xz. Let W .x/ and W .z/ be open subsets containing x and z, respectively, such that W .x/  N.x/; so that

W .x/W .z/  N.y/



W .z/1 W .x/1 W .x/ D ;:

118

2 Probability Measures on Topological Semigroups

There is a positive integer k such that Pr .Zk 2 W .z// > 0: But using (2.104), we have   Pr .Zk 2 W .z// D Pr Zk 2 W .z/, Zn 2 W .x/1 W .x/ infinitely often 

 Pr Zk 2 W .z/, XkC1 Xn 2 Wz1 W .x/1 W .x/ infinitely often/ D0 which is a contradiction. Thus for every y 2 xS , x 2 xS and x 2 yS whenever x 2 H . This means that xS is a minimal right ideal. Similarly, for x 2 H , S x is a minimal left ideal. By Proposition 1.6 and Theorem 1.7, S has a closed completely simple kernel I , which is the union of all minimal right ideals. It is clear that I D H . To complete the proof of Theorem 2.11, we show that the sequence .nk / in (2.102) is tight. By Lemma 2.4, given 0 < ı < , there exists a compact subset A, elements an in S , and a positive integer n0 such that for n > n0 ,   n Aan1 > 1  ı:

(2.105)

By (2.102) and (2.105), we have \ K ¤ ;; Aan1 k

nk > n0 :

 1 This means that ank 2 K 1 A for nk > n0 . Thus, if B  A K 1 A , then B is compact, and for nk > n0 , nk .B/ > 1  ı: In other words, .nk / is tight. By Lemmas 2.2 and 2.3, there is a subsequence .mk /  .nk / such that mkC1 mk !  D    2 P .S / weakly as k ! 1. For x 2 S./ and any open set N.x/ containing x, 0 < .N.x//  lim inf mkC1 mk .N.x//: k!1

This implies that S./  H . Also note that    D   , so that cl.S./S.// D cl.S./S.//. Since ! 1 [ n S./ ; S D cl nD1

2.3 Weak Convergence of Convolution Products of Probability Measures

119

it follows that clŒS S./ D clŒS./ S   H , where H is an ideal of S . Since cl.S S.// is also an ideal of S and H is simple, cl.S S.// D H . Thus, the closed completely simple semigroup H is generated by the support of   . Also .  /mkC1 mk D mkC1 mk   !    weakly, so that .  /n does not converge to zero vaguely. By Theorem 2.10, the group factor of H is compact. Then if e is an idempotent element in H , eHe must be a compact group. Since eSe D eHe, we have

S D e 1 .eHe/ e 1 ; which must be compact by the assumptions in Theorem 2.11. Theorem 2.11 follows easily. t u The problem of determining when the sequence .n / for a given  2 P .S / is tight is difficult. An easy way of checking simple conditions necessary and sufficient for the tightness of .n / is still not known (except in some special cases). However, we have the following interesting Theorem 2.12. Theorem 2.12. Let  2 P .S / and " S D cl

1 [

# S./

n

:

nD1

Then the sequence .n / is tight if and only if S has a closed completely simple minimal ideal H with a compact group factor such that for every open set V  H , lim n .V / D 1:

n!1

(2.106)

Proof. The only if part follows from Theorem 2.7(i) and Assertion (2.35). The if part is proven in two steps. Step 1. Suppose there is a positive integer k such that k .H / > 0. In this case, by Proposition 2.7, lim n .H / D 1: n!1

Given  > 0, there exists a positive integer m such that for n  m, n .H / > 1  : Let us write H D X G Y , the usual product representation of H with G compact. Let A  X and B  Y be compact subsets such that k .A G B/ > 1  :

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2 Probability Measures on Topological Semigroups

Thus if n  m, then we have nC2k .A G B/  k .A G B/n .X G Y /k .A G B/ > .1  /3 ; since A G B D .A G B/.X G Y /.A G B/. It follows that .n / is tight. Step 2. Let us now assume that k .H / D 0;

k  1:

(2.107)

Even though it is a standard result, let us first show that the sequence .n / is tight whenever every weak limit point of .n / is a probability measure. To this end, we assume that for some subsequence .nk / of positive integers, w

nk !  )  2 P .S /;

(2.108)

Since S is second countable and locally compact, there exists an increasing sequence of open sets .Vn / such that V n is compact for each n and SD

1 [

Vn :

nD1

We claim that given  > 0, there exists a positive integer k./ such that for all m  1, 0 1 k./ [ m @ Vn A > 1  : (2.109) nD1

If (2.109) is false, then there exist .pk / and .mk / such that pk < pkC1 and mk < mkC1 , and for each k  1, mk



pk [

! Vn

 1  :

(2.110)

nD1

Let  be a weak limit point of the sequence Œmk . Then  2 P .S / by (2.108). Let .nk /  .mk / be such that w nk !  and let .qk /  .pk / be such that nk



qk [ t D1

! Vt

 1

(2.111)

2.3 Weak Convergence of Convolution Products of Probability Measures

121

for k  1. If s is any positive integer, then 

qs [

!  lim inf 

nk

Vt

k!1

t D1

qs [

! Vt

 1  ;

t D1

which implies that .S /  1. This contradicts  2 P .S / and establishes (2.109). It follows that .n / is tight, since each Vn has compact closure. Thus, it is enough to show that each weak limit point of .n / is in P .S /. To show this, let us suppose w

nk ! ;

.S / D b < 1:

(2.112)

Choose p > 0 such that b C 2p < 1. Write H D

1 \

Wn

Wn  WnC1 ;

nD1

Wn open and n .Wn / < p;

n  1:

(2.113)

This is possible since (2.107) holds. Since S is sigma-compact, we can write H D

1 [

Km ;

Km compact;

Km  KmC1 :

mD1

Let Mi be an open set with compact closure such that Mi C1  Mi  Ki . Since we have   lim sup nk .Mi /   Mi  b k!1

there exists a positive integer k.i / such that for k  k.i /, nk .Mi / < b C p: This means there is a subsequence .mi / such that for i  1, mi .Mi / < b C p: Write Ci D Mi \ Wmi , and let C D

1 [ nD1

Ci :

(2.114)

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2 Probability Measures on Topological Semigroups

Then C is open and C  H . For all i  1, C  Mi [ Wmi and thus,   mi .C /  mi .Mi / C mi Wmi  b C 2p < 1; t u

which contradicts (2.106).

Under the conditions of Theorem 2.11, we saw that either the sequence n converges vaguely to the zero measure or the sequence n is tight. This behavior does not always hold as we already saw in Example 2.2. However, as we will show in a discrete semigroup, the convolution sequence n always has this property; also in such a semigroup, it is much simpler to describe the limit behavior of this sequence with respect to weak convergence. First, let S be a discrete semigroup and  2 P .S / such that SD

1 [

S./n :

nD1

Since S./ is countable, S is countable. Suppose n does not converge vaguely to the zero measure. Then there is a compact set (which in this case must be finite) K such that n .K/ does not converge to 0 as n ! 1. This means that there exists an element y in K such that lim sup n .y/ > 0: (2.115) n!1

Notice that for the element y in (2.115), we have 1 X

n .y/ D 1:

(2.116)

nD1

Let

ˇ1 ) ˇX ˇ n  .z/ D 1 : H D z2Sˇ ˇ (

nD1

Then from the proof of Theorem 2.11, it follows easily that H is a completely simple ideal of S . Since the set of all z in S that has the property lim sup n .z/ > 0;

n!1

is an ideal of S and a subset of H , H can also be identified by n o H D z 2 S j lim sup n .z/ > 0 : n!1

(2.117)

Since H is an ideal and S is discrete, it follows from Proposition 2.7 that lim n .H / D 1:

n!1

(2.118)

2.3 Weak Convergence of Convolution Products of Probability Measures

123

Using (2.118) and proof of Theorem 2.10 and Lemma 2.4, we can also establish that the completely simple semigroup H must have its group factor compact (and therefore finite). Thus, we can now write H DX G Y

(2.119)

(the usual product representation), where G is a finite group. Using (2.118) and (2.119), given  > 0, we can then find a positive integer N and finite subsets A  X and B  Y such that for n  N , n .H / > 1  ;

N .A G B/ > 1  :

This means that for n  3N , n .A G B/  N .A G B/n2N .H /N .A G B/ > .1  /3 ; which establishes that the sequence n is then tight. Now we use Theorem 2.7. Let D 2 be the idempotent weak limit point of .n /. Then we have S. / D X H Y;

(2.120)

where H is a normal subgroup of G. Suppose that the sequence n does not converge weakly. Then by Theorem 2.7,  D ¤ and S.  / D X gH Y;

g 2 G  H:

Since G is finite, there is a smallest positive integer d such that g d 2 H . Notice that  S d  D X H Y: (2.121) Write   d  . Then we have   D   D :

(2.122)

It follows from the idempotence of , Theorem 2.2, (2.120)–(2.122) that for B  S , x 2 S./,    Bx1 D

Z

    Bx1 y 1 .dy/ D Bx1

and therefore, using (2.122) again, Z .B/ D

   Bx1 .dx/ D

Z

  Bx1 .dx/ D .B/

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2 Probability Measures on Topological Semigroups

so that d  D . Now let  be any weak limit point of .n /. Then there is a subsequence .nk / of positive integers such that nk !  weakly:

(2.123)

Write nk D mk d C sk ;

0  sk < d:

Then there is a subsequence .pk /  .mk / and some r, 0  r < d such that pk d Cr !  weakly: Therefore,  pk D r  D   D : pk d Cr  D r  d  In other words, the only weak limit points of .n /, when n does not converge weakly, are ;   ; : : : ; d 1  , and  S k  D X g k H Y;

0  k  d:

(2.124)

Thus, we have proven Theorem 2.13. Theorem 2.13. Let S be a discrete semigroup and  2 P .S / such that S D 1 S S./n . Then the following assertions are valid: nD1

(i) n does not converge to zero vaguely if and only if S has a completely simple (minimal) ideal H as described by (2.117) and (2.119) with its group factor finite (if and only if the sequence n is tight). (ii) If n does not converge vaguely to the zero measure, then there is a positive integer d such that for each integer r such that 0  r < d , w

ndCr ! r ; where r D r  0 for 0 < r < d , the supports of r are pairwise disjoint for 0  r < d and given by (2.124), and 0 D , with its support given by (2.120). As we saw in Theorem 2.8 (and in the remark following this theorem), when S is a noncompact group such that S has no proper closed subgroup containing S./ with  2 P .S /, then exactly one of the following two possibilities exist: (i) For each compact subset K  S , ˚   

lim sup n Kx 1 W x 2 S D 0:

n!1

(2.125)

2.3 Weak Convergence of Convolution Products of Probability Measures

125

(ii) There exist elements xn 2 S such that n  ıxn converges weakly:

(2.126)

In what follows, we discuss when (2.125) occurs. First, let us look at Example 2.6: Example 2.6. (This is an example of a noncompact group where (2.125) does not hold.) Let 0 < d < 1 and S be the multiplicative group of 2 2 matrices given by  k   d r SD W k is an integer, r real : 0 1 Let S have the usual topology of R4 and let  2 P .S / such that the support of  is the set    d r W r real 01

 d X and EjX j < 1, where has distribution . Notice that for an i.i.d. sequence 0 1 

d Xn Xn such that has distribution , 0 1

   n  d X1 d X2 d Xn d Zn D ; 0 1 0 1 0 1 0 1 where Zn D X1 C dX2 C C d n1 Xn . Since E jXi j < 1 and 0 < d < 1, the sequence Zn is a Cauchy sequence in L1 , so there exists Z such that EjZj < 1 and lim E jZn  Zj D 0:

n!1

In other words, the sequence Zn has a limiting distribution. Since we have

n  n   d Zn d 0 1 Zn ; D 0 1 0 1 0 1 it follows that the sequence n  ıan converges weakly if we take

n  d 0 : an D 0 1 We will show that the nonvalidity of (2.125) in Example 2.6 is due to the nonconnectedness of S there and to the fact that S is neither discrete nor abelian. The problem when (2.125) holds is a difficult problem and has attracted the attention of many mathematicians. In a group S , the expression   ˚  fn .K/ D sup n Kx 1 W x 2 S ;

126

2 Probability Measures on Topological Semigroups

where  2 P .S / and K is a compact set, is usually referred to as the concentration function of n . (This is a term originally introduced by Paul Levy.) If the support of ,  2 P .S /, is contained in a coset of a compact subgroup H of a group S such that S./  H x D x H , where x 2 S./, then it is obvious that fn .H / D 1 for all n. This support condition on  is also necessary for fn .K/ not to converge to zero for some compact set K, when S is an abelian group, as Proposition 2.9 [31] shows, and also when we assume that     N D N  , where .A/ N D  A1 instead of assuming that S is abelian, as Proposition 2.10 (due to Riddhi Shah) shows. Proposition abelian group. Let  2 P .S / such that 

12.9. Let S be a noncompact

S n . Then (2.125) does not hold (or equivalently, S./ [ S./1 S D cl nD1

(2.126) holds) if and only if the following two conditions hold: (i) S is topologically isomorphic to Z H , where Z is the discrete group of integers and H is a compact abelian group; (ii) S./ where A is some compact subset of H such that H D

1D f1g A,   S  1 n . cl A[A nD1

Proof. The if part is easy. For the only if part, suppose that there exist elements xn in S such that n  ıxn converges weakly to some  in P .S /. But this means that N ıxn1  N n ! ;   where for  2 P .S /, .B/ N D  B 1 . It follows that w  .  / N n D .n  ıxn /  ıxn1  N n !   N  ˇ, say: w

Then, ˇ D ˇ  ˇ and by Theorem 2.2, S.ˇ/ is a compact abelian group; also,   N  ˇ D ˇ so that S./  S.ˇ/ x, for some x … S.ˇ/, and consequently S is compactly generated. By [88, p. 90], S is topologically isomorphic to the direct product Rn

Z m H , where n; m are nonnegative integers, R is the additive group of reals, Z is the additive group of integers, H is a compact abelian group. If m; n are both positive, then identifying S with Rn Z m H , it follows that S./  fx1 g fx2 g H but in this case S./ cannot generate all of Rn Z m H , unless n D 0 and m D 1. The rest is clear. t u

2.3 Weak Convergence of Convolution Products of Probability Measures

127

Proposition 2.10. Let  2 P .S / such that S is the smallest closed group containing S./. Suppose that   N D N  : Then, for any compact set A, lim fn .A/ D 0 if and only if there does not exist a n!1

compact subgroup H such that for each x in S./, S./  H x and H x D x H . Proof. The only if part is trivial. For the if part, notice that if (2.125) does not hold, then as in the proof of Proposition 2.9,   n  N n  .  / N n , since   N D N   converges weakly to some ˇ D ˇ  ˇ in P .S /. Then, S.ˇ/ ( H , say) is a compact subgroup such that (i)   N  ˇ D ˇ (ii)   ˇ  N D ˇ D N  ˇ  . This means that for any x in S , S  H x and xH x 1 [ x 1 H x  H: t u

The proof is now clear.

The problem when (2.125) holds was looked into detail by Hofmann and Mukherjea in [92]. They defined a locally compact group S to be strange if: (i) S has a co-compact normal subgroup H1 ; (ii) there is a compact subgroup H of H1 , and an element z such that for any open set V containing H , H1 

1 [

zn V zn :

nD1

A locally compact group S , which is not strange, was called neat. They proved that if S is a neat group,  2 P .S / and S is generated by S./ as a closed semigroup, then (2.125) holds. Hofmann and Mukherjea also showed that the class of neat groups includes all noncompact groups which are either almost connected, Lie projective, discrete, or maximally almost periodic. Their main result, which we will not prove here as the proof is too technical, is the following: Theorem 2.14. Let S be either a noncompact abelian group or a group with a noncompact quotient which is pro-Lie or maximally almost periodic. Let  2 P .S / such that S is the smallest closed semigroup generated by S./. Then (2.125) holds. It is relevant to point out that if  is the unit mass at 1 and S is the discrete additive group of integers, then (2.125) does not hold. Hofmann and Mukherjea also conjectured that all locally compact groups are neat. This conjecture has been proved in the affirmative by Jaworski, Rosenblatt, and Willis.

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2 Probability Measures on Topological Semigroups

When S is a noncompact group and  2 P .S / is strictly aperiodic (that is, the support of  is not contained in a coset of a proper closed normal subgroup of S ), and if  is also adapted (that is, S is the smallest closed subgroup containing S./) and S is either a Lie group or an almost connected group, then (2.125) holds. This was proven by Jaworski, Rosenblatt, and Willis, and earlier in some special cases by Lin and Wittman and also by Bartoszek. This result was also obtained in [60] with the additional condition that  is spread out (that is, n is not singular for some positive integer n). Derriennic and Guivarc’h [59] proved the following result: If S is a nonamenable group and  is adapted on S , then (2.125) holds. Dani and Riddhi Shah have proven Theorem 2.15, which we only state below. Let G be a locally compact topological group. Then if T W G ! G and lim T k .g/ D e D e 2 2 G 8g 2 G, T is called a contraction on G. G is called a k!1

C -group if it has closed normal subgroups M , N , and H such that M  N  H , satisfying the following conditions: (i) M and H=N are compact and G=H is cyclic; (ii) There exists x 2 G such that for all y 2 N , the map x , called the conjugation action of x on N=M , defined by x .yM / D xyx 1 M , is a contraction on N=M: A coset H x, above, where x is such that the conjugation action on N=M is a contraction is called a contractive coset of G. A closed subgroup of a Lie group is called a C -subgroup if it is a C -group with respect to the induced topological Lie group structure. Theorem 2.15. (a) Let S be a real Lie group and let  2 P .S / such that the concentration functions of  do not converge to zero. Then S1 , the smallest closed subgroup of S containing the support of , is a C -subgroup of S and S./ is contained in a contractive coset of S . (b) Let S be a Lie group with finitely many connected components. Suppose also that the center of S 0 =R, where R is the radical of S 0 , is finite. Let  2 P .S / be such that the concentration functions of  do not converge to zero. Then there exists a closed subgroup C containing S1 and a simply connected nilpotent closed normal subgroup U of C such that C =U is a direct product of a compact subgroup K and a (possibly trivial) one-parameter subgroup ˚ and for g 2 S./ the conjugation action of g on U is a contraction. Bougerol [14] studied the rate of convergence in (2.125) for spread-out probability measures  (that is, for some n  1, n is not singular) when S is a compact extension of a closed normal compactly generated abelian subgroup of rank r and has no proper closed subgroup containing S./ and such that S./ is not contained in a coset of a proper closed normal subgroup. He showed that for such  and S , given any compact set K  S , there exists a constant M > 0 such that for n  1, sup fn .xKy/ W x; y 2 S g 

M : nr=2

2.3 Weak Convergence of Convolution Products of Probability Measures

129

Clearly, the problem of the rate of convergence, though very interesting, is far from being solved even in compact groups. However, a number of interesting results exist in the literature. The next two results are due to Bhattacharya [12]. Theorem 2.16. Let  2 P .S /, where S is a compact group such that for some positive integer p, the absolutely continuous component of p (with respect to the normed Haar measure m on S ) has a support whose m measure exceeds half. Then there exists M > 0 and 0 < ˛ < 1 such that for all n  1, kn  mk  M˛ n : Proof. With no loss of generality, we assume p D 1. Let  be the absolutely continuous component of  with respect to m and f be the density d=d m. Let 1 [ Sf D An ; nD1

where

  1 : An D x W f .x/  n

Then there exists 0 < c < 1 such that m.A/ >

1 ; 2

A D fx W f .x/  cg:

Now notice that if B  A and D  Ac , then .cm/.B/  0 and .cm/.D/  0. This means that k  cmk  k  k C k  cmk  1  .S / C .A/  cm.A/ C cm .Ac /   .Ac / D 1  2 .Ac /  cŒ2m.A/  1  r; where r D 1 D cŒ2m.A/  1 < 1. Then, if we take  D .  cm/=Œ1  c,   m D m, and n D Œ.1  c/ C cmn D .1  c/n  n C

n1 X kD0

! n .1  c/k c nk m k

D .1  c/n  n  .1  c/n m C m so that since k.1  c/k D k  cmk  r, we have: kn  mk  .1  c/n C r n  2 Œmaxfr; 1  cgn : t u

130

2 Probability Measures on Topological Semigroups

Corollary 2.5. Let  2 P .S /, where S is a compact connected group such that  has a nonzero absolutely continuous component with respect to m, the normed Haar measure on S . Then, as n ! 1, kn  mk converges to zero exponentially fast. Proof. Let f be the density of the absolute continuous component  of . Let A be such that h  f . IA is bounded and the measure ˇ, given by Z ˇ.B/ D

hdm B

is nonzero. Let g D h  h. Then g is continuous (prove it) and Z 2 .B/ D Z

   Bx1 .dx/   ˇ Bx1 ˇ.dx/

 Z

  ˇ Bx1 h.x/m.dx/ Z  Z D h.x/m.dx/ IB .yx/h.y/m.dy/  Z Z  1  m.dy/ D h.x/m.dx/ IB .y/h yx  Z Z   h yx 1 h.x/m.dx/ m.dy/ D ZB D g.y/m.dy/  .B/, say: D

Thus, 2 .B/  .B/ and S./  fx W g.x/ > 0g  V , say. Note that g is continuous, so V is open. If B is open and B contains the identity e, then 1 [  n B \ B 1 nD1

is an open subgroup and, therefore, also closed. Since S is a connected compact group, there is a positive integer p such that 1 [  n B \ B 1 D B p :

SD

nD1

Thus, if we take z 2 V , then V z1 D W is an open set containing e and since S is compact, there is an open set U  W such that e 2 U;

x 1 Ux  W 8x 2 S:

2.3 Weak Convergence of Convolution Products of Probability Measures

131

Thus, for n  1, zn U znC1  V . By what we proved earlier with B above, there is a positive integer k such that U k D S and, consequently, Vk DV „ Vƒ‚ V…  k

k Y  n nC1  z Uz D z1 U k zkC1 D S: nD1

  Thus, S  k D V k D S ; therefore, S is the support of the absolutely continuous component of 2k . Corollary 2.5 follows from Theorem 2.16. t u Theorem 2.17. Let  2 P .S /, where S is a compact group such that it has no proper closed containing S./. Suppose that there is a positive integer  subgroup  k such that k a , the absolutely continuous component of k (with respect to the normed Haar measure m on S ), is nonzero. Then there exists a positive integer s such that ksn  mH k ! 0, as n ! 1, exponentially fast, where H is the smallest closed subgroup containing S .s /. Moreover, the factor group G=H is finite. Note that if in Theorem 2.17 we also assume that S./ contains the identity e, then H D S and kn  mk ! 0 exponentially fast. Proof. We may and do assume that k D 1. Let f be the density of a . Let A  S be such that m.A/ > 0, and f is strictly positive and bounded on A. Let g D .f IA /  .f IA /. Then g is continuous. Let d > 0 be such that fx 2 S W g.x/ > d g  Jd .g/ is nonempty and open. Since S is a compact group, the closed semigroup generated by S./ is a compact cancellative semigroup (and therefore, a group) so that "1 # [ n S D cl S./ ; nD1

and thus, there exists p  1 and there is at least one element z in ŒJd .g/1 \ S./p : We claim:

  Jd .g/  S 2 :

(2.127)

To prove (2.127), let y 2 Jd .g/ and W be an open neighborhood of y. Then as in the proof of Corollary 2.5, Z 2  .W /  g dm  dm .W \ Jd .g// > 0; (2.128) W

which establishes (2.127). Let s D p C 2 and V D zJd .g/, with z as before. Then (2.127) implies that   V  S./p S 2  S./s :

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2 Probability Measures on Topological Semigroups

Since z1 2 Jd .g/, V contains the identity and V 

1 \

S./sn :

nD1

By assertion (iv) of Theorem 2.7, sn converges weakly as n ! 1 to , the normed Haar measure on the compact subgroup " H D cl

1 [

# S./ns :

nD1

Now we claim there is a positive integer r such that for any y 2 H ,  rsCs2 y ŒJd .g/1 > 0:

(2.129)

To prove (2.129), notice that there exist x1 ; x2 ; : : : ; xq in H such that H D

q [

xi V:

i D1

Since e 2 S./s  S./ns  S./.nC1/s ; there is a positive integer r such that xi 2 S./rs for each i , 1  i  q. Since V D zJd .g/ and z 2 S./p , it follows that H  S./rsCp Jd .g/ which implies (2.129).   Using (2.128), for t D .r C 1/s, H D S t , and any Borel set B  H , Z

  2 u1 B t 2 .d u/  Z Z   g u1 v t 2 .d u/ mH .d v/  B Z  d t 2 v ŒJd .g/1 mH .d v/

 .B/ D t

B

  and because of (2.129), the support of t a is all of H . Now Theorem 2.17 follows from Theorem 2.16. t u We end this section showing how the weak convergence of a tight sequence .n / on a semigroup can be reduced to a similar problem for an induced sequence .Q n / on a group.

2.3 Weak Convergence of Convolution Products of Probability Measures

133

In the rest of this section,  is a probability measure on the Borel subsets  of 1 S S./n . We a locally compact second countable semigroup S . Let S D cl nD1

assume that .n / is tight. By Theorem 2.7 ((i) and (ii)), S then admits a completely simple minimal ideal K, where K D X H Y;

Y:X  H;

H D eSe is a compact group with identity e, e is an arbitrarily chosen, but fixed idempotent in K, and X (resp. Y ) is the closed left-zero (resp. right-zero) semigroup of idempotents in Ke (resp. eK). Let N be the compact normal subgroup generated by the set YX ( H ). Now we define a continuous homomorphism ˚ from S to the factor group H N as follows. Let x 2 S , y 2 S . Set ˚.x/ D .exe/N and ˚.y/ D eyeN . Write: ex D agb and ye D a0 g 0 b 0 , where a and a0 belong to X , b and b 0 belong to Y , and g and g 0 to H . Then we have:         e.xy/e D ea g ba0 g 0 b 0 e D e g ba0 g 0 e   D g ba0 g 0 2 gN:N:g 0 N D .exe/N:.eye/N: , the continuity This proves that ˚ is a homomorphism. With quotient topology in H N of ˚ follows since it is the composition of the canonical homomorphism from H to H N and the continuous map s ! ese from S to H . Define for any Borel subset B  H N,   .B/ Q D  ˚ 1 .B/ : It is verified easily that S ./ Q D ˚.S.//; and

./ Q n D .n / :   Here, .n / .B/ D n ˚ 1 .B/ . All of these lead to

Theorem 2.18. The sequence .n / on S converges weakly if and only if the sequence .Q n / on H converges weakly. N n Proof. Suppose n converges weakly.  If Q is not convergent, then, by Theorem H1 H 2.7, S ./ Q lies in a proper coset r N of N . This will then imply that eS./e 

rH1 , since ˚.S.//  S ./, Q and HN1 is a proper closed normal subgroup of H N. This contradicts Theorem 2.7(iii). Conversely, suppose that Q n converges weakly. Then, if n is not convergent, by Theorem 2.7(iii), the set eS./e lies in a proper coset gH1 of H , where H1 is a

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2 Probability Measures on Topological Semigroups

proper closed normal subgroup of H , g … H1 , and YX  H1 . Since N  H1  H and g … H1 , we must have: S ./ Q D ˚.S.// D eS./eN  gH1 D ˚.g/ HN1 , which is a proper coset of H . This is a contradiction by Corollary 2.1. t u N Here is an example illustrating Theorem 2.18. Let I , A, and B be the 2 2 blocks

 

 10 01 00 ; , and ; 01 10 00 respectively. Consider the set of 4 4 matrices given by S D fa; b; c; d; e; f; g; hg; where 





 BA BA BI B I ; ; bD ; cD ; d D B I BA BA BI







 I B I B AB AB eD ; f D ; gD ; hD : I B AB I B AB aD

Note that fa; d; e; f g is the set of idempotents of S , and af D h ¤ h2 D e. Also, choosing a as the distinguished idempotent in S , we have: H  aSa D fa; cg;

c 2 D aI

the set of idempotents in Sa  X D fa; d g, ad D a, da D d ; the set of idempotents in aS  Y D fa; eg, ae D e, ea D a. It is easily verified that S is a completely simple semigroup and S D XHY. The subgroup N here turns out to be H . Thus, for any probability measure  whose support generates S as a semigroup, the sequence n converges weakly since Q n converges by Theorem 2.7.

Section 2.3 Exercises Exercise 2.16. Let S be a locally compact Hausdorff second countable group and .n /  P .S /. Suppose that .n / is a periodic sequence (that is, for k  1,k D kCp for some fixed positive integer p). Show that for k  1, the sequence k;n can only have either zero or probability measures as weak -cluster points. Exercise 2.17. Let S , .n / and p be as in Exercise 2.16. Show that 1  2   n converges to zero vaguely (that is, in the weak -sense) iff S .s  sC1    sCp1 , for each s, 1  s  p, is not contained in a compact subgroup.

2.3 Weak Convergence of Convolution Products of Probability Measures

135

Exercise 2.18. Let  2 P .S /, where s D fx j 0  x  1g with usual topology and multiplication. Show that either  is the unit mass at 1 or n converges weakly to the unit mass at 0. Exercise 2.19. Suppose that S./ D

   a 1a c 1c ; ; b 1b d 1d

where 0  b < a  1, 0  d < c  1. Show that the kernel of the closed semigroup (with

respect  to multiplication) generated by S./ consists of matrices x 1x of the form , 0  x  1, and thus, the group H in Theorem 2.7, in x 1x this case, is a singleton. It follows that n converges weakly as n ! 1. Show that the same

 remains true as long as S./ contains at least one matrix of the form a 1a , 0  b < a  1. b 1b Exercise 2.20. Show that S./, where  D .w/ lim n , where  is as in n!1

Exercise 2.19 with a D

b D 0 and c D 1, d D 23 is the Cantor subset of

 x 1x Œ0; 1, where x in Œ0; 1 is identified with the matrix . x 1x 1 3,

Exercise 2.21. Show that in Exercise 2.19 if for some probability measure  whose support S./ consists of 2 2 stochastic matrices with rank one and    D , then  D .w/ lim n . n!1

Exercise 2.22. Let  and  be in P .S /, where S is a locally compact second countable semigroup, such that  D  and  D   D . Let c > 0 and 0  . Consider 1 X ck k ˇc D  : e c kŠ kD0

Show that ˇc is a well-defined probability measure on S such that it is the nth convolution power of ˇ.c=n/ . (Note: ˇc is called a compound Poisson probability. It is infinitely divisible, meaning that for each natural number n, there is a probability measure (namely, ˇ.c=n/ ), whose nth convolution power is itself.) Exercise 2.23. Consider the discrete semigroup of rationals under addition. Show that on this semigroup, the unit mass at 1, though an infinitely divisible probability, is not a compound Poisson probability. (See Exercise 2.22.) (Note: It was, however, shown in [138] that a probability measure on a finite semigroup is infinitely divisible iff it is compound Poisson.)

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2 Probability Measures on Topological Semigroups

Exercise 2.24. Consider the stochastic matrices 0 1 0 1 0 a 1a 0 1a a 0 0 0 e D @a 1  a 0A ; e 0 D @1  a a 0A ; g D @0 0 0 0 1 0 01 a 1a 0 1 0 1 0 01 010 g 0 D @ 0 0 1A , and x D @1 0 0A : 1a a 0 001

1 1 1A ; 0

Here, 0  a  1. Let S./ D fx; gg and S be the semigroup generated by S./. Show that the weak limit  D .w/ lim n exists. Further, show that n!1

S./ D fe; e 0 ; g; g 0 g, and if fgg D b, fxg D 1  b, 0 < b < 1, then 1 .e/ C  .e 0 / D .g/ C  .g 0 / D 12 and .e/ D .g/ D 2.2b/ . Exercise 2.25. Let  be a group (with respect to convolution) of probability measures on a discrete semigroup S . Let be the identity of  . Then show that S. / D [fS./ W  2  g is a completely simple semigroup, and the product representations of S. / and S. / are, respectively, given by S. / D X G Y;

S. / D X H Y:

Show also that the elements of  are of the form 1 2 3 ; where 1 2 P .X /, 3 2 P .Y /, and 2 is the uniform distribution on a coset of the finite subgroup H of the group G. Exercise 2.26. Let  2 P .S /, where S D

1 S

ŒS./n is a locally compact

nD1

Hausdorff second countable semigroup. Suppose that S has a kernel which is completely simple and has a product representation X G Y , where G is a compact group and YX D G. Show that n converges weakly iff .n / is a tight sequence. Exercise 2.27. Let  be a probability measure such that 80 9 1 x x1x1 < = S./  @x  1 x x  1A W x real : ; x1x1 x

2.4 Weak Convergence of Convolution Products of Nonidentical Probability Measures

137

1 2 1 1   B 3 3 3C C B C B 1C B 1 2 e D B  C; B 3 3 3C C B @ 1 1 2 A   3 3 3 n Prove that if  .e/ > 0 for some n  1, then n converges weakly to the unit mass at e. (Hint: Note that ye D ey D e for y 2 S./.) 0

Let

Exercise 2.28. State and prove a result similar to that in Exercise 2.25 when S is nondiscrete, but compact. Exercise 2.29. Let  2 P .S /, where S is a commutative semigroup (with respect to multiplication) of d d real matrices (with usual topology) such that S is the 1 S closure of S./n . Suppose that the sequence .n / is tight. Show that n connD1

verges weakly iff there does not exist a closed subgroup H of K, the kernel of S (which is, in this case, a compact abelian group), and an element g 2 K  H such that eS./  gH , where e is the identity of K. Use this to discuss the weak convergence of the tight sequence .n /, when S consists of 3 3 real circulant matrices; that is, matrices of the form 0 1 abc @c a b A bca (Hint: Consider three cases separately – when the rank of the matrices in K is 1, when it is 2, and when it is 3.) Exercise 2.30. Let 1 and 2 be two idempotent probability measures on a locally compact second countable topological semigroup S such that S .1 / D S .2 / and 1  2 D 2  1 D 1 . Show that 1 D 2 .

2.4 Weak Convergence of Convolution Products of Nonidentical Probability Measures In this section, we consider the problem of weak convergence of the convolution products k;n  kC1   n ;

n>k

as n ! 1, where n s are in P .S / and S is a topological semigroup. This problem is much more complicated than the corresponding problem when the n s are identical and, not unexpectedly, it is far from being solved. However, we present a

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2 Probability Measures on Topological Semigroups

number of interesting results in this context when S is a compact or discrete group, and when S is a compact or discrete semigroup. Our first two results are due to Kloss [128, 129]. Theorem 2.19. Let S be a compact connected group and .n /  P .S /. Suppose there exist  > 0, ı > 0 such that m.B/ <  ) n .B/  1  ı 8n;

(2.130)

where m is the normed Haar measure in P .S /. Then, .w/ lim k;n D m; n!1

k  1:

Proof. First suppose that  2 P .S / and  satisfies (2.130). We claim that n converges weakly to m. To prove this, since Z   n .B/ D  Bx1 n1 .dx/; it is clear that for each n  1, n satisfies (2.130). If nk converges weakly to , and B is a Borel set with m.B/ < , there is an open set V  B such that m.V / <  and, consequently, .B/  .V /  lim inf nk .V /  1  ıI k!1

this means that  satisfies (2.130). If, furthermore,  is idempotent, then S./  H is a compact subgroup and  is the normed Haar measure on H . If H is proper, then m.H / D 0, since m.H / > 0 ) H has nonempty interior ) H is an open and (therefore, also) a closed subgroup ) H D S , since S is connected. Since  satisfies (2.130),  cannot be a probability measure if H is proper. Thus,  D m and, consequently, n converges weakly to m. Now let n s satisfy (2.130). By Theorem 2.8, there exist elements an 2 S such that w 1  2   n  ıan ! ˇ 2 P .S /: Write ˇk D ıa1  k  ıak ;

k > 1;

k1

and ˇ1 D 1  ıa1 . Then w

ˇ1  ˇ2   ˇn ! ˇ: Let  be a weak limit point of .ˇn /. Then ˇ   D ˇ; therefore, ˇ  n D ˇ for n  1. Since each n satisfies (2.130), each ˇn does and therefore,  also satisfies w (2.130). By what we have already established, then n ! m, so ˇ D ˇ  m D m. This means that w 1   n  ıan ! m:

2.4 Weak Convergence of Convolution Products of Nonidentical Probability Measures

139

For any subsequence .nk / such that ank ! a, w

1   nk  ıank  ıa1 ! m  ıa1 D m: n k

t u

Theorem 2.19 now follows. The next result is interesting, though very simple. Proposition 2.11. Let S be a compact group and .n /  P .S / such that n .B/  ˛n m.B/;

˛n > 0

for each Borel set B  S , where m is the normed Haar measure in P .S /. Then for any Borel B  S , n Y

j1  2   n .B/  m.B/j 

.1  ˛i / :

i D1

Proof. Notice that there are probability measures ˇn on S such that n D ˛n m C .1  ˛n / ˇn . It is also clear that for k  0, k;n  kC1   n n Y

D

.1  ˛i / ˇk;n

i DkC1

0

C @1 

n Y

1 .1  ˛i /A m;

i DkC1

t u

which implies Proposition 2.11.

We now present two results along the same lines as Theorem 2.16 and Corollary 2.5. Theorem 2.20. Let S be a compact group and n 2 P .S /, n  1. Let ˇ 2 P .S / and ˛n > 0 such that n  ˛n ˇ: (2.131) Suppose also that m .S .ˇa // > 1=2, where m is the normed Haar measure in P .S / and ˇa is the absolutely continuous part of ˇ. Then there exists 0 < d < 1 such that for n > k  1, n Y   k;n  m  2 .1  ˛r d / : rDkC1

(2.132)

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2 Probability Measures on Topological Semigroups

Proof. Let f D dˇa =d m. Then there exists 0 < c < 1 such that m.A/ >

1 ; 2

A  fx W f .x/ > cg:

Notice that if B  A, then .ˇa  cm/ .B/  0, and if B  Ac , then .ˇa  cm/ .B/  0. Thus, kn  ˛n cmk  kn  ˛n ˇa k C ˛n kˇa  cmk  1  ˛n ˇa .S / C ˛n ˇa .A/  ˛n ˇa .Ac /  ˛n cŒ2m.A/  1  1  ˛n cq; where q D 2m.A/  1 > 0. Now taking n D Œn  ˛n cm = .1  ˛n c/, we have: n D ˛n cm C .1  ˛n c/ n so that for k < n, 2

n Y

k;n D

.1  ˛r c/ k;n C 41 

rDkC1

Y

3 .1  ˛r c/5 m;

rDkC1

t u

which implies Theorem 2.20.

Theorem 2.21. Let S be a compact connected group and n 2 P .S /, n  1. Let ˇ 2 P .S /, ˇa ¤ 0, and a > 0 such that n  a:ˇ. Then there exists a positive integer p and 0 < r < 1 such that k1   n  mk  2 r Œn=pC1 for n  1, where m is the Haar probability on S .

  Proof. As in Corollary 2.5, there is a positiveinteger p such that S ˇap D S . Let  us write k  k;kCp . Then if f  .d=d m/ ˇap , as in Corollary 2.5, there exists 0 < c < 1 such that m.A/ > 1=2, where A D fx 2 S W f .x/ > cg, and for k  0, kk  ap cmk  kk  ap ˇap k C ap kˇap  cmk  1  ap cŒ2m.A/  1  1  ap c  r:

As before, write ˇk  .k  ap cm/ = .1  ap c/ so that jˇk j  1, and k D rˇk C .1  r/m;

k  0:

2.4 Weak Convergence of Convolution Products of Nonidentical Probability Measures

141

Thus, k  kCp   kCp.s1/ D r ˇk  ˇkCp   ˇkCp.s1/ C .1  r s / m: s

This means that if p.s  1/  n < ps; then k1   n  mk    1   p.s1/  m < 2r s :

t u

Note that under the conditions of Theorem 2.20, we see that when 1 X

˛n D 1;

nD1

for k  1, k;n ! m weakly and the speed of convergence is given by (2.132). In what follows, we present similar results on convergence (rather than the speed of convergence) on semigroups. We assume that S is a compact semigroup and n 2 P .S / for n  1. However, compactness is not always needed, even though the assumption that ˚  k;n W 0  k and n > n.k/ > k , where fn.k/ j k  0g is a subsequence of positive integers, is tight is always in force. By following the proof of Lemma 2.2, we observe that for any given sequence .pi / of positive integers, there exists a subsequence .ni /  .pi / such that for k  0, k;ni ! k ;

ni !  D  2 ;

k D k  ;

(2.133)

where k s and  are in P .S /. In what follows, whenever we write k ( k .ni /), we imply that (2.133) holds. First we state and prove Lemma 2.5. Lemma 2.5. Given any two subsequences of positive integers .mi / and .ni /, there exist subsequences .pi /  .mi / and .qi /  .ni / such that if the probability measures k  k .pi / and k0  k0 .qi / are in the sense of (2.133) and such that qi !  and p0 i ! 0 , then the following convolution equations hold: (i) (ii) (iii) (iv) (v) (vi)

   D , where pi !  D   ; 0   0 D 0 , where q0 i !  0 D  0   0 ;   0 D  0 ;  0   D ; 0   D ;   0 D  0 .

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2 Probability Measures on Topological Semigroups

Proof. We show the derivation of two of the convolution equations (derivation of the others is similar). As in (2.133), we know there exist subsequences .pi /  .mi / and .qi /  .ni / such that for each k  0, we have k;pi ! k  pi !  D   

k;qi ! k0

q0 i !  0 D  0   0 p0 i ! 0

qi ! 

(2.134)

k0   0 D k0 :

k   D k Let j be fixed and qi > pj . Then

k;pj  pj ;qi D k;qi implies that

k;pj  p0 j D k0

for each j . Taking j to infinity, we have k  0 D k0 ;

k  0:

(2.135)

Equation (2.135) immediately gives   0 D  0 ;

  0 D 0 :

Interchanging the roles of pj and qi , we obtain (iv) and (v). Equations (i) and (ii) follow from the equations k   D k and k0   0 D k0 in (2.30). t u Lemma 2.6, among other things, shows that when S is abelian, local behavior of the convolution products k;n , as n ! 1, at a single point, as described by (2.136) below actually guarantees the weak convergence of these products. Lemma 2.6. Suppose there exists an element x in S such that given any open set N.x/ containing x, there exists a positive integer k0 and some ı > 0 such that for each k  k0 , lim inf k;n .N.x// > ı: (2.136) n!1

Then there is a compact subgroup H such that given any sequence .mi / of positive integers, if k  k .pi / are as in Lemma 2.5 (and as in (2.134)), then we have H D eS./e D eS./e; where e is the identity of H and  is any arbitrary weak limit point of the k s. Moreover, the measures ıe   and   ıe are, respectively, ` -invariant and r  -invariant. Furthermore, over and above (2.136) if we also assume that S is abelian (or every completely simple subsemigroup of S is a group, a condition that, for example,

2.4 Weak Convergence of Convolution Products of Nonidentical Probability Measures

143

holds when S is the multiplicative semigroup of d d bistochastic matrices), then the sequence k;n converges weakly for all k  0 as n ! 1. Proof. Assume k , k0 , ,  0 , , and 0 are as in Lemma 2.5. Since  and  0 are both idempotent probability measures, their supports are both completely simple subsemigroups of S . By Condition (2.136), the element x belongs to     S./ \ S 0 \ S./ \ S  0 : Now xS./x is a compact subgroup (since  D   ), since S./ is completely simple. Let e be its identity. Then xS./x D eS./e. By Lemma 2.5(i), eS./e D xS./x  xS./  S./S./  S./;

(2.137)

which implies that eS./e  eS./e:

(2.138)

Similarly, by Lemma 2.5(iii),

so that

    eS./e D xS./x  S./S 0  S 0

(2.139)

  eS./e  eS 0 e:

(2.140) 0

Thus, (2.137) and (2.139) imply that e 2 S./ \ S . /. Now by Lemma 2.5(v), we have eS./e  eS./e (2.141) and

  eS 0 e  eS./e:

(2.142)

Then from (2.138), (2.140)–(2.142), similarly we have     eS./e D eS./e D eS 0 e D eS  0 e:

(2.143)

In the same manner, we also obtain the following equalities eS./ D eS./I     eS  0 D eS 0 I   S./e D S 0 eI   S  0 e D S./e: Let A  eS./  S./. Since  is idempotent, for y, z in S./,      y 1 z1 A D  z1 A :

(2.144) (2.145) (2.146) (2.147)

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2 Probability Measures on Topological Semigroups

By Lemma 2.5(i), .ıe  /   D ıe  ; and thus, for z 2 S./, we have Z       ıe   z1 A D  y 1 z1 A ıe  .dy/ D  z1 A : For z 2 eS./ and A  eS./, e 1 A \ eS./ D A and z D ez1 with z1 2 S./, so that     ıe   z1 A D  z1 A   1 D  z1 A ; z D ez1 1 e   D  e 1 A   D ıe   e 1 A D ıe  .A/: This means that ıe   is ` -invariant and, therefore, it is also idempotent. Similarly,   ıe is r  -invariant and idempotent. Finally, let us assume that S is a semigroup with the property that S./ and S . 0 /, being completely simple, are groups. Then by (2.143), we have     S./ D eS./e D eS  0 e D S  0  H: Consequently,  D  0  !H , the Haar probability on the compact group H . By Lemma 2.5(i),   ıe D   .  ıe / D    D  since !H  ıe D !H . Thus, S./ D S./ e D H by (2.147). Similarly using Lemma 2.5(ii) and (2.146), it follows that S .0 / D H . We then have  D    D   !H D !H : Similarly, 0 D !H . Thus, we have  D  0 D  D 0 D !H :

(2.148)

Now it follows from (2.135) and (2.148) that k   D k0 ;

k > 0:

k   D k ;

k > 0:

However, (2.134) gives Thus, k D n ! 1.

k0

for all k  0. This means that for k  0, k;n converges weakly as t u

2.4 Weak Convergence of Convolution Products of Nonidentical Probability Measures

145

Lemma 2.7 below finds conditions that guarantee Condition (2.136). It then leads to the most interesting theorem of this section (Theorem 2.22). Lemma 2.7. Suppose there is a compact, completely simple subsemigroup S0 with the following two properties: (i) Given any open set U  S0 , there exists ı > 0 and a positive integer k0 such that for k  k0 , lim inf k;n .U / > ı: n!1

(ii) There exists some s 2 S0 such that for each z 2 sS0 s and any open set N.z/ containing z, there exists some ı > 0 such that lim inf n .N.z// > ı:

n!1

Then for z 2 sS0 s and any open set N.z/ containing z, there exists ı > 0 and a positive integer k0 such that k  k0 ) lim inf k;n .N.z// > ı: n!1

Proof. Let us write the usual product representation of the completely simple subsemigroup S0 as S0 D XHY . X H Y /; where YX  H , H is a compact subgroup of S0 , X  S0 , Y  S0 and the map .x; h; y/ ! xhy is one-to-one. Write s D x1 x2 x3 , where s 2 S0 , x1 2 X , x2 2 H , x3 2 Y . Let h 2 H and y 2 Y . Then for x20 2 H and u 2 X ,   .x1 hy/ u .yu/1 h1 x20 x3 D x1 x20 x3  z: Then z 2 sS0 s and given open N.z/ containing z, there exist open sets N .x1 hy/ 1 1 0 containing x1 hy and also N x1 .yx1 / h x2 x3 containing the element x1 .yx1 /1 h1 x20 x3 (in sS0 s) such that  N .x1 hy/ N x1 .yx1 /1 h1 x20 x3  N.z/: Since sS0 D x1 H Y is compact, there exist hi 2 H , yi 2 Y (1  i  n) such that sS0 D x1 H Y 

n [

N .x1 hi yi /  Us :

i D1

Let us write:

0 zi  x1 .yi x1 /1 h1 i x2 x3 ;

1  i  n:

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2 Probability Measures on Topological Semigroups

Then zi 2 sS0 s, 1  i  n. Let zi1 ; zi 2 ; : : : ; zi r (r  n) be all the distinct elements of the set fz1 ; z2 ; : : : ; zn g. Choose pairwise disjoint neighborhoods N .zi1 / ; N .zi 2 / ; : : : ; N .zi r / such that   N x1 hj yj N .zi k /  N.z/;

1kr

for all those j such that zj D zi k . Thus for t 2 N .zi k /, [˚

   N x1 hj yj W zj D zi k  N.z/t 1 ;

it then follows that k;n .N.z// 

Z r X

  k;n1 N.z/t 1 n .dt/

sD1 N .zis /

so that there exists ı > 0 such that lim inf k;n .N.z//  ı lim inf k;n1 .Us /

n!1

n!1

since by hypothesis, lim inf n .N .zi k // > 0;

1  k  r:

Now sS0  Us . Since S0 is compact and Us is open, there exist open subsets N.z/ containing z and V  S0 such that N.z/V  Us : Since there exists k0 and ı 0 > 0 such that k  k0 ) k .N.z// > ı 0 ; we now have for k  k0 , n > k C 1, k;n1 .Us /  kC1 .N.z//kC1;n1 .V / > ı 0 kC1;n1 .V /: It follows that for k  k0 , n > k C 2, lim inf k;n .N.z//  ı ı 0 lim inf kC1;n .V /:

n!1

Lemma 2.7 follows.

n!1

t u

2.4 Weak Convergence of Convolution Products of Nonidentical Probability Measures

147

Theorem 2.22. Let S be a compact abelian semigroup and K its kernel. Suppose that for each x 2 K and any open set N.x/ containing x, we have lim inf n .N.x// > 0:

n!1

()

Then for all k  0, the sequence k;n converges weakly as n ! 1. Note that Theorem 2.22 implies that if for some ˛ > 0 and m 2 P .S /, n  ˛m for all n  1 and if K  S.m/, then k;n converges weakly as n ! 1. Proof. Notice that for x 2 K, xK D xS , so that for any open set U  K, there exists an open set N.x/ containing x such that N.x/ S  U:

(2.149)

(This is possible since S is assumed compact.) By the assumption (), there exists a positive integer k0 and ı > 0 such that k  k0 ) k .N.x// > ı:

(2.150)

Then for n  1 > k  k0 , k;n .U /  kC1 .N.x//kC1;n .S / > ı because of (2.149) and (2.150). Thus, the conditions (i) and (ii) of Lemma 2.7 hold. Theorem 2.22 now follows from Lemmas 2.6 and 2.7. t u The next result is our convergence theorem in a compact semigroup. As we do not assume the abelian property here, the proof is more involved. It is also relevant to point out that this theorem says that when S is a compact group, then if for each x 2 S and every open set N.x/ containing x, lim inf n .N.x// > 0;

n!1

then for all k  0, k;n converges weakly as n ! 1. Theorem 2.23. Let S be a compact (not necessarily abelian) semigroup with kernel K. Suppose that the following conditions hold: (i) There exists x 2 K such that for each z 2 xKx and any open set N.z/ containing z, lim inf n .N.z// > 0: n!1  ˚ (ii) For any closed subset C  y 2 S W y D y 2 and y 2 xK with x as in (i), lim n .S C / exists. n!1

Then for all k  0, k;n converges weakly as n ! 1.

148

2 Probability Measures on Topological Semigroups

Proof. Notice that given any open set U  K, xK  K  U so that xS D xK  U and, consequently, there exists open N.x/ containing x such that N.x/ S  U so that for k large and n > k C 1, 9 ı > 0 such that k;n .U /  kC1 .N.x//kC1;n .S / D kC1 .N.x// > ı: Thus, conditions (i) and (ii) of Lemma 2.7 hold with K replacing S0 . By Lemma 2.7, for any open set N.z/ containing z, an arbitrary element in xKx, there is a ı > 0 such that for k sufficiently large, lim inf k;n .N.z// > ı:

n!1

By Lemma 2.6, if .ni / and .mi / are two given sequences of positive integers, then there exist subsequences .pi /  .ni / and .qi /  .mi / such that for k  0, k;pi ! k  pi !  D   

k;qi ! k0 ;

q0 i !  0 D  0   0 ; k0   0 D k0 ; p0 i ! 0 :

k   D k qi ! 

(2.151)

We claim that for k  0, S .k /  K: Let k  0 and let U and V be open sets containing K such that S:U  V . Then given an positive integer m, there exist open subsets Vi , 1  i  m, and closed W  U such that K  Vi  V i  Vi C1  W and S Vi  Vi C1 , 1  i  m. By condition (i), there exists ı > 0 and a positive integer p such that n .V1 / > ı for n > p. Let r > maxfk; pg. Then Z  c   r;rCm Vm D rC1 Vmc x 1 rC1;rCm .dx/ c Vm1

 c   c   rC1 Vm1 rC1;rCm Vm1 < .1  ı/m : It follows that r .W c / D 0 so that r .U / D 1. Since k;r  r D k , k .V / D 1. This establishes that S .k /  K. It is clear that for some ˇk 2 P .S /, S .ˇk /  K and 0 2 P .S /, where pij ! 0 ;

k;pi

j

1

! ˇk

(for each k  0), we have ˇk  0 D k ;

k  0:

2.4 Weak Convergence of Convolution Products of Nonidentical Probability Measures

149

Now let C be a closed subset of E.xK/ (that is, the set of idempotents in xK). Then S C D K C and Z   k .S C / D ˇk KC w1 0 .d w/: Since K, S./, and S . 0 / are all completely simple subsemigroups and ,  0 are both idempotent, we have K D XGY  X G Y I S./ D X1 H Y1  X1 H Y1 I   S  0 D X 2 H Y2  X2 H Y 2   1 !H 2 ;  0  10 !H 20  D 1  !H  2 ;

 0 D 10  !H  20 ;

where 1 2 P .X1 /, 2 2 P .Y1 /, 10 2 P .X2 /, 20 2 P .Y2 /, !H is the Haar probability on H . These decomposition results have already been established in earlier sections. Now notice that for C  Y , we have .S C / D .KC / D .XGC / D 2 .C / and similarly,

 0 .S C / D 20 .C /:

This means that 2 D 20 . By Lemma 2.5, we have 0   0 D 0 I also by (2.135),

k  0 D k0 ;

k  0:

(2.152)

(2.153)

Then using (2.134), for all k  0, we have k0 D k  0

  D .k  /  0   0

  D k  .1  !H  2 /  0  10  !H  20   D k  1  !H  2  0  10  !H  20 D k  1  !H  20 D k  .1  !H  2 / D k   D k   since S 2  0  10  H and 2 D 20 . (Note that by Lemma 2.7, eS .0 / e D H .) It follows that for all k  0, k;n converges weakly. t u

150

2 Probability Measures on Topological Semigroups

In what follows, we describe some of the consequences when for k  0, the sequence k;n converges weakly to some k as n ! 1. In this context, our next result describes the structure of the weak limit points of the sequence .k /. Theorem 2.24. Suppose that for each k  0, k;n converges weakly to k 2 P .S / and fk W k  0g is tight. Let us define the sets F , J1 and J2 as follows: F D f 2 P .S / W  is a weak limit point of .k /g ; [ fS./ W  2 F g; J1 D   J2 D cl J1 : Then the following assertions hold: (i) J1 and J2 are both completely simple subsemigroups of S . (ii) Let J1 D XH Y  X H Y be the usual product representation for the completely simple subsemigroup J1 . Then for any  2 F , S./ D X H Y  X H Y and   1 !H y ; where !H is the Haar probability on the compact subgroup H , 1 2 P .X /, and y 2 P .Y /. The proof of Theorem 2.24 is omitted. See [26] for a proof. Now we look into a useful connection between the weak convergence of the sequence k;n for k  0 and the almost sure convergence of the random walk X1 X2 Xn , where .Xi / is an independent S -valued sequence such that Xi has distribution i . To this end, let us assume that S is either abelian or a group, for k  0, k;n converges weakly as n ! 1 to k and furthermore, the sequence fk W k  0g is tight. First we claim that in this case, as k ! 1, the sequence k converges weakly. To prove this claim, let .pi / and .qi / be sequences of positive integers such that pi !  and qi !  0 . For k < s < n, k;s  s;n D k;n and it follows after writing s D pi and then s D qi , respectively, k   D  k ;

k   0 D k ;

k  0:

This means that  D  2 ,  0 D  02 ; also  0   D  0;

   0 D :

(2.154)

2.4 Weak Convergence of Convolution Products of Nonidentical Probability Measures

151

By Theorem 2.2, S./ and S . 0 / are both compact subgroups of S . When S is abelian, clearly  0 D  0   D    0 D , and when S is a group, it is clear from (2.154) that S . 0 / S./ D S . 0 / and S./ S . 0 / D S./, so that S./ D S . 0 / and ;  0 (where both are Haar measures on their supports which are both compact subgroups of S ) are equal. Thus, k converges weakly to some  2 P .S / as k ! 1. Let H be the support of . Then H is a compact subgroup of S . Consider the collection S=H of cosets fgH W g 2 S g. Note that S=H , when equipped with quotient topology, is an abelian Hausdorff topological semigroup if S is abelian. When S is a group, S=H with quotient topology is a Hausdorff topological space. Since S is assumed to be second countable, it follows easily that S=H is also second countable. Let p be the natural map from S to S=H , i.e., p.x/ D xH . Then we have Theorem 2.25, which may be skipped after the first reading. Theorem 2.25. Let S be either a topological group or a compact abelian semigroup. Let n 2 P .S /, n  1. Suppose that for each k  0, k;n converges weakly to k . Then the following assertions hold: (i) There is a compact subgroup H such that k ! !H weakly as k ! 1, where !H is the Haar probability on H . (ii) For each open set U  H , 1 X

 

1  n U U 1 < 1:

nD1

Note that when S is a group, given any open set V  H , there is an open set U  H such that U U 1  V and therefore, in this case 1 X

Œ1  n .V / < 1:

nD1

(iii) If p.x/ D xH for x 2 S , then p .X1 X2 Xn / almost surely converges, where the Xi s are S -valued independent random variables such that Xi has distribution i . Proof. Part (i) was proved earlier. We prove (ii) in several steps. Step 1. Let S0 be a compact subsemigroup of S . Let U be an open set containing S0 . Then there is an open set V such that S0  V , V V  U and an element z 2 U such that V S01  U z1 : (2.155) When S is a group, S0 is then necessarily a group and (2.155) is then trivial. Now we prove (2.155) when S is an abelian semigroup. Since S0  U , S0 S0  U , and S0 is compact, there exists an open V such that S0  V , V is compact, and V V  U . Now the class of sets  ˚ y V j y 2 S0

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2 Probability Measures on Topological Semigroups

is contained in the compact space S0 V and it has the finite intersection property since n \ y1 y2 yn 2 yi V i D1

if each yi is in S0 . Hence, there exists z2

\

y V:

y2S0

Let w 2 V S01 . Then there exists w0 2 S0 such that w w0 2 V . Now z 2 w0 V , so that z D w0 w00 for some w00 2 V , hence   wz D ww0 w00 2 V V  U or w 2 U z1 , this establishes (2.155). Step 2. We do not need S to be abelian; however, in this step S must be either a group or a compact semigroup. For x; y in S , and an open set N.x/ containing x, there exist open sets N 0 .x/ and N 0 .y/ containing x and y, respectively, such that N 0 .x/N 0 .y/1  N.x/y 1 :

(2.156)

This is trivial when S is a group, and left as a problem when S is a compact semigroup. Step 3. Let S0 be a compact subset of S and V an open subset such that V  S0 . Then there exists an open subset W  S0 so that W W 1  V S01 : The proof of Step 3 follows routinely from Step 2, using the compactness of S0 . Step 4. Let 1  i  m. Let fA1 ; A2 ; : : : ; Am g and fB1 ; B2 ; : : : ; Bm g be two families of events such that for each i , Bi is independent of each Ac1 ; Ac2 ; : : : ; Aci1 ; Ai . Then we have ! !   m m [ [ P .Ai \ Bi /  inf P .Bi / P Ai : 1i m

i D1

i D1

To see this, notice that P

m [

!

   .Ai \ Bi /  P .A1 \ B1 / [ A2 \ B2 \ Ac1 [

i D1

  [ Am \ Bm \ Ac1 \ \ Acm1

D

"m X



P .Bi / P Ai \

i D2

C P .B1 / P .A1 / : The proof of Step 4 is now clear.

Aci1

\ \

Ac1



#

2.4 Weak Convergence of Convolution Products of Nonidentical Probability Measures

153

Step 5. Let us assume that for each k  1, k;n ! k as n ! 1 and k ! !H as k ! 1, where H is a compact subgroup of S . Given  > 0, let U be an open set with compact closure containing H . Then there exists k0 such that for k  k0 , k .U / > 1  :

(2.157)

Since for k < m, k D k;m  m , it follows that for m > k  k0 ,   k;m U U 1 > 1  2:

(2.158)

Let X1 ; X2 ; : : : be a sequence of independent random variables with values in S such that P .Xn 2 B/ D n .B/. From Steps 1–3, there exists z0 2 U and an open set V  H such that V V 1  U z1 (2.159) 0 : By the same reason, there exists z 2 V and an open set W  H such that W W 1  V z1 :

(2.160)

Notice that m1 [

˚  Xk XkC1 XkCi … V V 1 and XkCi C1 XkCm 2 V z1

i D0

 ˚  Xk XkC1 XkCm … V z1

(2.161)

since Xk XkC1 XkCm 2 V z1 ; XkCi C1 XkCm 2 V z1  1  ) Xk XkC1 XkCi 2 V z1 V z1  V V 1 : Using (2.160) and the same method as was used in (2.158), there exists k1 such that r > k  k1 implies   k;r V z1 > 1  : (2.162) Using Step 4, (2.161) and (2.162), for k  k1 , we have P

! m [ ˚  1 Xk XkC1 XkCi … V V  i D0

 < 2; 1

0 1  2: P Xk 2 U z1 0 ; Xk XkC1 2 U z0 ; Xk XkC1 XkCm 2 U z0 (2.163) We can now write Y   mC1   1 P Xk 2 U z1  X X X 2 U z kCi U U 1 : (2.164) k kC1 kCmC1 0 0 i D0

  1 1  U z0  U U 1 , also (Notice that U z1 0   1 P Xk 2 U z1 0 ; Xk XkC1 2 U z0   1 1  1   P Xk 2 U z1 U z0 0 ; XkC1 2 U z0     1 1  P Xk 2 U z0 P XkC1 2 U U : Thus, we can use induction.) The proof of Part (ii) is now complete from (2.163) and (2.164). This step is also clear in the group case. t u Proof (of part (iii)). There exists a sequence of open sets Vi  Vi C1 such that V 1 is compact and 1 \ Vi : (2.165) H D i D1

Let V be any open set containing H . Then there exists i such that   Vi Vi Vi1  V:

(2.166)

Equation (2.166) is trivial when S is a group, however it is not completely obvious when S is a compact abelian semigroup, and is left as an exercise. Let us now write A.i / D

1 1 \ [ ˚

 Xk XkC1 XkCm 2 Vi Vi1 ;

kD1 mD0

where the Vi s are as in (2.165). By (2.163) and (2.164), P .A.i // D 1 for i  1. Let AD

1 \

A.i /

i D1

so that P .A/ D 1. We show that for ! 2 A, p .X1 .!/X2 .!/ Xn .!// converges as n ! 1.

2.4 Weak Convergence of Convolution Products of Nonidentical Probability Measures

155

Notice fX1 .!/ Xn .!/ W n  1g is relatively compact since Vi Vi1 is relatively compact for each i  1. Let n.j / be a subsequence of positive integers such that X1 .!/X2 .!/ Xn.j / .!/ ! g:

(2.167)

Let U be an open set containing H . By (2.166), there exist i0 and i1 > i0 such that for all i  i1 ,

Vi0 Vi0 Vi1 U 0 and



Vi Vi Vi1  Vi0 :

This means that when S is a compact abelian semigroup, for i  i1 , h˚  h 1  i 1 i Vi Vi1 Vi D Vi Vi Vi Vi1 Vi Vi1 Vi 1

 V i 0 Vi 0 V i 0

D Vi0 Vi0 Vi1 0  U: When S is a group, it is easy to see that this inequality holds, namely, there exists i1 such that for i  i1 , h 1  i Vi Vi1 (2.168) Vi Vi1 Vi  U: Now choose i  i1 . Then ! 2 A.i /, and therefore there exists some k such that for each m  k, Xk .!/XkC1 .!/ Xm .!/ 2 Vi Vi1 : Choose m > n.j / > k. Then we have 1    Vi Vi1 : Xn.j /C1 .!/ Xm .!/ 2 Vi Vi1

(2.169)

By (2.167), given an open set Vg containing g, there exists j0 such that n .j0 / > k, and for j  j0 , X1 .!/X2 .!/ Xn.j / .!/ 2 Vg : Therefore, for m > n .j0 / > k, from (2.169), we have X1 .!/X2 .!/ Xm .!/ 2 Vg



1   Vi Vi1 Vi Vi1 ;

so for m > n .j0 /,   1  

Vi Vi1 Vi  p Vg U : p .X1 .!/ Xm .!// 2 p Vg Vi Vi1 It follows that for each ! 2 A, p .X1 .!/ Xn .!// converges.

t u

156

2 Probability Measures on Topological Semigroups

In the remainder of this section, we present sufficient conditions for the weak convergence of k;n for k  0 when S is either a discrete group or a discrete abelian semigroup. Lemma 2.8. Let S be a discrete group and .n /  P .S /. Let H be a subgroup of S and X1 ; X2 ; : : : independent S -valued random variables such that the distribution of Xn is n . Then if p is the natural map (that is, p.x/ D xH for x 2 S ), p .X1 X2 Xn / converges almost surely if and only if 1 X

n .S  H / < 1I

nD1

furthermore, in this case, lim inf k;n .H / D 1:

k!1 n>k

Proof. The first part of Lemma 2.8 is immediate by the Borel–Cantelli Lemma (and since S is discrete). For the second part, notice that if 1 X

n .S  H / < 1;

nD1

then by the Borel–Cantelli Lemma, P .Xn … H infinitely often/ D 0 so that given  > 0, there exists a positive integer N0 such that for n > N > N0 , n \

P

! fXk 2 H g > 1  

kDN

so that lim inf k;n .H / D 1:

k!1 kk

(2.180)

By (2.179) and (2.180), the sequence .hn / must be finite and, therefore, since k;n ! k and k ! !H , there is some h in S such that hn D h for infinitely many n and   !H hH 0 h1 D 1;

2.4 Weak Convergence of Convolution Products of Nonidentical Probability Measures

159

so that H  hH 0 h1 . This is a contradiction, since H 0 is assumed to be a proper subgroup of H , which means that the cardinality of hH 0 h1 is less than that of H . t u Corollary 2.6. Let S be a discrete group with identity e and .n /  P .S /. Suppose there exists ı > 0 such that for each n  1, n .e/ > ı. Then the following two conditions are equivalent: (i) There is a finite subgroup G  S such that 1 X

n .S  G/ < 1:

nD1

(ii) For k  0, the sequence k;n converges to some k in P .S /. Proof. Note that Condition (ii) implies Condition (i) by Theorem 2.26. For the converse, let us assume Condition (i). Then there is a smallest integer p with the property that there is a finite subgroup G 0 with cardinality p such that 1 X

  n S  G 0 < 1

nD1

If k;n does not converge for some k, as n ! 1, then by Theorem 2.26, there exist elements hn in S and a proper subgroup H of G 0 such that 1 X

  < 1: n S  hn1 H h1 n

nD1

Then there exists a positive integer N such that n  N implies that e 2 hn1 H h1 n or hN 1 H D hN H D hn H for n > N ; in other words, for n  N , hn1 H h1 n is a subgroup and it is the same subgroup for all such n. But since H is proper in G 0 , the cardinality of this subgroup is smaller than p. This is a contradiction. t u Our last result in this section concerns the weak convergence of k;n for all k  0, when S is a discrete abelian semigroup. For this, we assume the following condition: ./ For .n /  P .S /, fn W n  1g˚ is tight and there exists  a sequence n.k/ of positive integers such that the set k;n W k  0; n > n.k/ is tight. Theorem 2.27. Let S be a discrete abelian semigroup and .n /  P .S /. Suppose that condition ./ above holds. Suppose also that lim inf n .B/ > 0;

n!1

where  ˚ B D x 2 S j f x D .f x/2 , whenever f is an idempotent in S : Then for k  0, k;n converges weakly.

(2.181)

160

2 Probability Measures on Topological Semigroups

Notice that under condition ./, as in (2.133), given any sequence .pi / of positive integers, there exists a subsequence .ni /  .pi / such that for k  0, 9 k;ni ! k ; = ni !  D  2 ; ; k   D k :

(2.182)

We refer to k as k .ni / and  as the tail idempotent corresponding to .ni /. Let  0 be another tail idempotent corresponding to a different subsequence .mi / such that k;mi ! k0 ;

0 m !  0 D  02 : i

  Let us now assume that S is abelian. If  is a weak limit point of mi and 0 is a weak limit point of n0 i , then as in Lemma 2.5,  D 0   D    0 D  0 ;

(2.183)

and  is the Haar probability on its support that is a compact group, say, H . Thus, the tail idempotent  is independent of whatever sequence .ni / or .mi / is chosen, and, therefore, unique in this sense. Also,    D ;

0   D 0 :

(2.184)

If e is the identity of H , then   ıe D  and   ıe D     ıe D    D :

(2.185)

For x 2 S./ and y 2 S .0 /, the element y x is in H by (2.183), and, therefore, there exists z in H such that .zy/x D z.yx/ D e:

(2.186)

Since zy 2 S .0 / by (2.184), from (2.183) and (2.185), we have S./ D S./ e D S./.zy/x  H x: By (2.184), xH  S./; therefore, for x 2 S./, S./ D xH:

(2.187)

ue D u whenever u 2 S./:

(2.188)

Now we claim that

2.4 Weak Convergence of Convolution Products of Nonidentical Probability Measures

161

To this end, let u 2 S./. By (2.187), there exists v 2 H such that u D vu. Let w 2 H such that vw D wv D e. Then wu D w.vu/ D eu so that u D vu D .ve/u D .vw/u D eu D ue: Thus, (2.188) is established. It is now clear that if S1 is the subsemigroup of S generated by AD

[

S./ W

 there is a subsequence .ni / such that (2.182) ; holds and  is a weak limit point of .k /

then S1 is a group. This follows from (2.183) and (2.188). We can now prove Lemma 2.9, needed for the proof of Theorem 2.27. Lemma 2.9. Let S be abelian and .n /  P .S /. Then the subsemigroup S1 (as just defined) is a group. There is a unique compact subgroup H of S1 such that !H , the Haar probability on H , is the unique tail idempotent (as defined in (2.182)); moreover, if  is a weak limit point of the k s in (2.182) and  ¤ !H , then there exists x 2 S1  H such that  D !H  ıx . Furthermore, the following assertions hold: (i) Given  > 0 and any open set D such that D  V S1 , V is open and V  S1 , there exists a positive integer k0 such that for each k  k0 , there exists a positive integer n.k/ such that for n  n.k/, k;n .D/ > 1  :

(2.189)

 ˚ (ii) Let B D x 2 S j f x D .f x/2 whenever f is an idempotent in S . If 0 is a weak limit point of .n /, then ey D e whenever y is an element of B \ S .0 /, where e is the identity of S1 . (iii) Suppose S is discrete. [Note that discreteness is not assumed in (i) or (ii).] Then we have 1 X Œ1  ıe  n .S1 / < 1; nD1

where e is the identity of S1 . Proof. Assertion (i): Let S1  V , V S1  D, where V and D are open subsets of S . We know that given any sequence of positive integers, there is a subsequence .ni / of positive integers such that as i ! 1, for k  0, k  !H D k ; ni !   !H ; k;ni ! k :

(2.190)

162

2 Probability Measures on Topological Semigroups

Let  > 0. Then there exists a positive integer k0 such that k  k0 ) k .V / > 1  

(2.191)

since all weak limit points of .k / have their supports inside S1 . Now if (2.189) is false, there exists some p  k0 and a subsequence .mi / of positive integers such that p;mi .D/  1  ; k;mi !

k0

i  1I

(2.192)

8 k  0:

  It follows from (2.190) and (2.192) that for some weak limit point 0 of n0 i , p  0 D p0 : Since V S1  D and S .0 /  S1 , this implies that p0 .D/  p .V / > 1  ; which contradicts (2.192). This proves Assertion (i). Assertion (ii): If 0 is a weak limit point of .n /, then noting (2.190) and observing that k;ni  ni C1 D k;ni C1 ; k < ni ; we have S .!H  0 /  S1 . Thus if x 2 B \ S .0 /, assuming that B \ S .0 / is nonempty, then ex is an idempotent in S1 by definition of B and, therefore, ex D e. Assertion (iii): Here, we assume that S is discrete. First, we make the following observations: By Assertion (i), there exist positive integers k and n.k/ such that k;n .S1 / > y 2 S1 e 1 x 1 ; y 2 S1 e

1 1

x

;

2 ; n  n.k/ > k; 3 x 2 S1 e 1 ) y 2 S1 e 1 ; y 2 S1 e

1

) x 2 S1 e

1

:

(2.193) (2.194) (2.195)

Let p be any positive integer and n > n.k/. Then, using (2.193), (2.194), and (2.195), we have the following:   k;nCpC1 S1 e 1 D

X

  k;nCp S1 e 1 x 1 nCpC1 .x/

x2S1 e 1

C

X

x…S1 e 1

  k;nCp S1 e 1 x 1 nCpC1 .x/

     k;nCp S1 e 1 nCpC1 S1 e 1    

C 1  k;nCp S1 e 1 1  nCpC1 S1 e 1

2.4 Weak Convergence of Convolution Products of Nonidentical Probability Measures

   k;nCp S1 e 1 

163

1

 

1  nCpC1 S1 e 1 3 X   1 nCpC1  

 k;n S1 e 1  1  r S1 e 1 : 3 rDnC1 t u

The claim in Assertion (iii) follows from this inequality.

Proof (of Theorem 2.27). By (2.181), there exist a positive integer n0 and ı > 0 such that n  n0 ) n .B/ > ı: Let e be the identity of the group S1 . If ıe  n .e/ < ı=2 for infinitely many n, then there is a weak limit point 0 of .n / such that ıe  0 .e/  ı=2. However since .n / is tight, there is a finite subset B1  B such that n .B1 / > 2=3ı for n  n0 . By Assertion (ii) of Lemma 2.9, this means     ıe  0 .e/ D ıe  0 e B1 \ S 0 2  0 .B1 /  ı; 3 which is a contradiction. Thus, there exists a positive integer n1 such that n  n1 ) ıe  n .e/ 

ı : 2

(2.196)

For n  n1 , we define the probability measure ˇn 2 P .S1 / such that for Q  S1 , ˇn .Q/ ıe  n .S1 / D ıe  n .Q/: Notice that for Q  S , by Assertion (iii) of Lemma 2.9 1 X nDn1

jˇn .Q/  ıe  n .Q/j 

1 X

2 Œ1  ıe  n .S1 / < 1;

nDn1

since if Q \ S1 D ;, ˇn .Q/ D 0 by definition, and

1 P

ıe  n .Q/ < 1. This

nD1

means that whenever ˇk;n converges weakly as n ! 1, the sequence k;n  ıe D .ıe  kC1 /   .ıe  n / also converges weakly and, consequently, k;n converges weakly. This last statement follows, since in (2.190) k  !H D k ) k  ıe D k

164

2 Probability Measures on Topological Semigroups

˚  by Theorem 2.1. Since for each k, ˇk;n W n > k is tight and since we can consider the ˇk s as probability measures on the group S1 , it follows from Theorem 2.8 (using arguments similar to those used in (2.171) and (2.172)) that there exist elements xn in S1 such that if k D ıx 1  ˇk  ıxk ; k1

then for each k > 0, k;n converges weakly so that by Theorem 2.25, there is a finite subgroup H0 such that 1 X

Œ1  n .H0 / < 1;

nD1

which implies that

1 X

Œ1  ˇn .H0 gn / < 1;

(2.197)

nD1

where gn D xn1 xn1 . By (2.196) and (2.197), e 2 H0 gn for n large, so that 1 X

Œ1  ˇn .H0 / < 1:

(2.198)

nD1

Theorem 2.27 now follows from (2.196), (2.198) and Corollary 2.6.

t u

Section 2.4 Exercises Exercise 2.31. Let m be the Haar probability measure on a compact (Hausdorff) group G and H be a compact subgroup of G such that m.H / > 0. Show that H is also an open subgroup. (Hint: Consider the function f given by Z f .x/ D

IH .xy/IH .y/ m.dy/:

Given ı > 0, let V be an open set and N an open set such that H  V , m.V / < m.H / C 12 ı, and N:H  V . Use the function g defined by Z g.x/ D

IV .xy/IV .y/ m.dy/

to show that f is continuous at x D e.) Exercise 2.32. The well-known Birkhoff theorem states that a d d bistochastic matrix is a convex combination of d d permutation matrices (though this representation is not unique). (For a proof, see [147] or the paper by L. Mirsky, Proc.

2.4 Weak Convergence of Convolution Products of Nonidentical Probability Measures

165

  Amer. Math. Soc. 9 (1958), 371–374.) Note that if P  Pij is a d d stochastic matrix, then we can write: X P D P1k1 P2k2 Pd kd Ak1 k2 kd ; k1 ;k2 ;:::;kd 1ki d

where Ak1 k2 kd is a 0–1 stochastic matrix with 1 appearing on the i th row and the ki th column for 1  i  d , and zeros elsewhere. In other words, a stochastic matrix is a convex combination of 0–1 stochastic matrices. If S is the finite multiplicative semigroup of d d 0–1 stochastic matrices, and we define the probability measure  on S by    Ak1 k2 kd D P1k1 P2k2 Pd kd ; then we write: P $ . If Q is another d d stochastic matrix, and Q $ ,  the probability measure corresponding to Q, then show that P:Q $   . Also, show that if Pk $ k , for k  1, then the product Pk;n  PkC1 Pn converges pointwise if and only if k;n converges weakly (that is, elementwise in this case). Use the finite group of d d permutation matrices, the Birkhoff theorem, and Corollary 2.6 to prove the following: Let .Pn / be a sequence of d  d bistochastic matrices such that each of the d diagonal entries of each Pn is greater than some ı > 0. Then the sequence Pk;n converges elementwise as n ! 1 for each k 1.

Exercise 2.33. Let .Pn / be a sequence of d d stochastic matrices such that for each positive integer k, the sequence Pk;n converges elementwise to the matrix Qk . Show that lim n ! 1Qk D Q1 exists, where Q1 is an idempotent bistochastic matrix. (Hint: Pk;m Pm;n D Pk;n implies Pk;m Qm D Qk . Thus if Q0 and Q00 are two limit points of .Qk /, then Qk Q0 D Qk for k  1 and therefore, Q00 Q0 D Q00 , Q0 Q0 D Q0 and Q0 Q00 D Q0 . The probability measure 0 (corresponding to Q0 ) on the finite group of d d permutation matrices is idempotent, and thus, the Haar (or uniform) probability measure on a finite subgroup, and so is 00 , the probability measure corresponding to Q00 . Thus, 00  0 D 00 and 0  00 D 0 .) Exercise 2.34. Let S be a locally compact Hausdorff second countable semigroup and n 2 P .S /, n  1. Let k D .w/ lim k;n . Let F be the set of all weak limit n!1

points of fk W k  1g. Suppose that F is nonempty. Show that every element in F is an idempotent in P .S /, and so, the closure of the union of supports of elements in F is a completely simple subsemigroup of S . Exercise 2.35. In Problem 2.34, let 1 and 2 be two elements in F such that S .1 / \ S .2 / is nonempty. Let the usual product representations of S .1 / and S0 be given by: S .1 / D X0 G0 Y0 ;

S0 D X G Y;

where G0 D eS .1 / e, G D eS0 e, e D e 2 2 S .1 /. Show that S .2 / D X1

G0 Y0 , where Y0 .X0 [ X1 /  G0 .

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Exercise 2.36. Let G be a compact Hausdorff second countable group and for each n  1, n 2 P .G 0 /. Suppose that .w/ lim k;n D k for k  1. Show n!1

that .w/ lim k D H exists, where H is the Haar probability measure on k!1

some compact subgroup H , and also H is the largest compact subgroup such that k  H D k for all k  1. Exercise 2.37. In Exercise 2.36, show also that lim k;kCm .V / D 1;

k!1

uniformly in m  1, whenever V is open and V  H . Exercise 2.38. Let .Xn / be a sequence of real-valued independent random variables. Set Sn D X1 C X2 C C Xn . The classical L´evy equivalence theorem says that Sn converges almost surely iff Sn converges in distribution. This theorem was generalized to group-valued random variables in [50] and in [72]. Write a report on this extension. Exercise 2.39. Let X1 ; X2 ; : : : be a sequence of independent random variables taking values in a discrete completely simple semigroup S D X G Y (usual product representation). Show that the following conditions are equivalent: (i) The sequence Zn D X1 X2 Xn converges almost surely to Z as n ! 1; (ii) There exists y in Y such that 1 X

 

1  P Xn 2 Iy < 1;

nD1

  where Iy is the set of idempotents in X G fyg, that is, elements x;.yx/1; y , x 2 X. (Hint: (i) implies that there exists y in Y such that P .Z 2 X G fyg/ D ı > 0. Note that for k  1, Zk 2 X G fyg iff Xk 2 X G fyg, and Zn D ZnC1 2 X G fyg only if XnC1 2 Iy .) Exercise 2.40. Consider S D G Y , where G D feg, Y D fy1 ; y2 g such that .e; y1 / .e; y2 / D .e; y2 / D .e; y2 / .e; y2 / and .e; y2 / .e; y1 / D .e; y1 / D .e; y1 / .e; y1 / : Define: for n  1, n f.e; y2 /g D n1 , n 2 P .S /. Show that:   (i) lim inf n;m Iy1 D 1, Iy1 as in Problem 2.39; n!1 m>n

(ii) For k  1, k;n converges weakly to the unit mass at .e; y1 /; (iii) Zn does not converge almost surely (use Problem 2.39). (Hint: Notice that Iy1 sN1 D ;, s … Iy1 D S , s 2 Iy1 .)

2.4 Weak Convergence of Convolution Products of Nonidentical Probability Measures

167

Exercise 2.41. Let S and .Xn / be as in Exercise 2.39. Show that the following are equivalent: (i) The sequence Zn converges almost surely; (ii) There exists y 2 Y such that X

Œ1  n .X G fyg/ < 1;

nD1

  and lim inf n;m Iy D 1, where Iy is as in Exercise 2.39 and for k  1, n!1 m>n

k .x/ D P .Xk D x/. Exercise 2.42. Consider Exercise 2.39 in the nondiscrete locally compact second countable situation. Show that as before, when Zn converges almost surely to Z, then there exists y 2 Y such that P .Z 2 X G fyg/ D 1: (This and other relevant results appear in [147].)   Exercise 2.43. Let two sequences .n / and 0n in P .S /, S is a locally compact second countable semigroup, be called equivalent if 1 X   n  0  < 1; n nD1 0

where for  and  in P .S /,  ˇ  ˚ˇ    0  D sup ˇ.B/  0 .B/ˇ W B a Borel subset of S : Show that k;n converges weakly as n ! 1 for k  1 iff 0k;n does the same when   kCm   P   .n / and 0n are equivalent. (Hint: k;kCm  0k;kCm   kn  0n k.) nDkC1

Exercise 2.44. Let S be a compact second countable semigroup and .n /  P .S /. An element x in S is called inessential if there is an open set N.x/ containing x such that 1 X n .N.x// < 1: nD1

An element, when not inessential, is called essential. Show that the set E of essential points is the smallest closed set C such that 1 X nD1

whenever V is open and C  V .

Œ1  n .V / < 1;

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2 Probability Measures on Topological Semigroups

Exercise 2.45. Suppose in Exercise 2.44 S is also a group, and that for k  1, k;n converges weakly as n ! 1 to k . By Theorem 2.25, k ! !H weakly as k ! 1. Show that the set E of essential points is a subset of the compact subgroup H . Show also that if the homogeneous space H=ŒE, ŒE being the smallest closed subgroup containing E is finite, then H D ŒE. Use Example 2.5 to show that ŒE can be a proper subgroup of H . (The notion of essential points and the above result are due to [136].) Exercise 2.46. Prove assertion (2.156) when S is a compact Hausdorff semigroup. Exercise 2.47. Prove assertion (2.166) when S is a compact Hausdorff abelian semigroup.

2.5 Notes and Comments Section 2.2 Basic results, such as Propositions 2.1, 2.2, and 2.3, are standard and wellknown (see [172], [182] or [77]). Theorem 2.1 is due to many authors (independently); for instance, see [58] and [216]. A complete analog of this theorem in the nonabelian case is still unsolved. Proposition 2.4 is taken from [143]. Proposition 2.5 (note that  is not assumed to be finite here) is taken from [168]. Theorem 2.2 in locally compact (or complete metric) semigroups was first given in [168]; earlier in compact semigroups, it was given in [87] and [187]. Theorem 2.3 is also contained in [168]. It was given in [108] for infinite  when S is a semigroup of matrices. A related result in other special semigroups is given in [105]. It is relevant to point out that an analog of Theorem 2.3, when  is not finite, is still an unsolved problem. Theorem 2.4, given in [169], sheds some light on this open problem. Theorem 2.5 appears in [170]. For characterizations (and results) concerning other types of invariant measures, we refer the reader to [3, 9, 222, 223]. For analogs of Theorems 2.2 and 2.3 for semitopological semigroups (where the semigroup multiplication is separately continuous and not necessarily jointly continuous), the reader is referred to [75, 170, 188]. The paper [231] was probably the first survey paper in this area. The paper [1] is the first paper to introduce r  -invariant measures on semigroups. Let us mention [36] for idempotent measures, the papers [35, 58, 216, 235] and Raugi (1983) in connection with results related to Theorem 2.1 and Proposition 2.4. The paper of Gelbaum and Kalisch (1952) is among the first in this area.

Section 2.3 Theorem 2.7 is a key theorem in this section. Ideas used in the proof of this theorem are due to many authors. Parts (i) and (ii) of this theorem appeared in [147], and parts

2.5 Notes and Comments

169

(iii) and (iv) in [133]. Parts (i), (ii), and the first half of (iii) were given earlier by Rosenblatt for the case of compact semigroups. (See also the papers of [47, 48, 196, 198, 207, 210, 225] in this context.) Corollary 2.1, most of which was originally due to Kawada and Ito, is given in this form in [205]. (The part “(i) equivalent to (ii)” was given earlier in [47].) Corollaries 2.3 and 2.4 are taken from [145]. Proposition 2.6 is due to Rosenblatt. Lemma 2.2 is due to Csiszar. Theorem 2.8, in the case of a group, was given in [50]; see also [218]. This version of Theorem 2.8 is taken from [147]. Theorem 2.9 appeared in [145]; it was also obtained independently in [57]. Theorem 2.10 is taken from [146]. Proposition 2.8 appeared in [145] and Theorem 2.11 in [149]. Theorem 2.12 is due to H¨ogn¨as and Mukherjea. Theorem 2.13 first appeared in [138]; the proof presented here is somewhat different. Obtaining an analog of this theorem in a nondiscrete semigroup is much more difficult and still an unsolved problem. Example 2.6 is due to Harry Kesten (personal communication). Theorem 2.16 and Corollary 2.5 are given in [12]. Theorem 2.17 was communicated by Mindlin [140–142]. Theorem 2.18 appeared in [110]. It was, however, first initiated by [33]. Exponential convergence to Haar measure with respect to other probability metrics have also been studied. H. Horst, in Math. Nachrichten 104 (1981) 49–59, considers a compact abelian group and a wider class of measures than the spread-out ones. He finds one particular norm (called the BL-norm) for which there is exponential convergence. If  admits a density with respect to the Haar measure, then so does n for each positive integer n. O. Johnson and Yu Suhov, in J. Theoret. Probability 13 (2000) 843–857, investigate the exponential convergence of the successive densities to the uniform density in the Kullback–Leibler distance. In connection with Theorem 2.14, let us also mention [13,14,52,114–116,232–235]. Examples 2.2 and 2.3 appeared in [166]. Let us also mention [45, 46, 51] as some of the first papers in the area, and [53, 55] for some related results.

Section 2.4 Theorems 2.22 and 2.23 are due to Budzban and Mukherjea [27]. Theorem 2.25 is given in [150]. Theorem 2.26 appeared in [31]. See also papers by Maksimov in the context of all these results. Corollary 2.6 is a result of Maksimov. Theorem 2.27 appeared in [26]; however, a related and very similar result, with a completely different proof, first appeared in [203]. The papers [72, 113] are also relevant. For different types of results in the algebraic context, we mention [204]. The papers [135–137] are relevant in this section.

Chapter 3

Random Walks on Semigroups

3.1 Introduction The term random walk suggests stochastic motion in space, a succession of random steps combined in some way. In Chap. 3, we interpret the term very narrowly: We require the steps to be independent and to have the same probability distribution. The walk is then a succession of products of those steps. Later on, we apply our results to slightly more general situations, e.g., to cases where the steps depend on each other in a Markovian way. Thus, our study of random walks is synonymous with the study of products of independent identically distributed random elements of a semigroup. We study the most basic notions for these processes which are of course discrete-time Markov chains with the semigroups as state spaces. We deal with, for example, communication relations, irreducibility questions, recurrence vs. transience, periodicity and ergodicity. Generally speaking, these probabilistic notions have an algebraic counterpart, in the sense that the probabilistic properties of a random walk cannot be satisfied unless the semigroup supporting the random walk has a certain algebraic structure. The situation is very similar to that in Chap. 2 where we saw, for example, that only completely simple semigroups with a compact group factor support limit points of a tight convolution sequence of measures (see Theorem 2.7). Let S be a locally compact, second-countable Hausdorff topological semigroup. Let B denote the Borel sets of S , i.e., B is the -algebra generated by the open sets of S . Our basic measurable space ˝ is the set S 1 of sequences !1 ; !2 ; : : : of elements of S , equipped with the -algebra F generated by coordinate mappings ! 7! !i and the Borel sets of S . The sub--algebra generated by the n first coordinate mappings will be denoted by Fn . We refer to the first two sections of Chap. 2 for the basic properties of probability measures on S and the convolution operation. Let 1 ; 2 ; : : : 2 P .S /, the set of probability measures on S . The infinite product 1  2  : : : – the factor indexed by i is i – is well defined on the basic measurable space ˝. The coordinate functions Xi .!/  !i are measurable and the Xi ’s are independent S -valued random variables with the distribution i .

G. H¨ogn¨as and A. Mukherjea, Probability Measures on Semigroups: Convolution Products, Random Walks and Random Matrices, Probability and Its Applications, c Springer Science+Business Media, LLC 2011 DOI 10.1007/978-0-387-77548-7 3, 

171

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3 Random Walks on Semigroups

We almost exclusively take equal i ’s for i  1. The product measure on ˝ described above is then denoted by P . For the sake of simplicity, the subscript is often dropped. The symbol E denotes expectation with respect to the probability measure P. Of paramount importance is the fact that the product of independent S -valued random variables is again a random variable: Its distribution being the convolution of the distributions of the factors. This follows from the joint continuity of the multiplication and the second countability of S : The set f.x; y/jxy 2 Bg (where B 2 B) is a Borel set in S  S and belongs to the product -algebra generated by the rectangles A1  A2 , A1 ; A2 2 B. The formulas Z 1  2 .B/ D

1B .s1 s2 /d1 .s1 /d2 .s2 / Z Z D 2 .s11 B/d1 .s1 / D 1 .Bs21 /d2 .s2 /;

(3.1)

proved in Proposition 2.2 show how to calculate the distribution of the product of two independent factors with distributions 1 and 2 , respectively. Note that it is possible to dispense with the second countability assumption and still obtain a workable definition of random walks on S , see [61]. Definition 3.1. Let X1 ; X2 ; : : : be independent S -valued random variables with L (Xi ) = , i  1, where L y.X ), law of X , denotes the distribution of the random variable X . Let x 2 S be arbitrary. Then the collection of random variables Zn ; n D 0; 1; 2; : : : defined by Z0 D x;

Zn D Zn1 Xn .Zn D Xn Zn1 /; n D 1; 2; : : :

is called the right (left) random walk on S generated by  with starting point x. To distinguish random walks with different starting point, we also write Znx for the random walk started at x. If Y1 ; Y2 ; : : : is another set of random variables of law , independent of the X -variables as well as of each other, then we can define the bilateral walk generated by  as the set Wn ; n D 0; 1; 2; : : : of random variables satisfying W0 D x;

Wn D Xn Wn1 Yn ; n D 1; 2; : : : :

Note that the basic probability space is ˝  ˝ endowed with the -algebra F  F and the measure P  P. We occasionally look at a mixed random walk Mn ; n D 0; 1; 2; : : : where M0 D x and Mn D Mn1 Yn with probability ˛ 2 .0; 1/ and Mn D Xn Mn1 with probability 1  ˛, the choices are independent of each other and of the X and Y variables.

3.1 Introduction

173

Consider the right random walk Znx generated by  with starting point x. Its transition probability function Pr .x; B/; x 2 S; B 2 B, is defined by Pr .x; B/ D PfZ1x 2 Bg: Hence, Pr .x; B/ D .ıx  /.B/ D .x 1 B/;

(3.2)

where ıx is the unit mass at the point x. We see that Pr .x; B/ is a measurable function of x for each fixed B 2 B (see Proposition 2.1). On the other hand, Pr .x; / is a regular probability measure on B. Thus, we can define the transition probability operator of the right random walk Zn by the formula Z Z (3.3) f 7! Pr .x; dy/f .y/ D f .xy/d.y/ D Ef .Z1x /; where x ranges over the semigroup S and f over the bounded B-measurable real functions on S . We use the same symbol Pr for the transition probability operator. Notice that the transition probability function Pr .; B/ is the transition probability operator acting on the characteristic function of B: Pr 1B . Since Z Pr2 1B .x/ D Pr .x; dy/Pr .y; B/ D .ıx    /.B/ D .ıx  2 /.B/ the two-step transition probability PfZ2x 2 Bg is Pr2 1B .x/ D .ıx 2 /.B/. Similarly, the n-step transition probabilities are given by iterating the transition probability operator n times or equivalently convolving the measure  with itself n times. Definitions are analogous the left random walk. The transition probability Pl .x; B/, x 2 S , B 2 B, is, for example, given by Pl .x; B/ D .  ıx /.B/:

(3.4)

The bilateral random walk has transition probability function Pb .x; B/ D .P  P/fW1x 2 Bg D .  ıx  /.B/:

(3.5)

For the mixed random walk, we have simply Pm .x; B/ D .1  ˛/Pr .x; B/ C ˛Pl .x; B/: Remark 3.1. Note that Pb D Pr Pl D Pl Pr . To see this, consider Pb 1B .x/ D .E  E/1B .W1x / D .  /f.s; t/jsxt 2 Bg Z D fsjsy 2 BgPr .x; dy/ Z D .Pl 1B /.y/Pr .x; dy/ D Pr .Pl 1B /.x/:

(3.6)

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3 Random Walks on Semigroups

The random walks just defined are Markov chains on the measurable space (S; B) with the initial point x and the transition probabilities given by (3.2), (3.4), (3.5), and (3.6), respectively. This all-important fact is used extensively throughout the rest of the book. We show this for the right walk. The other three proofs are similar. Let the right random walk Z0x D x; Z1x D xX1 ; Z2x D xX1 X2 ; : : : be generated by . In the sequel, the superscript x is dropped for ease of notation. Following [195], Chap. 1, Sect. 2, and [86], p. 2, we consider the expectation E

n Y

fi .Zi /;

i D1

for any bounded measurable real-valued functions f1 ; f2 ; : : : fn and any nonnegative integer n. We obtain Z

Z :::

f1 .xx1 /    fn1 .xx1    xn1 /fn .xx1    xn /d.x1 /    d.xn /;

which by first integrating out xn , equals Z

Z :::

f1 .xx1 /    fn1 .xx1    xn1 /Pr fn .xx1    xn1 /d.x1 /    d.xn1 /:

This in turn is E

n1 Y

fi .Zi /Pfn .Zn1 /:

i D1

In other words, the conditional expectation of fn .Zn / given the past, Effn .Zn /jZ1 ; : : : Zn1 g, equals the conditional expectation of fn .Zn / given the immediate preceding random variable, or state in Markov chain terminology, Effn .Zn /jZn1 g, which in turn is Pr fn .Zn1 / P-almost surely. Actually, one can show slightly more: Effn .Zn /jFn1 g D Pr fn .Zn1 / P-almost surely. The intuitive meaning of the calculation can perhaps be explained more simply as follows. Let B be a Borel set in S and recall that z1 B denotes the set fyjzy 2 Bg. Then PfZnC1 2 BjZ0 D x; Z1 D z1 ;    Zn D zn g D PfXnC1 2 z1 n BjZ0 D x; Z1 D z1 ;    Zn D zn g D PfXnC1 2 z1 n Bg.D Pr 1B .zn // D PfZnC1 2 BjZn D zn g: Hence, the right random walk is a Markov chain. Thus, we have Theorem 3.1.

3.1 Introduction

175

Theorem 3.1. The right, left, bilateral, and mixed random walks are Markov chains on (S; B). We remark that our random walks are not canonical Markov chains in the sense of [195], p. 18. Also, the -algebras Fn are in general strictly larger than the ones generated by the random variables Z1x ; Z2x ; : : : Znx (where Znx is now any one of the random walks defined above). This is due to the fact that in a semigroup, knowledge of the first factor z1 D x1 and the product z2 D x1 x2 does not imply that the second factor x2 is known as well. In a group, we have of course z1 1 z2 D x2 . Consider one of the random walks; call its transition probability kernel P . Given x and P there is a unique probability measure P0 x on ˝ D S 1 equipped with the -algebra F 0  the smallest -algebra making the coordinate maps measurable, with the property that P0 x f! 2 ˝j!i 2 Ai ; i D 1; 2 : : : ng D Z

Z D

Z

P .x; dx1 / A1

P .x1 ; dx2 / : : : A2

P .xn2 ; dxn1 /P .xn1 ; An /; An1

where n is any positive integer and the sets Ai are arbitrary Borel sets of S , (see [195], p. 20, and [86], p. 8). The random variables (coordinate maps) Z00 .!/ D x; Zn0 .!/ D !n ; n > 0; constitute a Markov chain, the canonical Markov chain on S with the initial point x and the transition probability kernel P . The two different approaches to random walks lead to the same thing in the following sense: P0 x fZi0 2 Ai ; i D 1; 2; : : : ng D PfZix 2 Ai ; i D 1; 2; : : : ng:

(3.7)

In other words, all probabilistic statements about the Z 0 -chain can be translated into a corresponding one for the Z x -chain and vice versa, (see [195], pp. 20 and 31). Briefly stated P0 x .A/ D Pf.x; Z1x ; Z2x ; : : :/ 2 Ag for any A 2 F 0 . Both approaches have their advantages (and drawbacks) so we will not hesitate to choose the most convenient one for the particular problem we are trying to solve. Definition 3.2. Let Z 0 be a canonical random walk with transition probability kernel P and starting point x. Define the shift or translation operator  on ˝ by ..!//n D !nC1 ; n D 0; 1; 2; : : :

0 or Zn0 ı  D ZnC1 ; n D 0; 1; 2; : : : (3.8)

The iterates of  are denoted by 2 ; 3 ; : : : A stopping time of the canonical random walk Z 0 is a random variable (defined on (˝; F 0 ) with values in N (the nonnegative integers with C1 added) such that for every n 2 N the set fT  ng 2 Fn0 , where Fn0 is the -algebra generated by the first n coordinate maps Z10 ; Z20 ; : : : Zn0 . The - algebra FT0 of events A 2 F 0 such that A \ fT  ng 2 Fn0 for all n 2 N is called the -algebra associated with T . (Intuitively, these are the events occurring no later than T .)

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3 Random Walks on Semigroups

ZT0 is defined to be Zn0 on the set fT D ng, n D 0; 1; 2; : : : Similarly, T is defined to be n on fT D ng. For a Borel set B 2 B, we define the first hitting time TB of B and the first return time SB of B by TB D inffn  0jZn0 2 Bg and SB D inffn > 0jZn0 2 Bg; respectively, where the infimum of the empty set is taken to be C1. Proposition 3.1. The canonical random walk satisfies the relation (the strong Markov property) E0 x fX ı T jFT0 g D E0 ZT0 .X / on

fT < 1g P0 x  almost surely 8x

for any random variable X defined on (˝; F 0 ). Let Tn be the successive return times to set B 2 B. Then the sequence Yn  ZT0 n is a Markov chain with respect to the -algebras FT0 n . For a proof and a detailed discussion of these issues, see [195], Chap. 1, Sect. 3. In view of the almost equivalence, see (3.7), of the two definitions of our random walks, we henceforth drop the primes when using the canonical random walks. The transition probability operators P just defined are Feller (in the sense of [195]) or weakly continuous, [224], i.e., Pf is bounded and continuous whenever f is. This fact was used early in Chap. 2; it is an important link in the argument leading to the definition of the convolution operation (see the proof of Proposition 2.1). If S is a group and  is absolutely continuous with respect to Haar measure, then Pf is continuous for all bounded, measurable f . This will be shown later. For the discussion of these and related questions, we need some additional concepts. Definition 3.3. The transition probability operator P is said to be strongly continuous if Pf is bounded and continuous for all bounded measurable real-valued functions f . (Equivalently, we could require P1B to be continuous for all B 2 B.) Let T .x; B/ be a nonnegative kernel on S  B, i.e., T .; B/ is a measurable function on S and T .x; / is a nonnegative measure, not necessarily a probability measure, on B. T is called a component of P if T .x; B/  P .x; B/ for all x 2 S and all B 2 B. The component T is said to be nontrivial at x if T .x; S / > 0 and continuous at x if T .x; B/ is lower semi-continuous at x for all B. T is a continuous component of P if it is continuous at all x 2 S . If the measures T .x; / and P .x; / are equivalent, then we say that T is equivalent to P at x. If this is the case for all x 2 S , then T is an equivalent component of P . The Markov chain Zn with transition probability kernel P isPsaid to have T as n n its continuous if T is a continuous component of 1 P , i.e., nD1 2 P1component n n T .x; B/  nD1 2 P .x; B/; x 2 S; B 2 B.

3.1 Introduction

177

Let S be the support of the measure . If the starting point lies in S as well (or more generally, the initial distribution has its support inside S ), then the right (or left) random walk will P -almost surely never leaves the subsemigroup ! ! 1 1 [ [ n S D closure S n ; closure nD1

nD1

(see formula (2.8)). Another common assumption is that the random walk starts at the identity element of S (adjoined if need be). In that case too, the effective state space of the walk is the smallest closed subsemigroup containing the supports of all the convolution powers n ; n D 1; 2; : : : : We will henceforth make the following blanket assumption ! 1 [ n S D closure (3.9) S : nD1

In other words, we will restrict our attention to the closed subsemigroup generated by the support of . This set can also be described, e.g., as the support of the probability measure 1 X 2n n : nD1

In other words, any open set in S has positive n -measure for some n. Note that by restricting the topological space to a closed subspace as above we preserve all the basic properties, i.e., S remains a locally compact second countable topological semigroup. The closure of a set B is also written B. Before beginning the detailed analysis of our random walks, let us see how the elementary communication relations from the theory of Markov chains can be formulated algebraically. We use the generic Sn for any of the four random walks introduced in Definition 3.1 and P for any of the corresponding transition probability functions or transition probability operators, as the case may be. Definition 3.4. Let P be the Feller transition probability operator of a Markov chain on S . In particular, we take P to be the transition probability of one of the walks of Definition 3.1. For x; y 2 S , we say that x leads to y, x ! y if for all neighborhoods N.y/ of y, P n .x; N.y// > 0 for some n  1. If x ! y and y ! x, then x and y are said to communicate (notation: x $ y). x is a return point if x ! x. The return point x is said to be an essential element if x ! y implies y ! x. x is said to lead to y infinitely often, (x ! y infinitely often,) if Px fSn 2 N.y/ infinitely ofteng D 1 for any neighborhood N.y/ of y. If x ! x infinitely often, then x is said to be recurrent for the walk Sn . The element x is conservative (positive) if for every neighborhood N.x/ of x 1 X

P n .x; N.x// D 1

nD1

(lim supn!1 P n .x; N.x// > 0).

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3 Random Walks on Semigroups

If the point x is not recurrent for a random walk, then it is said to be transient. If a random walk admits a recurrent point then it is said to be a recurrent random walk. If the recurrent random walk admits an invariant probability distribution , i.e., Z .A/ D .dx/P .x; A/; A 2 B and .S / D 1; then it is said to be positive recurrent. Otherwise, it is called null recurrent. If a random walk admits no recurrent points, it is called transient. Proposition 3.2. (i) If S satisfies our basic assumption (3.9), the following holds for the random walks on S : (a) For the right walk on S , x ! y if and only if y 2 xS , (b) For the left walk, x ! y if and only if y 2 S x, n n (c) For the bilateral walk, x ! y if and only if y 2 [1 nD1 S xS , (d) For the mixed walk, x ! y if and only if y 2 S x [ xS [ S xS . (ii) The relation ! is transitive. The relation $ is an equivalence relation on the set of return points. Proof. We first recall that if O is an open set then the sets x 1 O and Ox 1 are open in S and the set f.s; t/jsxt 2 Og is open in S  S (see Proposition 1.16). The sets xS; xS ; Sn are countable unions of compact sets (and thus Borel sets) because S is, by assumption, locally compact and second countable. (i) We begin by proving the Claim (ia). Clearly, Px fZn … xS g D 0 for all n. So any z … xS will have a neighborhood N.z/ such that Px fZn 2 N.z/g D 0 for all n. This proves that x ! y implies y 2 xS . To prove the converse, let y 2 xS and let N.y/ be any neighborhood of y. N.y/ \ xS is nonempty; thus, we conclude that x 1 N.y/ is a nonempty open set. By our basic assumption in (3.9), Prn .x; N.y// D n .x 1 N.y// > 0 for some n. This holds for all neighborhoods N.y/ of y, hence x ! y. The proof of Claim (ib) is similar and therefore omitted. In Claim (ic), we note that Pbn .x; N.y// D .n  n /f.s; t/jsxt 2 N.y/g > 0 for a neighborhood N.y/ of y if and only if N.y/ \ Sn xSn ¤ ;: Reasoning as above, we conclude that x ! y if and only if y belongs to the closure of the union of the sets Sn xSn . The basis of the proof of Claim (id) is the fact Pmn .x; N.y// > 0 if and only if N.y/ has nonempty intersection with any of the sets xSn ; Sn x; Sk xSnk ; k D 1; 2; : : : n  1: (ii) The transitivity of ! is a consequence of (i) and the continuity of the multiplication. For the bilateral walk, the argument runs as follows. If x ! y and y ! z, then y is the limit point of a sequence sn xtn and z the limit point of a 0 0 0 0 sequence sm ytm (where sn ; tn 2 Sn ; sm ; tm 2 Sm ). Hence, z is a limit point of 0 0 nCm nCm t u sm sn xtn tm 2 S xS ; n; m ! 1. The rest of (ii) follows easily.

3.1 Introduction

179

n n Remark 3.2. The set [1 nD1 S xS is not necessarily a subsemigroup of S , while the sets mentioned in the statements (ia), (ib), and (id) are.

The behavior of Markov chains on general nondiscrete topological spaces is markedly different from the discrete case, see, for example, [200]. As for our random walks on semigroups, we will find in the subsequent sections that only careful analysis reveals to what extent the notions originating from discrete Markov chain theory are transportable to the topological semigroup setting. Example 3.1. Let S be the closed semigroup 0 [ f2n jn D : : : ; 2; 1; 0; 1; 2; : : :g generated by the probability measure  on the multiplicative semigroup Œ0; 1 defined by f2g D f 12 g D 12 . There is only one essential element, 0, for the random walk with transition probability .x 1 B/. 0 is, of course, also recurrent. The nonzero elements form a communicating class of nonessential recurrent elements. This is because the random walk on S n f0g is the isomorphic image of a symmetric Bernoulli random walk on the integers, which is known to be recurrent. We now proceed to a more detailed study of various special types of semigroups. We try to determine the communication structure and characterize the recurrent elements. As we have seen throughout the book, a certain probabilistic behavior may only be possible in a semigroup with a very particular structure. As far as possible, we point out the differences and similarities with classical random walk theory on 0g. Since S is assumed to be generated by S , see (3.9), we conclude that the semigroup S under study is a countable set. The random walks on S are thus particular Markov chains on the countable set S and we have at our disposal the notions and the results of Markov chain theory. Basic references are, for example, the first chapters of [37, 118, 199]. General definitions from the previous section are considerably simplified since the subsets of S are both open and closed. The neighborhood N.y/ of y is interpreted as fyg and xS as xS , etc. The notation x 1 y is used to denote the set fsjxs D yg  x 1 fyg. Accordingly, the transition probability function for the right random walk will be written Pr .x; y/ D .x 1 y/ instead of Pr .x; fyg/. Similar changes in notation will be made for the other random walks. We emphasize at this point that these notational simplifications will be used in Sect. 3.2 only. Proposition 3.3. Let S be a countable semigroup generated by the support of the probability measure . Then the following assertions are true (i) x is essential for the right random walk if and only if x belongs to a minimal right ideal. (ii) x is essential for the left random walk if and only if x belongs to a minimal left ideal. (iii) x is essential for the bilateral walk if and only if x belongs to a minimal twosided ideal. (iv) x is essential for the mixed walk if and only if x belongs to a minimal two-sided ideal. Furthermore, all essential elements communicate. Proof. Assertion (i): By Proposition 3.2 x ! y if and only if y 2 xS . If x is essential and x ! y, then x 2 yS and xS D yS , i.e., the ideal generated by x is minimal. On the other hand, if the ideal fxg[xS is minimal, then x 2 xS (implying that x is a return point) and yS D xS 3 x for all y 2 xS (implying that y ! x for all y with x ! y). We omit the proof of Assertion (ii). To see that Assertion (iii) holds, consider a bilateral walk on S and let x be an essential element. If x ! y, then y ! x, i.e., n n x 2 [1 nD1 S yS , so x and y generate the same bilateral ideal of S : fxg [ xS [ S x [ S xS D fyg [ yS [ Sy [ SyS . For u 2 Sk , the element xu is essential, too. To prove this assertion, let xu ! y. Then y D sn xutn for some appropriate sn ; tn 2 Sn . x ! uy since uy 2 SnCk xSnCk . x is essential by assumption so uy ! x, i.e., x D sm uytm for some appropriate sm ; tm . But then xu D .sm u/y.tm u/ 2 SmCk ySmCk implying y ! xu. Similarly, ux and uxu0 are essential for any u; u0 2 S . Hence, the set of essential elements form a two-sided ideal of S .

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181

Let x; x 0 be two essential elements. Then x 0 ! xx 0 x and xx 0 x ! x 0 which implies that x 0 2 S xS . Hence, all essential elements lie in the ideal generated by x. Consequently, the set C of essential elements is contained in the ideal S xS generated by x. On the other hand, we just saw that C is an ideal so S xS must equal C . Since SyS D C independently of the choice of y 2 S xS , it must be a minimal ideal. Conversely, let x 2 S generate a minimal ideal I D S xS D I xI D IyI (where y in any element of I ). Suppose x ! y. We have to show that y ! x. By assumption x 2 SyS , i.e., x D sm ytn for some sm 2 Sm \I and tn 2 Sn \I . If m D n, then y ! x and the proof is finished. If not, let h D n  m. For definiteness, take h > 0. In the following calculations, we use the convention that a subscripted element si belongs to the set Si . For some k, y D ak xbk since x ! y. Then x D sm ak xbk tn and x D .sm ak /˛ x.bk tn /˛ for all ˛ D 1; 2; : : : : Furthermore, there are i and j such that ˇ ˇ sm D ci sm dj whence sm D ci sm dj for ˇ D 1; 2; : : : : Let n > ˛1 > 0 and ˛1 C ˛2 D ˛. Then x D .ci sm dj ak /˛1 .sm ak /˛2 x.bk tn /˛ D .ci sm dj ak /˛1 .sm ak /˛2 1 sm ytn .bk tn /˛1 ˇ

ˇ

ˇ

ˇ

If we now can arrange ˛1 ; ˛2 ; ˇ to satisfy ˛1 Œˇ.i C j / C m C k C .˛2  1/.m C k/ C m D n C .˛  1/.k C n/; or, equivalently, then x 2

S

˛1 Œˇ.i C j / D ˛h; n n n1 S yS ,

i.e., y ! x. To satisfy (3.10) we can choose, for example,

ˇ D 1; ˛1 D h; ˛ D i C j and ˇ>

(3.10)

if h  i C j

h ; ˛1 D h; ˛ D ˇ.i C j / i Cj

if h > i C j:

This concludes the proof of (iii). Assertion (iv): Let x be essential for the mixed random walk and let x ! y. Then y 2 xS [ S x [ S xS is a return point and x 2 yS [ Sy [ SyS , i.e., x lies in the bilateral ideal generated by y. Thus, the ideal generated by x must be minimal. In particular S x; xS S xS , yS; Sy SyS and S xS D SyS . Conversely, if x belongs to a minimal bilateral ideal then fxg[S x[xS [S xS D S xS and for any y 2 S xS we have SyS D S xS . Since x 2 S xS this means that x ! y implies y ! x. The above argument also shows that the essential elements (D the elements of the minimal bilateral ideal) communicate. t u Remark 3.3. Propositions 3.2 and 3.3 show that the communicating essential elements form equivalence classes. These are called essential classes. Inessential classes are made up of communicating elements that are not essential.

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3 Random Walks on Semigroups

Corollary 3.1. (i) Let x be essential for one of the unilateral walks. Then x is essential for the bilateral and mixed walks as well. (ii) x is essential for both the right and the left random walk if and only if x belongs to the completely simple minimal two-sided ideal K of S . Proof. (i) Let x be essential for the left random walk. Let I be the set of all essential elements for the left random walk. I is the union of the minimal left ideals of S . To show that I is a minimal two-sided ideal, let us consider the elements si and i t (i 2 I; s; t 2 S .) Clearly, si 2 I , since S i D S si is a minimal left ideal. Also, i s 2 I , since i s is essential for the left walk whenever i is. (i s ! t ” t D ui s for some u 2 S . But i 2 S ui so i s D .vui /s D vt for some v. In other words, t ! i s.) Thus, S iS D I for all i 2 I . (S iS ¤ I is impossible because if j 2 I then j 2 S ij and, further, S ij S iS .) Hence, I is a minimal two-sided ideal. The claim then follows from Assertion (iii) of Proposition 3.3. (ii) Let x be essential for both the right and the left random walks. Then x belongs to a minimal right ideal and a minimal left ideal of S . Let K be the union of the minimal left ideals of S . By (i), K is a minimal two-sided ideal and hence a simple subsemigroup of S . Proposition 1.6(ii) shows that K is completely simple since it contains at least one minimal right ideal. Conversely, let K be the completely simple minimal ideal of S . Then K is a union of minimal left (right) ideals, so any x 2 K is essential for the left and right random walks. t u Example 3.2. (i) Let S be QC  QC endowed with the multiplication .a; b/.a0 ; b 0 / D .aa0 ; ba0 C b 0 /; where QC is the set of positive rational numbers. Then S does not admit any minimal left or right ideals. Nevertheless, S is simple since S xS D S for any x 2 S . Example 3.2 was studied in Sect. 1.4 in a slightly different form. (ii) Let S be the set of strictly increasing polygonal lines of the form ( a.t/ D

sm1 C

t rm1 .s rm rm1 m

sn C t  rn ;

 sm1 /;

if rm1  t  rm ; m D 1; 2; : : : n; if t > rn ;

where 0 D r0 < r1 < : : : < rn and 0 < s0 < s1 : : : < sn are rational and n D 0; 1; 2; : : :. The multiplication in S is ordinary composition of functions. S is left simple since the equation xb D c is solvable for any b; c 2 S . One solution is ( (¸b 1 .t//; if b.0/  t; x.t/ D c.0/ t 2 .1 C b.0/ /; if 0  t  b.0/: S is not completely simple, however, since it contains no idempotent. To see this, suppose that there is an e 2 S with ee D e. In particular,

3.2 Discrete Semigroups

183

e.e.0// D e.0/ D s0 > 0. But this is impossible since the elements of S are strictly increasing functions. Thus, S cannot contain any minimal right ideals. Definition 3.5. Let P be the transition matrix of one of the random walks defined in Sect. 3.1. The period of a return point x 2 S (in the random walk in question) is the greatest common divisor (gcd) of the set fn  1jP n .x; x/ > 0g: The definition of the period is the usual one in Markov chain theory. It also follows from the general theory that communicating elements have the same period. Proposition 3.4. Essential classes for the right/left/bilateral/mixed random walk have the same period pr ; pl ; pb ; and pm , respectively. If both the right and the left walk admit essential elements x, then pr D pl D gcdfn  mjn .x/; m .x/ > 0g; for any essential x. Proof. Let x and y be essential with respect to the right random walk. Call their periods pr .x/ and pr .y/, respectively. If Prn .x; x/ is strictly positive, then so is Prn .yx; yx/. This implies that pr .y/ (which is equal to pr .yx/) is a divisor of pr .x/. Similarly, pr .x/ can be shown to be a divisor of pr .y/. Hence, pr .x/ D pr .y/. The case of the left random walk is analogous. Consider the bilateral random walk on S . Let I be the set of essential elements and assume that it contains more than one communicating class. Let x and y be 0 00 noncommunicating elements of I . We have y D sxt for some s 2 Sn and t 2 Sn where n0 ¤ n00 . Let z belong to the same communicating class as x. Then z D s1 sxts2 ; for some s1 2 Sn1 and s2 2 Sn2 with n1 C n0 D n2 C n00 D n. Furthermore, since z ! x in the bilateral walk, we have x D uzv D u.s1 sxts2 /v D us1 ys2 v; where u; v 2 Sm for some m. Then y D .sus1 /y.s2 vt/, i.e., y 2 SnCm ySnCm . If x D axb with a; b 2 Sk , we can write y in the form .saus1 /y.s2 vbt/ which means that y 2 SkCnCm ySkCnCm . We conclude that if Pbk .x; x/ > 0 then PbkCmCn .y; y/ > 0. Hence, the period of y divides that of x. By interchanging x and y, we obtain the result pb .x/ D pb .y/. As for the mixed random walk, we note that it admits at most one essential class. The period is, then, the same for all the essential elements.

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3 Random Walks on Semigroups

If both unilateral random walks admit essential elements, then these elements form the completely simple minimal ideal K of S . Let x belong to K and assume that k .x/ > 0. Denote the positive integer gcdfn  mjn .x/; m .x/ > 0g; by q. If Prn .x; x/ > 0 (Pln .x; x/ > 0), then clearly kCn .x/ > 0. Hence, q is a divisor of pr .x/ (pl .x/). Since x 2 K there is a y 2 K with xyx D x. k .y/ > 0 for some k. Then n  .x/; m .x/ > 0 implies that PrkCn .x; x/; PrkCm .x; x/ > 0. pr being a divisor of kCn and kCm it is also a divisor of nm. Hence, pr divides q, which shows that pr , in fact, equals q. Since pr is constant on K we conclude that q is independent of the choice of x 2 K. A similar argument shows that pl D q. t u We saw in Corollary 3.1(i) that the existence of essential elements for both unilateral walks furnishes us with a completely simple minimal ideal K. The very detailed knowledge that we have about the algebraic structure of K (see Sect. 1.4) is immensely helpful as shown in Theorem 3.2. Theorem 3.2. Let K be the completely simple minimal ideal of S . Suppose that the unilateral walks are periodic with period p. Define the sets Ck  fx 2 Kjm .x/ > 0 ) m  k .mod p/g for k D 0; 1; : : : p  1. Then (i) The sets xCk .Ck x/ are the cyclic classes for the right (left) random walk starting at x 2 K. They satisfy Ck Cj D CkCj .mod p/ for all j and k. (ii) If K has the Rees–Suschkewitsch representation E  G  F then C0 D E  H  F where H is a normal subgroup of the group G and Ck D E  Hk  F for k D 0; 1; : : : p  1 (where H0 D H ). The sets Hk are cosets of H and they satisfy Hk Hj D HkCj .mod p/ . Hk may also be written g k H where g is any element of H1 . (iii) The bilateral walk has period p2 if p is even and p if p is odd. The number of essential classes with respect to the bilateral walk is 2 if p is even and 1 if p is odd. Recall that a set of communicating states (of a periodic Markov chain with transition probability matrix P and period p) is partitioned into cyclic classes by the equivalence relation x y , P mp .x; y/ > 0 for some positive integer m. We also know that P mp .x; y/ is in fact strictly positive for any integer m large enough, larger than some mx;y , say. Proof. The completely simple minimal ideal K is exactly the set of essential states (for all walks). The Proposition 3.4 implies that the sets Ck form a partition of K. Clearly, Ck Cj CkCj .mod p/ . Any idempotent e of K must belong to C0 , since m .e/ > 0 implies 2m .e/ > 0. For any element z of CkCj .mod p/ , there are e idempotent, x 2 Ck and y 2 K such that exy D z. y must belong to Cj . Hence C0 Ck Cj D Ck Cj D CkCj .mod p/ .

3.2 Discrete Semigroups

185

x 2 K can be written x D xe for some idempotent e; hence, x 2 xC0 . If y D xs D xt, then s and t belong to the same Ck since fn  mjn .y/; m .y/ > 0g; must consist of multiples of p. Then, xS D xK D

p1 [

xCk

kD0

(disjoint union). If y 2 xCk and y z (for the right random walk starting from x), then y D xs; z D xst for some s; t 2 K; s 2 Ck . But we have necessarily t 2 C0 ; hence, z 2 xCk . This completes the proof of (i) for the right random walk case. The proof for the left random walk is similar. Random walks generated by p are aperiodic and admit essential elements forming a completely simple subsemigroup K 0 of the completely simple minimal ideal K of our original semigroup S . Actually, K 0 D C0 . Let the Rees–Suschkewitsch representation of K be E  G  F and that of C0 be E 0  H  F 0 . Since all idempotents of K belong to C0 the side factors must coincide: E D E 0 ; F D F 0 . We must also have .F; E/ H , where  is the sandwich function (see Sect. 1.4). The relation Cj Ck D CkCj shows that Ck is of the form EHk F , using here the product form of the representation. Hj FEHk D Hj Ck . Recalling the fact that FE H we see that HHk D Hk H D Hk and Hk Hk ; Hk Hk H . The last inclusion is, in fact, an equality because the Hk ’s form a partition of G. Similarly, one can then show that cH D Hc D Hk if c 2 Hk , so Hk is a left and right coset of H . (Clearly cH Hk . But any d 2 Hk may be written cc 1 d . There is no other alternative for c 1 d than to belong to H .) This is valid for any k, which implies that H is a normal subgroup of GD

p1 [

Hk :

kD0

Writing H1 D gH for a g 2 H1 one sees that Hk D g k H and g p H D H . To prove part (iii), notice that the idempotent e 2 K may be reached from itself in a multiple of p2 bilateral steps if p is even. On the other hand, only the elements of p 2 1

E

[

H2k  F

kD0

are reached from e. Similarly, the elements of E  Hk  F for odd k communicate (in the bilateral walk) only with eachS other. If p is odd the disjoint union E  p1 H2k  F equals all of K. Hence, e comkD0 municates in the bilateral walk with all elements of K. The p cyclic classes are of the form E  H2k  F . t u

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3 Random Walks on Semigroups

The concept of recurrence plays a central role in random walk theory. There are several ways in which one can express the phenomenon and we shall shortly see that, fortunately, a few other possible definitions turn out to be equivalent to our Definition 3.4: A point x 2 S is recurrent with respect to a random walk Sn if x ! x infinitely often, viz., Px fSn D x infinitely ofteng D 1. In a more general topological setting, this equivalence between the different definitions may no longer hold, however, as will be shown in later sections. Proposition 3.5. Let Wn (Zn ) be the right (left) random walk generated by the probability measure  on the discrete semigroup S . Then the following are equivalent: (a) (a’) (b) (b’) (c) (c’) (d)

x ! x infinitely often for Wn . x ! x infinitely often for Zn . PfWn D x infinitely ofteng > 0. PfZ P1n D nx infinitely ofteng > 0. Pr .x; x/ D 1. PnD1 1 Pln .x; x/ D 1. PnD1 1 n nD1  .x/ D 1.

Furthermore, any of the above conditions imply that the unilateral, bilateral, and mixed walks all admit essential elements. The essential elements form the completely simple minimal ideal of S . We remark that x satisfying one of the conditions (b) is said to be unconditionally recurrent for the random walk in question. Proof. To fix the notation, let Wn D X1    Xn1 Xn be the right random walk generated by . Condition (d) being symmetric, we need only show the equivalence of Conditions (a), (b), and (c). Conditions (a) and (c) are equivalent. This is a standard fact in Markov chain theory, see [118], p. 66. Condition (a) implies Condition (b) since PfWn D x infinitely ofteng is no less than PfWk D x; xXkC1    Xn D x infinitely ofteng. This last probability is simply k .x/ which, by our blanket irreducibility assumption (3.9), is positive for some k. PfWn D x infinitely often g D PfWk D x for some positive integer k; xXkC1    Xn D x infinitely ofteng D PfWk D x for some positive integer kgPx fWn D x infinitely ofteng. According to Markov chain theory, Px fWn D x infinitely ofteng is either 0 or 1, see [118], p. 72. If Condition (b) holds so does Condition (a), because then the preceding probability is 1. For any x 2 S , there exists k such that k .x/ > 0. Then n .x/  k .x/Prnk .x; x/;

(3.11)

if k < n. Summation over n shows that Condition (c) implies P Condition (d). n Assume that x is nonconservative for Wn , i.e., the series 1 nD1 Pr .x; x/ converges. We can write (see [199], pp. 8 and 15)

3.2 Discrete Semigroups

187

Prn .y; x/ D

n X

k fy;x Prnk .x; x/

(3.12)

kD0

and

1 X

 Prn .y; x/ D fy;x

nD1

1 X

Prn .x; x/;

(3.13)

nD0

k  (fy;x ) is the probability that the right random walk starting from y 2 where fy;x  S; y ¤ x will hit x at exactly the kth step (will ever visit x); fx;x and Pr0 .x; x/ are defined to be 1. Using this relation, we obtain 1 X

n .x/ D

nD1

1 X X

.y/Prn1 .y; x/

nD1 y2S

D

1 X

Prn .x; x/

nD0



1 X

X

 .y/fy;x

(3.14)

y2S

Prn .x; x/

nD0

This shows that Condition (d) implies Condition (c). x cannot be recurrent unless it is essential, cf. [37], p. 20. Furthermore, it is recurrent, thus essential, for both unilateral walks. Corollary 3.1 shows that the essential elements for all the walks considered are exactly the same, making up the completely simple minimal ideal K of S . t u The existence of essential elements for the bilateral or mixed random walks was not enough to guarantee the essentiality of the unilateral walks (see Example 3.2). Recurrence is a strong enough condition: Existence of recurrent elements for the bilateral or mixed walks is actually equivalent to any of the conditions mentioned in Proposition 3.5. Proposition 3.6. x is recurrent for the bilateral or mixed random walks if and only if it is recurrent for any (thus both) of the unilateral ones. The set of recurrent elements then coincides with the completely simple minimal two-sided ideal K of S . Proof. If x is transient for the right random walk and k .x/ > 0, then X n

k .x/Pbn .x; x/ 

X n

2nCk .x/ < 1;

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by Proposition 3.5, Condition (d). Hence, x is transient for the bilateral walk. Also X X k .x/Pmn .x; x/  nCk .x/ < 1; n

n

so the same conclusion can be drawn for the mixed random walk. Reasoning as in the proof of Proposition 3.5, see formulas 3.13 and 3.14, we can write X 2nC1 .x/ D .y/Pbn .y; x/ y

and 1 X

2nC1 .x/ D

nD1

1 X nD0

Pbn .x; x/

X

 .y/fy;x 

y

X

Pbn .x; x/ < 1

n

 is the probability that the bilateral walk if x is transient for the bilateral walk. (fy;x starting from x will ever reach y ¤ x.) xs 2 K is also transient for the bilateral walk, for any s 2 S . This will be proved below. Then X X 2n .x/.s/  Pbn .xs; xs/ < 1; n

n

which implies that x is transient for the unilateral random walks. Let x 2 K; ˛ .x/ > 0, be transient for the bilateral walk and take s 2 S . Then yxst D x for some t; y with k .t/; kC1 .y/ > 0. This is possible since x 2 K is essential for the bilateral walk. Let u; v be such that n .u/; n .v/ > 0 and uxsv D xs. Hence, uxsv D xs ) yuxsvt D x (3.15) and so ˛ .x/.s/Pbn .xs; xs/k .t/kC1 .y/  ˛ .x/PbnCkC1 .x; x/:

(3.16)

Summing over n proves that xs is transient for the bilateral walk. The same argument goes through for all elements of K D S xS . We note that X nC1 .x/ D .y/Pmn .y; x/ y

so the same reasoning as for the bilateral case can be employed to prove that if x is transient for the mixed random walk, then it is transient for the unilateral ones as well. Suppose that the right random walk admits a recurrent point x. Then all points of the completely simple minimal ideal K are recurrent for the unilateral walks. To see this, consider X X k .s/n .x/  nCk .sx/: n

n

3.2 Discrete Semigroups

189

Points satisfying Condition (d) of Proposition 3.5 form a two-sided ideal of S . On the other hand, we know that points outside the minimal ideal are inessential and hence transient. t u In Definition 3.4, we introduced the concept of a positive point. In the context of a Markov chain P on a countable state space x is said to be positive if lim sup P n .x; x/ > 0: n!1

In other words, x is positive if and only if it is positive recurrent in the usual Markov chain sense. A point which is recurrent but not positive recurrent is said to be null recurrent. Definition 3.6. The type of a point with respect to one of the random walks is the property of being positive recurrent, null recurrent or transient. (These possibilities are exhaustive and mutually exclusive.) In other words, x is positive recurrent iff P lim supn P n .x; x/ > 0, null recurrent iff limn P n .x; x/ D 0 and n P n .x; x/ D 1, transient iff limn P n .x; x/ < 1. A random walk (or the generating probability measure ) is called positive (null) recurrent if there are points x 2 S which are positive (null) recurrent with respect to the random walk in question. (Theorem 3.3 shows that positive and null recurrent points do not coexist.) A semigroup S is called (positive) recurrent if there is at least one (positive) recurrent probability measure  on S (satisfying the irreducibility condition (3.9)). Justification for the above definitions is given in Theorem 3.3. Theorem 3.3. (i) Consider the two unilateral, the bilateral, and the mixed walks on S generated by the probability measure . An element x of S is of the same type for all the four random walks: x is positive recurrent iff lim supn n .x/ >P 0, n n x is null recurrent iff lim  .x/ D 0 and n n  .x/ D 1, P n x is transient iff n  .x/ < 1. (ii) If one element of S is positive (null) recurrent, then all essential elements are positive (null) recurrent. (iii) There are positive recurrent points if and only if the semigroup S contains a completely simple minimal ideal K with a finite group factor in its Rees representation. In that case, the type is independent of the generating measure  as long as the irreducibility condition (3.9) is met. Proof. Methods developed in the proofs of Propositions 3.5 and 3.6 are used fairly directly.

190

3 Random Walks on Semigroups

(i) We first show that x positive recurrent for the right random walk is equivalent to lim sup n .x/ > 0. Equation (3.11) shows that the condition is necessary. To show the sufficiency, we use (3.12) to obtain nC1 .x/ D

X y

.y/

n X kD0

k fy;x Prnk .x; x/ 

N XX y2S 0

k fy;x Prnk .x; x/ C "; (3.17)

kD0

where 0 < "  lim sup n .x/, S 0 is a large finite subset of S with .S 0 / very close to N is some large fixed number making the tail of the convergent series P 1, and k n k fy;x small. Then necessarily lim supn Pr .x; x/ > 0. By symmetry, the same recurrence criterion works for the left random walk as well. As in the proof of Proposition 3.6, we see that x null recurrent for the right random walk implies x null recurrent for the bilateral one. All essential elements are positive recurrent for the bilateral walk if one is. This may be demonstrated exactly as in the proof of the corresponding statement for recurrence (see formulas (3.15) and (3.16)). Using (3.17) here as (3.14) was used in the proof of Proposition 3.6, we conclude that if x is positive recurrent for the right random walk, then it is positive recurrent for the bilateral walk, too. The proof for the mixed random walk is analogous. The mixed walk Pm satisfies (3.11), hence lim sup n .x/ > 0 is a necessary condition for positive recurrence. Pm also satisfies (3.17) (if probabilities are interpreted with respect to the mixed random walk), so the condition is also sufficient. Proposition 3.6 showed that if x is recurrent for one walk, x is recurrent for the other three as well. Thus, we have proved Part (i). (ii) The relation k .s/n .x/  kCn .sx/; can be used to show that the recurrent elements form a two-sided ideal (stated in Proposition 3.6) and the positive recurrent elements form a two-sided ideal. Hence, if one element is positive (null) recurrent, then so are all elements of K, the completely simple minimal ideal of S . Outside the minimal ideal, all the elements are inessential. (iii) We first note that n .I / ! 1; (3.18) if I is any twosided ideal of S . To prove this assertion, consider x 2 I with k .x/ > 0 and a right random walk Wn on S . Write Wnk D X0 .X1    Xk /.XkC1 XkC2    X2k /    .   Xnk /: If Wnk … I , then necessarily none of the n blocks of k factors X each can belong to I . Hence, nk .I c /  .k .I //n  .1  k .x//n :

3.2 Discrete Semigroups

191

Let K be the completely simple minimal ideal of S , K D E  G  F with G finite. There are a positive integer k and a finite set E 0  G  F 0 D E 0 GF 0 such that k .E 0 GF 0 / > 0. (This number can be chosen arbitrarily close to 1, by (3.18).) Our set E 0 GF 0 is the intersection of a union of minimal right ideals with a union of minimal left ideals, so .E 0 GF 0 /S.E 0 GF 0 / D E 0 GF 0 KE 0 GF 0 D E 0 GF 0 : This means that 2kCn .E 0 GF 0 /  2k .E 0 GF 0 / > 0 for all n. Since E 0 GF 0 is finite this proves (by Part (i)) that positive recurrent points exist. By Part (ii) they constitute the whole minimal ideal K. Note that this argument holds true for any probability measure generating a semigroup S with the same completely simple ideal K. Now assume that there are positive recurrent points. These points form the completely simple minimal ideal K D EGF. Put ˛.e/ D lim n .eGF /; e 2 E n

and ˇ.f / D lim.EGf /; f 2 F: n

(3.19)

˛ and ˇ are welldefined as limits of increasing sequences. (The n -measure of any right (left) ideal is an increasing function of n.) Also, they are probability measures on E and F , respectively, because ultimately all the mass in concentrated on the minimal ideal K (see (3.18)). Random walks on S are thus eventually absorbed by the essential classes, with probability 1. Consider the right random walk on S begun at some point x 2 S . Suppose the walk is periodic with period p. There is a probability distribution x  on xK, with support xK such that p 1 X ıx  npCk .y/: x .y/ D lim n p kD0

(In fact ıx   .y/ converges for each k, k D 0; 1; : : : p  1:) This follows from (3.18) using the general theory of positive recurrent Markov chains (see [37], p. 33). Then there is a finite set S 0 with .S 0 / > 1  " and a finite subset K 0 of K such that X nC1 .K 0 / > .x/n .x 1 K 0 / > .1  "/2 ; npCk

x2S 0

for n larger than some n0 . This means that the sequence n ; n D 1; 2; 3; : : : is tight. Theorem 2.7 can then be invoked to complete the proof of Part (iii). t u Actually, the theory in Chap. 2, for example, Theorem 2.7 gives us much more: Corollary 3.2. In the positive recurrent case we have for all y 2 K p1 1 X npCk  .y/ D .y/; lim n p kD0

(3.20)

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3 Random Walks on Semigroups

where  is a probability measure supported by K with  D  D  and .e; g; f / D

˛.e/ˇ.f / ; .e; g; f / 2 E  G  F  K; jGj

(3.21)

with jGj denoting the order of the finite group G and the probability measures ˛ and ˇ are defined by (3.19). Furthermore, lim npCk .y/ D k .y/; n

(3.22)

with k a probability measure on E  g k H  F , g and H as in Theorem 3.2.   k D k   D kC1 and k .e; gh; f / D

.mod p/

˛.e/ˇ.f / ; .e; gh; f / 2 E  gH  F jH j

(3.23)

Remark 3.4. Note that, in our discrete setting, the pointwise convergence of a sequence of probability measures to a probability measure is equivalent to weak convergence. Proof. We know from Chap. 2 that the Ces`aro limit  of n exists. The probability measure  on K D E  G  F is a product measure where the middle factor, G , is the uniform or normed Haar measure on G. Clearly,  must then satisfy (3.21), i.e.,  D ˛  G  ˇ. In the proof of Theorem 3.3, we saw that sequences ıx  npCk ; n D 1; 2; : : : converge for k D 0; 1; 2; : : : p  1. From this, we can deduce that the sequences npCk converge pointwise, thus weakly, for k D 0; 1; : : : p  1. In particular, the Ces`aro limit  satisfies (3.20). The weak limit 0 of np D .p /n ; n D 1; 2; : : : is supported by E  H  F , where H is a normal subgroup of G (see Theorems 2.7, 2.13 and 3.2). 0 must satisfy (3.23). In other words, 0 D ˛  H  ˇ. The structure of the cyclic classes of the right random walk was studied in Theorem 3.2. The support of k D 0  k D 0  k is E  gk H  F , the set reached by the right random walk in np C k steps from some point in E  H  F . Since these supports are disjoint and the average of the ’s is equal to , they must satisfy (3.23) t u The ergodic theory of Markov chains describes the asymptotic behavior of a typical realization of the chain. In particular if there is only one essential class for a positive recurrent chain, then the time averages of a function h or the integrals of h with respect to the empirical distributions converge almost surely to the integral of h with respect to the unique invariant measure  for the chain (see [37], p. 92).

3.2 Discrete Semigroups

193

For our positive recurrent unilateral, bilateral, and mixed random walks on S with completely simple minimal ideal K D EGF, we have by Proposition 3.3 and Theorem 3.2 only one essential class (at least) in the following cases: For the right (left) random walk if jEj D 1 (jF j D 1) or if the starting point is in K, and for the bilateral walk, if p is odd or if the starting point is in K. There is always just one essential class for the mixed random walk. We can then state the following Corollary 3.3 Corollary 3.3. Let  be a positive recurrent probability measure on S . Let the completely simple minimal ideal K of S have the representation EGF. Denote by Sn (Tn ) the right (left) random walk generated by  and by Wn (Mn ) the bilateral (mixed) random walk generated by . If E is a singleton, then n 1X ıSk ! G  ˇ weakly almost surely; or n kD1 Z n 1X h.Sk / ! hd. G  ˇ/ almost surely n

(3.24)

kD1

where h is any bounded real-valued function on S and ıx denotes the point mass at x. Equivalently, ˇ.f / 1 jfk  njSk D .g; f /gj ! almost surely: n jGj

(3.25)

Similarly, if F is a singleton then n 1X ıTk ! ˛  G weakly almost surely: n

(3.26)

kD1

If p is odd it follows that n 1X ıWk ! ˛  G  ˇ weakly almost surely: n

(3.27)

kD1

For every positive recurrent , we have n 1X ıMk ! ˛  G  ˇ weakly almost surely: n

(3.28)

kD1

Theorem 3.3 shows that surprisingly enough, checking positive recurrence for random walks on a discrete semigroup is actually an easy task. It depends on the measure  only through its support S , or more exactly through the semigroup S generated by S . One only needs to verify that S admits a minimal two-sided ideal K D EGF, where the group factor G is finite. Trivially, any random walk on G is positive recurrent.

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3 Random Walks on Semigroups

The natural question is then whether there is a connection between the type of the semigroup S and that of the group factor G of its kernel K D EGF. (The discrete group G is said to be (positive / null) recurrent if there is an irreducible, in the sense of (3.9), (positive / null) recurrent random walk on G. If there is no such walk, G is a transient group.) As we show in Sect. 3.3, there are a number of recurrence criteria for groups that we would like to make use of in the semigroup context. Let us first look at some simple cases. If S is an abelian semigroup or if S is an inverse semigroup, then its kernel, if it exists, must be a group. If K D E  G  F is a direct product, i.e., the sandwich function .f; e/  u, where u is the identity of the group G, then for any right random walk Sn D .En ; Gn ; Fn / on K, Gn D uSn u is a right random walk on G. Hence, any probability measure of the form  D ˛   ˇ on E  G  F has the property that  is recurrent on K if and only if is recurrent on G. If K cannot be written as a direct product, then the projection Gn is not a random walk on the group component G. The relationship between the types of E  G  F and G is still not completely understood. We have, however, many partial results. Theorem 3.4. Suppose the semigroup S is recurrent. Let K D E  G  F be the kernel of S . Then G is a recurrent group. Proof. We first note that the positive recurrent case is covered by Theorem 3.3. Suppose that  generates a null recurrent right random Sn walk on S . Consider a maximal subgroup of the kernel K of S . This group is isomorphic with the middle factor G in the Rees–Suschkewitsch representation of K (see Sect. 1.4). For notational convenience, this group is also called G. We consider Sn starting at the identity u of G. Define the stopping times (or optional random variables) Tk ; k D 1; 2; : : : successively by T1 D minfn > 0jSn 2 Gg

TkC1 D minfn > Tk jSn 2 Gg;

k D 1; 2; : : : (3.29)

(Of course, we define Tk D 1 if the events do not occur.) The times are the successive return times to G and, since the random walk was assumed to be recurrent and G is included in the essential class uS D uK, they are all almost surely finite: Tk is no larger than the kth return time to u, which is almost surely finite, see [37], p. 81. Define the process Yn on G by Yn D STn ; n D 0; 1; 2;

(3.30)

(if Tn is infinite we can define Yn arbitrarily). Define a probability measure

on G by

.g/ D Pu fY1 D gg: (3.31) Evidently

.g/ D

1 X

.k/ G Pu;g

D

 G Pu;g ;

kD1 .k/

using the taboo probabilities of [37], pp. 45–46, i.e., G Pu;g is the probability of the right random walk Sn starting from the state u of entering the state g at the kth step

3.2 Discrete Semigroups

195

 under the restriction that none of the states of G is visited in between, and G Pu;g is the probability of first hitting the set G at the state g. is a probability measure since T1 is almost surely finite and Y1 is thus a well-defined random variable taking values in G. Note that Sn D uX1 X2    Xn is in G if and only if gSn D gX1 X2    Xn is, for any g 2 G. In fact, the event Sn 2 g1 h has the same probability as the event   1 gSn D h, which means that G Pg;h DG Pu;g h/. 1 h D .g The T s are stopping times; Yn is, by Proposition 3.1, a Markov chain on G. It has initial state u and transition probabilities

Q.g; h/ D Pg fY1 D hg D .g 1 h/:

(3.32)

Yn visits the state g 2 G exactly as many times as Sn does. Hence, the Y -chain is recurrent since the S -chain was assumed to be recurrent. But (3.32) shows that Yn is actually a right random walk on G. We have thus shown that G admits a recurrent random walk. In other words, G is a recurrent group. t u For the converse problem, we quote some of the best available results from the work of Larisse in [131]: A measure on a group G is symmetric if .g/ D .g 1 /; g 2 G. Proposition 3.7. Let G be a group and let K D E  G  F with sandwich function . Denote by N the normal subgroup of G generated by .F; E/. Then (i) If N is finite, then K is of the same type as G. (ii) If G is such that any symmetric probability measure on any generator of G is recurrent, then K is recurrent. (iii) If G is the limit of a sequence of finitely generated subgroups Gn with the property that any symmetric probability measure on any generator of Gn is recurrent, then both G and K are recurrent. In particular, if G is abelian then G and K are recurrent if and only if G is of rank at most two. (iv) If G is locally finite, i.e., any finite subset of G generates a finite subgroup of G, then G and K are recurrent. We will close this section with a useful application of the method of proof of Theorem 3.4 to Markov random walks. Proposition 3.8. Let the discrete semigroup S be given and assume that it admits a kernel K with a finite group factor. Let X0 ; X1 ; X2 ; : : : be a stationary Markov chain on S such that its invariant probability distribution is positive for some a 2 K. Define S0 D X0 and Sn D Sn1 Xn ; n D 1; 2; : : : (3.33) Then Sn is positive recurrent. Remark 3.5. Strictly speaking, positive recurrence for Sn is not defined since Sn is not a Markov chain. Here, we will adopt the following provisional definition: s

196

3 Random Walks on Semigroups

is positive recurrent for Sn if, for some a, .a; s/ is positive recurrent for the corresponding Koutsky process .Xn ; Sn /, see Sect. 3.3, i.e., lim sup PfXn D a; Sn D sjX0 D a; S0 D sg > 0:

(3.34)

n

Proof. Let a 2 K be given such that a is positive recurrent for the stationary Markov chain Xn . Consider the random return times to a: T0 D 0 and Tn D minfk > Tn1 jXk D ag; n D 1; 2; : : :

(3.35)

(and Tn D 1 if the defining set is empty). The return times are all almost surely finite. The mean interarrival time  is finite. Let Y0 D a and YnC1 D XTn C1 XTn C2    XTnC1 1  a; n D 0; 1; 2; : : :

(3.36)

To see what this means in practice take n D 1. Y2 is the block of factors in the product Sn occurring between the first occurrence of a, at time T1 , and the second occurrence of a at time T2 . The blocks Y1 ; Y2 ; : : : are independent and identically distributed since Xn is a Markov chain. A detailed description of this block, or excursion, technique is given, for example, in [8] (see also Proposition 3.1 and our treatment of Markov random walk at the end of Sect. 3.3 as well as [195], Chap. 1, Sect. 3). Consider the original process Sn started at a. STn D aY1 Y2    Yn is a random walk on S . Actually, its state space belongs to the set aSa which is a subset of K and also a finite group. Consequently, the random walk is positive recurrent and its state space is a subgroup H of aSa. Let u be the identity of H (and aSa). Then lim sup P.a;b/ fSTn D ug > ı > 0; n

for any b 2 H . Recall that XTn D a. (If the random walk with increments Yn is aperiodic, the limit exists and is strictly positive.) In particular, PfbX1 X2    XN 1 XN D u; XN D ag > ı 0 > 0 for some ı 0  ı and some (large) N D Nb . Let us assume, for simplicity, that the Markov chain Xn is aperiodic. Then PfXn D ag  " > 0Pfor all n large enough and P.a;u/ fXnCNj D a; SnCNj D u some j 2 H g  b 2 H P.a;u/ fXn D a; Sn D b; XnCNj D a; SnCNj D u some j 2 H g  "ı 0 > 0. H is finite so .a; u/ is positive recurrent. t u

Section 3.2 Exercises Exercise 3.8. Let S be a subset of a finite group G. Suppose that S satisfies the irreducibility condition (3.9). Prove that S is a subgroup of G. Exercise 3.9. Show that S in Example 3.2(i) is simple but not completely simple.

3.2 Discrete Semigroups

197

Exercise 3.10. Construct transformation semigroups S where the normal subgroup N of G, see Proposition 3.7, is nontrivial. Exercise 3.11. Consider the four random walks on the bicyclic semigroup C (see Definition 1.11). Determine the essential elements, if any. Is C recurrent for any of the random walks? Exercise 3.12. Show that the multiplicative semigroup of nonzero Gaussian integers fm C ni ¤ 0jm; n integers g is transient. Is the additive group of Gaussian integers recurrent? Exercise 3.13. Consider the additive group of integers Z. Let .1/ D .1/ D 12 : The random walk generated by  is called the symmetric simple random walk on Z. Calculate P n .0; 0/ and prove that the random walk is recurrent. Exercise 3.14. In Z2 , let  be the uniform distribution on the four nearest neighbors of the origin: .˙1; 0/; .0; ˙1/. Prove that the simple symmetric random walk on the additive group Z2 is recurrent by using the criterion in Proposition 3.5(c). Exercise 3.15. In Z3 , let  be the uniform distribution on the six nearest neighbors of the origin: .˙1; 0; 0/; .0; ˙1; 0/; .0; 0; ˙1/. Prove that the simple symmetric random walk on the additive group Z3 is transient by using the criterion in Proposition 3.5(c). Exercise 3.16. Suppose that the Rees product K D E  G  F is direct. Prove that G recurrent implies K recurrent. Exercise 3.17. Assume that xy 1 and y 1 x are finite for all x; y in S . Let  and

be probability measures on S and let   be their convolution. Prove that the convolution operation is continuous in the vague topology. Exercise 3.18. Let 0 < a < 1 and let the probability measure ˇ be given by ˇ D .1  a/

1 X

an1 n :

nD1

Suppose  is recurrent. Is ˇ also recurrent? Exercise 3.19. Let G D f."; n/j" D ˙1; n 2 Zg with the multiplication defined by ."; n/."0 ; n0 / D .""0 ; n C "n0 /: Prove that G is recurrent. (Hint: Use the visits to the subgroup f.1; n/jn 2 Zg as in the proof of Theorem 3.4.)

198

3 Random Walks on Semigroups

3.3 Locally Compact Groups In Sect. 3.3, we answer the basic questions about the asymptotics of the random walks defined in Sect. 3.1 under the assumption that the state space G is a locally compact second-countable group (or a subsemigroup of such a group). The treatment will be self-contained when general results and general criteria for recurrence/transience are discussed. There are many excellent expositions specializing in probabilistic analysis of random walks on groups, so we will treat particular group theoretical results rather summarily. Relevant literature on the subject include the books by Bougerol and Lacroix [17], Heyer [90], Revuz [195], Guivarc’h et al. [86], and the lecture notes by Roynette [202]. Let  be a probability measure on the locally compact second countable group G. The random walks (left, right, bilateral, mixed) on G are defined in Definition 3.1. If Zn D Zn1 Xn ; Z0 D x is the right random walk started at x we denote 1 by Zm Zn (m < n) the random variable XmC1    Xn1 Xn . Note that Zm and 1 1 Zm Zn are independent and distributions w. r. t. Pe of Znm and Zm Zn are the same. The same remarks hold for the left random walk and the random variables 1 Zn Zm  Xn Xn1    XmC1 . The measure  is said to be adapted to G if G itself is the minimal closed subgroup containing the support S of . Equivalently, G is the smallest closed subgroup containing the smallest closed subsemigroup S generated by the support of , 1 [ S Sn nD1

(see the irreducibility condition (3.9)). We shall henceforth assume that the probability measures generating our random walks are adapted to the group in question. In most cases of interest to us S and G actually coincide: Proposition 3.9. If one of the unilateral walks admits essential elements, then S is a group and all elements of S are essential for the right, left, bilateral, and mixed random walks. S then constitutes one essential class w. r. t. the unilateral and mixed walks and at most two essential classes for the bilateral walk. Proof. Suppose that x is essential for the right random walk on S . The sets sS are closed for all s 2 S (Proposition 1.16) so the proof of Proposition 3.3, Assertion (ia) can be applied without change to show that xS is a minimal closed ideal R of S : rS D R for any r 2 R. Clearly, R is a subsemigroup of S . Also, r n 2 rS for n D 1; 2; : : : so r 2 r n S . This implies r n 2 S and r nC1 2 R. Thus, r 1 2 R and R is a group. Since the identity e of G belongs to R we can conclude that R D eS D S . By Ellis’s theorem (Theorem 1.4), S is a topological group. Since S is closed, it must coincide with G. If S is a topological group, then xS D S x D S xS D S so the proof of Proposition 3.3 shows that all elements of S are essential for both the right and the left and the mixed random walks. There is only one essential class, all of S , for these walks.

3.3 Locally Compact Groups

199

To prove the assertions for the bilateral walk, we first notice that 2 generates a subsemigroup [ H  Sn2 n

of G. By considering that tn 2 with lim tn D e implies lim tn2 D ee D e we see that e 2 H . Assume that H ¤ S . For s; t 2 S sH is closed and hence Sn

S D sS D sH [ s.S n H / and S D stS D stH [ st.S n H /: Since stH H this means that stH D H or hH D H for all h 2 S2 . Similarly, hH D H for all h 2 H . As above one concludes that H is a group, a closed subgroup of S D G. Furthermore, H is normal in G. The essential elements for the bilateral walk form an ideal of S . To prove this, assume that x is essential and let xs be any element of xS . Since s2

[

Sn ;

n 0

we can find a sequence sn0 2 Sn converging to s. Let y be such that xs ! y, i.e., yD

[

Sk xsSk :

k

Then sy 2 s

[ k

Sk xsSk

[[ n0

0

SkCn xSkCn

0

k

so x ! sy. Since x is essential, sy ! x or x2

[ k

Sk sySk and xs 2

[[ n0

0

SkCn ySkCn

0

k

implying y ! xs. In other words, xs is essential. Similarly, sx and sxt are essential for all s; t 2 S . e ! h in the bilateral walk if and only if x 2 H . The argument in the preceding paragraph (xs ! y ) x ! sy) shows that e D hh1 ! h implies h ! h1 h D e. Hence, e is essential for the bilateral walk. Since the essential elements form an ideal all elements of the group S D SeS are essential for the bilateral walk. H is, of course, an essential class. If s … H , then still ss 1 ! h and s ! s 1 h for any h 2 H proving that sH D S n H is the other essential class. t u Note: Example 3.2(i) shows that the bilateral or mixed walks may admit essential elements without S being a group. According to Definition 3.4, x is recurrent for the random walk Zn if Px fZn 2 N.x/ infinitely often g D 1 for every neighborhood N.x/ of x.

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3 Random Walks on Semigroups

Lemma 3.1. If the right (left) random walk admits a recurrent element, then S D G. All elements of S are recurrent if one is. Proof. Consider the right random walk on S started at x 2 S . Clearly, xX1 X2    Xn 2 N.x/ if and only if eX1 X2    Xn 2 x 1 N.x/. N.x/ is a compact neighborhood of x if and only if x 1 N.x/ is a compact neighborhood of the identity e. Hence, x ! x infinitely often if and only if e ! e infinitely often. In particular, the existence of recurrent elements implies e 2 S . Actually, s 2 S is equivalent to e ! s. It remains to show that s ! e, i.e., e is essential and S is in fact a group. Suppose s does not lead to e (which is equivalent to saying that s 1 … S ). Then there are compact neighborhoods N.s/ of s, N.e/ of e and V .s 1 / N.s/1 N.e/ of s 1 such that k .N.s// > 0 for some k but n .N.s/1 N.e//  n .V .s 1 // D 0 for all n. Consider the probabilities Pe fZn 2 N.e/ finitely ofteng  Pe feX1 X2    Xn … N.e/ for n > kg  Pe feX1 X2    Xk 2 N.s/; Xk1 Xn … N.s/1 N.e/g  k .N.s//  .1  nk .N.s/1 N.e/// D k .N.s//  1 > 0; contradicting the assumption e ! e infinitely often. Hence, s 1 belongs to S and S coincides with the group G. t u Proposition 3.10. The left and right random walks are recurrent if and only if the return time to any neighborhood of the identity is Pe -almost surely finite. Proof. Let Zn be the right random walk generated by  and beginning at e. Let SN be the return time to the compact neighborhood N of e: SN D minfn  1jZn 2 N g. As usual, SN is defined to be infinite if the defining set is empty. If e is recurrent, then clearly SN is finite with probability one. Conversely, assume that SN is finite almost surely for all neighborhoods N of the identity. Let K and V be symmetric neighborhoods of e with K compact and V open and K V . (A is said to be a symmetric subset of G if A1 D A.) If V c denotes the complement of V , then K V c is a closed set (see Proposition 1.16) not containing e. U  .K V c /c is an open neighborhood of e. 1 Consider the event fZm 2 K; ZmCn … V g fZm 2 K; Zm ZmCn 2 K 1 V c g. 1 Since Zm ZmCn ; n D 1; 2; : : : are independent of Zm and have the same distribution as Zn ; n D 1; 2; : : :, Pe fZm 2 K; ZmCn 2 V c ; n D 1; 2; : : :g 1  Pe fZm 2 KgPe fZm ZmCn 2 U c ; n D 1; 2; : : :g

D m .K/Pe fZn … U; n D 1; 2; : : :g: The last probability is m .K/Pe fSU D 1g, which is 0 by assumption.

3.3 Locally Compact Groups

201

Let Ki ; i D 1; 2; : : : be an increasing sequence of compact neighborhoods of the origin such that [ Ki D V: i

Since 0 D lim Pe fZm 2 Ki ; ZmCn … V g D Pe fZm 2 V; ZmCn … V g i

we conclude that the last exit time T from V is infinite with probability 1: Put T D supfn  0jZn 2 V g and notice that 0  T  1. Pe fT < 1g D D

1 X mD0 1 X

Pe fT D mg Pe fZm 2 V; ZmCn … V; n D 1; 2; : : :g D 0

mD0

This means that Pe fZn 2 V finitely ofteng D Pe fT < 1g D 0 so Zn 2 V infinitely often Pe -almost surely. e is recurrent since for any neighborhood N of e we can find a symmetric open V contained in N . (If N0 is the interior of N , take V D N0 \ N01 .) t u Theorem 3.5. (i) For unilateral walks either all or none of the elements of S are recurrent. (ii) The following conditions are equivalent for a left or right random walk Zn : (a) x is recurrent. (b) Pe fZn 2 N.x/ infinitely ofteng > 0 for any neighborhood N.x/ of x (unconditional recurrence). (c) e is recurrent. (d) Equation (3.37) holds 1 X n .N / D 1 (3.37) nD1

for some compact neighborhood N of the identity element e. (e) Equation (3.37) holds for all neighborhoods N of e. (iii) The left random walk generated by  is recurrent if and only if the right random walk is recurrent. Proof. Part (i) was shown in Lemma 3.1. Part (ii). The equivalence of Conditions (iia) and (iic) was proved in Lemma 3.1. Clearly, Condition (iie) implies Condition (iid).

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3 Random Walks on Semigroups

Notice that the expression in (3.37) can be written as Ee f

1 X

1N .Zn /g D

nD1

1 X

Pe fZn 2 N g D 1:

(3.38)

nD1

The so-called direct half of the Borel–Cantelli lemma (see [19], p. 41) tells us that if the event fZn 2 N infinitely often g has positive probability then the sum in (3.37) diverges. Hence, Condition (iic) implies Condition (iie). Let us assume that (3.37) holds for all compact symmetric neighborhoods K of e. As in the proof of Proposition 3.10 let T be the last exit time from K. We note that 1 Zm 2 K and Zm ZmCn … K 2 implies ZmCn … K (because K 1  K K 2 ). Then 1 X

1 Pe fZm 2 K; Zm ZmCn … K 2 ; n D 1; 2; : : :g

mD0 1 X



Pe fZm 2 K; ZmCn … K; n D 1; 2; : : :g

mD0

D Pe fT < 1g  1:

(3.39)

Further 1 X

1 Pe fZm 2 K; Zm ZmCn … K 2 ; n D 1; 2; : : :g

mD0

D

1 X

Pe fZm 2 Kg  Pe fZn … K 2 ; n D 1; 2; : : :g

mD0

D

1 X

m .K/  Pe fSK 2 D 1g:

(3.40)

mD0

The divergence of the series in (3.37) shows that the return time SK 2 to the compact neighborhood K 2 is Pe -almost surely finite. Again, for any neighborhood N of e we can construct a compact neighborhood K with K 2 N . Hence, SN is almost surely finite for all neighborhoods N of e. By Proposition 3.10, e is recurrent. We have thus shown that Condition (iie) implies Condition P (iic). Assume that there is a compact set K satisfying n n .K/ D 1. By compactness, there is a point x 2 K such that 1 X nD1

n .N.x// D 1

(3.41)

3.3 Locally Compact Groups

203

for all neighborhoods N.x/ of S x. If not, each k 2 K has a neighborhood V .k/ for which the series converges. k2K V .k/ is an open covering of K and contains a finite subcovering V .k1 /; : : : V .km /. Then X n

n .K/ 

m X X

n .Ki / < 1;

i D1 n

a contradiction. We mimic the proof of the implication Condition (iie) ) Condition (iic) with K a compact symmetric neighborhood of e and T the last exit time from xK (which 1 is a compact neighborhood of x). Again, Zm 2 xK; Zm ZmCn … K 2 implies 1 2 ZmCn … xK (because .xK/  xK K ). Then the inequalities corresponding to (3.39) and (3.40) yield 1

1 X

Pe fZm 2 xKgPe fZn … K 2 ; n D 1; 2; : : :g

mD0

and Pe fSK 2 D 1g D 0 and e is recurrent. This proves that Condition (iid) ) Condition (iic). It remains to prove that Condition (iib) is equivalent to the other assertions in Part (ii). Condition (iib) ) Condition (3.41), again by the Borel–Cantelli Lemma. We just saw that Condition (3.41) implies Condition (iic). Conversely, let e be recurrent. Using the method of proof of Proposition 3.10 with xK substituted for K and xV for V , we get 1 Pe fZm 2 xK; ZmCn … xV g  Pe fZm 2 xKgPe fZm ZmCn 2 K 1 V c g:

The last exit time from the set xK is infinite Pe -almost surely and hence Pe fZn 2 xV infinitely ofteng D 1. This completes the proof of Part (ii). Part (iii) follows from (3.37). t u P n Corollary 3.4. The left and right random walks are transient if and only if n  is a Radon measure on S . In particular, any left or right random walk on a compact group is recurrent. P n Proof. If one of the random walks P is nrecurrent, then n  .K/ D 1 for many compact sets K.P In other words, n  is not a Radon measure. Conversely, n n .K/ D 1 for a compact set K implies the existence of a point (3.41), which in turn implies that e is recurrent. P satisfying n  .S / always diverges, so if S is compact then the random walk is necesn sarily recurrent. t u Corollary 3.5. The left and right random walks are recurrent if and only if the hitting time of a relatively compact open subset of S is Pe -almost surely finite.

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3 Random Walks on Semigroups

Proof. This was shown above in the proof of the equivalence of Conditions (iib) and (iic) in Theorem 3.5. t u A topological group G is a recurrent group if there is a probability measure  adapted to G satisfying (3.37). Hence, a group is recurrent if and only if it admits an irreducible (in the sense of (3.9)) recurrent right or left random walk. Remark 3.6. (i) A compact group is recurrent (see Corollary 3.4). An open subgroup H of a recurrent group G is recurrent, since the open neighborhoods of the identity in H are open sets in G as well and thus satisfy (3.37). (ii) 0 , m.A/ > 0 for all x 2 S . Choose n such that n1 i dominates a multiple of m on A. (To find n consider an open covering of A with sets Ox on which nx dominates P some multiple of a right Haar measure.) As in Lemma 3.2, Part (i), the kernel n1 i .x 1 A/ has a continuous component (¤ 0) at the identity e, so n X 1

i .x 1 A/  ı > 0

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3 Random Walks on Semigroups

on some neighborhood N of the identity. This means Px fZi 2 A; some i; 1  i  ng 

ı : n

for x 2 N . Let T1 ; T2 ; : : : be the almost surely finite successive return times to N separated by at least n steps: Ti C1 D inffj > Ti C njZj 2 N g: Then Pe fSA > T2 g  PZT1 fZ1 ; Z2 ; : : : Zn … Ag  1 

ı n

(by the strong Markov property, Proposition 3.1) and Pe fSA > Ti C1 g  .1 

ı i /: n

Hence, Pe fSA < 1g D 1 and the random walk is Harris recurrent.

t u

It is shown in [224] that recurrent and conservative are essentially equivalent properties in the presence of a continuous component. We formulate this important result as Proposition 3.12 Proposition 3.12. Let Zn be a Markov chain on the locally compact space S equipped with its Borel -algebra B. Call its transition probability kernel P . Assume P (i) There is a point x0 in S such that n P n .x; N / > 0 for all x 2 S and all neighborhoods N of x0 and (ii) The Markov chain has a nontrivial continuous component at x0 . Then the chain is (i) Recurrent if and only if x0 is conservative and (ii) Positive recurrent if and only if x0 is positive. For a proof see [224], Theorem 4.1. As an application, we obtain Theorem 3.7 Theorem 3.7. Suppose that  is spread out. Then the bilateral and mixed random walks on S are recurrent if and only if (3.37) holds. In particular, the four random walks are either all recurrent or all transient. Proof. As in the proof of Proposition 3.11, the bilateral and mixed walks have nontrivial continuous components everywhere on S . (Note that the powers Pmk dominate certain multiples of Prk and Plk and that Pbk D Plk Prk ; detailed calculations are omitted.)

3.3 Locally Compact Groups

209

Let us first assume that (3.37) holds. Then S S is the whole group G (Lemma 3.1). e ! x in the bilateral walk if and only if x 2 n Sn2 D H , a closed (and open) normal subgroup of G (see Proposition 3.9 and its proof). As far as the mixed walk is concerned, all elements of G P communicate. Equation (3.37) implies that n 2n .V / diverges for any compact neighborhood P V of P the identity. Assume the contrary. Then n 2nC1 .V / has to diverge and so does n 2nC2 .VK/, where K is a suitable compact subset of S , since the terms are  2nC1 .V /.K/. VK is compact so this means that X

2n .N.x// D 1

n

for all neighborhoods N.x/ of a point x 2 VK. (3.41) is equivalent to (3.37). Replacing  with 2 in (3.37) and (3.41), we conclude that the identity is conservative for the bilateral walk. Recall that Pbn .e; B/ D 2n .B/ and that x is conservative if X Pbn .x; N.x// D 1 n

for all neighborhoods N.x/ of x.) e communicates with all elements of H and the bilateral walk has a nontrivial continuous component at e, so by Proposition 3.12, e is recurrent. The same argument shows that all elements of H are recurrent. To prove that all elements of the essential class H c are recurrent, it suffices to prove that an x 2 Sd , d odd, is conservative. Without loss of generality, we can assume that d has a continuous positive density f at x (see Lemma 3.2). Let N.x/ be any neighborhood of x. (In the calculations that follow, a set with subscript m is an open set with positive m -measure.) Then there are open sets VkCd and Wk with VkCd xxWk N.x/ and kCd .VkCd /; k .Wk / > 0. Actually, there is an open set U.x 2 / 3 x 2 such that VkCd U.x 2 /Wk N.x/. Consider pairs of open sets An ; Bn such that An x 2 Bn U.x 2 /. We have An x 2 Bn U.x 2 / H) VkCd An x 2 Bn Wk N.x/; which permits us to compare Pbn .x 2 ; U.x 2 // with PbnCkC1 .x; N.x// (see the proof of Proposition 3.6). We get the following relation between the densities: kCd .VkCd /k .Wk /

d2d 2 n 2 .x /Pb .x ; U.x 2 //  f PbnCkCd .x; N.x// dm

Summing over n, we conclude that x has to be conservative since x 2 2 H is. We have thus proved that (3.37) implies that all elements of S D G are recurrent for the bilateral walk. Suppose now that some x0 is recurrent for the bilateral walk. Since the random walk has a nontrivial continuous component at x0 , it follows that all elements

210

3 Random Walks on Semigroups

communicating with x0 are recurrent. Since  is spread out the essential class containing x0 has nonempty interior. P Let N be a relatively compact open subset of the essential class of x0 . Then n Pbn .; N / diverges on the open set N . k .N / > 0 P P R for some k so n 2nCk .N /  n N k .y/Pbn .y; N / D 1, i.e., (3.41) holds for some x 2 N . Hence the equivalent statement (3.37) holds as well. Proofs for the mixed walk, which are similar, are omitted here. t u There is a vast literature on random walks on groups. We will briefly mention some highlights on the recurrence/transience question for these walks. In general, our probability measures will not be assumed spread out. If a random walk generated by  is recurrent, the question arises as to how much the generating measure  may be altered. For concrete groups, we have sometimes precise and easily verifiable criteria (as given, e.g., in Remark 3.6) for recurrence. For a measure  on the locally compact group S we define L to be the image of  under the inversion x 7! x 1 . If  coincides with , L it is said to be symmetric. Lemma 3.3. If the probability measure  is recurrent, i.e., satisfies (3.37), so is the symmetric measure 12 . C /. L Let be a symmetric probability measure on S and  any symmetric nonnegative function on S with compact P support and strictly n1 n positive on some open set. Define the probability measures  1

(with 0 2 0

 ıe ) and 1  c , i.e., the measure with density  with respect to normed to be a probability measure. The support of is all of S and that of 1 is a compact set. If is recurrent, and 1 are recurrent as well. For a proof see [4] where it is an important ingredient in the derivation of the main result – Theorem 3.8 Theorem 3.8. A connected Lie group is recurrent if and only if it is of polynomial growth of order at most 2. Theorem 3.8 is historically speaking one of the latest links in a development of recurrence criteria for groups. For abelian groups, the criteria came much earlier. For these groups, a Fourier transform theory is available and we have the following precise characterization, originating with [39] and [134]: Theorem 3.9. Let  be a probability measure on S and let  be its Fourier transform.  is recurrent if and only if Z lim supt !1 <e.1  t/1 D 1 N

for all neighborhoods N of 0 in the dual group of S . (The integration is understood to be with respect to Haar measure on the dual group.) 0

00

Corollary 3.6. 0 for some compact K. Corollary 3.7. If  is spread out, Theorem 3.11 holds for the bilateral and mixed random walks as well.

212

3 Random Walks on Semigroups

Proof. Consider the bilateral walk on S . If S is compact, then the normal subgroup H is compact, too. The Haar measure D H on H is an invariant probability measure for the bilateral walk (which is obviously recurrent). Hence, the walk is positive recurrent on H . Similarly, it is positive recurrent on the compact set H c D gH D Hg, where a translate ıg  H D H  ıg of the Haar measure on H serves as invariant probability measure: ıg  Pb D   ıg    D ıg  : If the bilateral walk is positive recurrent on one essential class, then an argument similar to the one given in the proof of Theorem 3.7 shows that it is positive recurrent on all of S . If the point e is positive recurrent, then it is positive (see Proposition 3.12). It follows that lim sup 2n .K/ > 0 for some compact set K. By Theorem 3.11, S is compact. The proof for the mixed random walk is omitted.

t u

By Corollary 2.3 and (2.54) of Chap. 2, the Ces`aro average n 1X k  n kD1

converges weakly to the Haar measure S on S . In terms of convergence of empirical distributions, we can formulate Theorem 3.12. Its proof is deferred to Sect. 3.4 (see Corollary 3.11). Theorem 3.12. Let  be a probability measure on the compact group S . Denote by Zn (Tn ) the right (left) random walk generated by  and by Wn (Mn ) the bilateral (mixed) random walk generated by . Let H be the compact normal subgroup of S generated by 2 . S and H are the Haar measures on S and H , respectively. Then the random walks are equidistributed on S in the following sense Z n 1X f .Zk / ! f d S almost surely n kD1

for any continuous real-valued function on S . The same limit holds for Tn and Mn . It also holds for Wn in case H and S coincide. If not, (R n 1X if W0 2 H I f d H almost surely f .Wk / ! R n f d.ıg  H / almost surely; otherwi se kD1 where g 2 S n H .

3.3 Locally Compact Groups

213

A Markov random walk was defined for discrete semigroups in Sect. 3.2 in a very intuitive way. The idea is to study the behavior of the product of random elements which are no longer independent but are drawn from a Markov chain. For topological groups and semigroups, we need to define these objects more carefully. Definition 3.9. Let S be a locally compact second countable semigroup and let C be a locally compact second countable Hausdorff space. Let Xn be a Markov chain on C , equipped with its Borel algebra C , with transition kernel P , which we will assume to be Feller. Px will denote the probability on the canonical probability space (path space) C 1 induced by P and the initial condition X0 D x. Let f be a continuous function from C to S . The process Sn 

n Y

f .Xk /; n D 1; 2; 3; : : :

kD1

is called a Markov random walk on S . A noteworthy special case is Sn D

n Y

Xk ;

kD1

that is, C S and f is the identity map. When the factors Xk are independent and identically distributed, Sn is an ordinary random walk on S . Example 3.4.

(i) If Xk is a Markov chain on the real line, then Sn  X1 C X2 C X3 C : : : C Xn ; n D 1; 2; 3; : : :

is a Markov random walk on 0, there exists a finite covering O1 ; O2 ; : : : On of S such that 8f 2 C W sup jf .s/  f .t/j < "; j D 1; 2; : : : n: s;t 2Oj

Let P be a bounded linear operator mapping the family C (S ) of continuous functions on S into itself. (The norm of such a function f is understood to be sups2S jf .s/j.) P is said to be an equicontinuous operator on C (S ) if the family P n f; n D 1; 2; 3; : : : ; is equicontinuous for all f 2 C (S ). Lemma 3.7. Let Pr ; Pl ; Pb ; Pm be the transition probability operators of the right, left, bilateral, and mixed random walks on the compact semigroup S generated by ; i.e., for f 2 C (S ) and x 2 S , Z Pr f .x/ D f .xs/d.s/; Z Pl f .x/ D f .sx/d.s/; Z Z f .sxt/d.s/d.t/; Pb f .x/ D Z Z Pm f .x/ D c f .xs/d.s/ C .1  c/ f .sx/d.s/; 0 < c < 1: (i) Pr ; Pl ; Pb ; Pm are equicontinuous. (ii) Let P be any of Pr ; Pl ; Pb ; Pm . Then the Ces`aro sum n 1X k P n kD1

converges to an operator P on C (S ) in the following sense sup s2S

n 1X k P f .s/  P f .s/ ! 0 n kD1

2

as n ! 1 for all f 2 C (S ). Furthermore, P P D P P D P D P : Proof. (i) Pr and Pl are shown to map continuous functions into continuous function in Sect. 2.2. As pointed out in connection with the definition of Pb , in Sect. 3.1 it

230

3 Random Walks on Semigroups

is the product of Pr and Pl . Hence, Pb f D Pr Pl f is continuous if f is. Similarly, Pm f is continuous, being a convex combination of Pr f and Pl f . Let f be a continuous function from S to < and let s; x; t 2 S . By the joint continuity of the multiplication in S , there are open sets O1 3 s; O2 3 x; O3 3 t such that sup jf .s 0 x 0 t 0 /  f .sxt/j < "; .s 0 ;x 0 ;t 0 /2O

where O D O1  O2  O3 . The sets O D O1  O2  O3 cover the compact set S  S  S and so there is a finite covering O i D O1i  O2i  O3i ; containing the points .s i ; x i ; t i /; i D 1; 2; : : : ; n, such that jf .sxt/  f .s i x i t i /j < " if .s; x; t/ 2 O1i  O2i  O3i : Let U D \x2O i O2i ; 2

i.e., the intersection of those members of the open covering O2i that contain a fixed element x 2 S . Then we have for x 0 2 U jf .sxt/  f .sx 0 t/j  jf .sxt/  f .s i x i t i / C f .s i x i t i /  f .sx 0 t/j and hence jf .sxt/  f .sx 0 t/j < 2" for every s; t 2 S . The set U D Ux is an open set containing x. The family of sets Ux forms an open covering of S . Hence, there is a finite subcovering Ui ; i D 1; 2; : : : ; n, such that sup jf .sxt/  f .sx 0 t/j < 2" 8s; t 2 S: (3.60) x;x 0 2Ui

Consider the transition probability operator Pb for the bilateral random walk. We have Z Z k Pb f .x/ D f .sxt/k .ds/k .dt/ and clearly for all k, jPbk f .x/  Pbk f .x 0 /j < 2" if x; x 0 2 Ui : Hence, Pb is an equicontinuous operator. Proofs for the other transition probability operators are similar, so we omit the details. Assertion (ii), which is valid for all positive equicontinuous operators, is proved in [199], pp. 134–135. t u Definition 3.12. Let Sn be one of the random walks. Call the transition probability operator P . Suppose that Sn is stationary; i.e., the initial distribution  is invariant with respect to P :  D P meaning Z

Z fd D

Z .Pf /d 

Z Z f .s/.ds/ D

or S

.ds/P .s; dt/f .t/ S

S

3.4 Compact Semigroups

231

Sn and P are said to be ergodic if for every bounded B-measurable function f Z n1 1X k P f .x/ D f d  lim n!1 n

  almost surely:

(3.61)

kD0

Equivalently (see the discussion in [199], Sect. 2 of Chap. 4), Sn is ergodic if for every pair of Borel sets A and B n Z 1X .ds/P k .s; B/ D .A/.B/: n!1 n A

lim

(3.62)

kD1

Remark 3.12. Rosenblatt [199] also shows that P is ergodic with respect to the stationary probability measure  if and only if the translation or shift operator  (see Definition 3.2) is ergodic on the space of trajectories ˝ D S 1 with respect to the measure P . Thus, for every bounded P -measurable function F on ˝, we have Z n1 1X F .k .!// D F d P n!1 n lim

P  almost surely:

(3.63)

kD0

Let F be a function depending only on one coordinate: F .!/ D f .!1 / D f .S1 .!//, say. We then get Z n 1X lim f .Sk .!// D f d  n!1 n

P  almost surely

(3.64)

kD1

for every bounded Borel function f on S . Theorem 3.16 characterizes the form of the stationary distributions for the four random walks. Theorem 3.16. (i) Stationary probability distributions for the right random walk are supported on the closed right ideals of K. A closed right ideal of K may be written A  G  Y , where A is a closed subset of X . The stationary probability measure then takes the form ˛0  G  ˇ, where ˛ 0 is a probability measure on X with support A. (ii) Stationary probability distributions for the left random walk are supported on the closed left ideals of K. A closed left ideal of K may be written X  G  B, where B is a closed subset of Y . The stationary probability measure then takes the form ˛  G  ˇ 0 , where ˇ 0 is a probability measure on Y with support B. (iii) For the bilateral walk, stationary probability distributions are convex combinations of ˛  H  ˇ supported on X  H  Y and ˛  gH  ˇ supported on X  gH  Y . Here, X  H  Y is the Rees–Suschkewitsch representation of

232

3 Random Walks on Semigroups

the kernel of the compact semigroup generated by 2 , H is the normed Haar measure on the compact group H and gH D ıg  H denotes its translate by an element g 2 G n H . (iv) The only stationary probability distribution for the mixed random walk is D ˛  G  ˇ. Proof. (i) If x 2 K, then ıx  is an invariant probability distribution for the right random walk. Its support is the minimal right ideal xK. Let  be an invariant distribution for the right random walk on S . Pr D  is equivalent to    D . The assertion then follows from Corollary 2.3. (ii) Is proved in a similar way. (iii) is an invariant probability measure for the bilateral walk since     D . By Theorem 2.7, all limit points of the sequence n have their support inside K. If  is an invariant probability measure, then n    n D  for all n. Passing to convergent subsequences, we then obtain S K. Using the support of the idempotent limit point  (see the proof of Theorem 2.7), we get .X  H1  Y /S .X  H1  Y / D S : Here, H1 is a closed normal subgroup of G, and H1 D G if and only if n converges (to  D ). Since the middle factor of the product representation of is the normalized Haar measure on G, the equation    D  implies that  D , when H1 D G. If  ¤ , by considering the limit point k   D   k , we see that .X  g k H1  Y /S .X  g k H1  Y / D S ; for some group element g … H1 . It also follows that  is a product measure ˛ 

 ˇ where the middle factor has a support which is either the closed group H generated by g 2 H1 or a coset gH of H or their union. H can also be characterized as the group factor of the kernel of the semigroup generated by 2 . H is a normal subgroup of G. We have, then, two cases: the odd case with H D G and the even case with H ¤ G; then G D H [ gH . Consider the limit points of the sequence of measures n  ıx  n for an x 2 K. Such limit points are, for example,   ıx   and .  /n    ıx    .  /n . Since the middle factor of  is a Haar measure on H1 analysis of the above expression reveals that n    ıx    n D   ıx    2n . The Ces`aro limit of this expression is either ˛  H  ˇ or ˛  gH  ˇ depending on whether or not the middle component is supported on H . Then, if  is invariant with respect to the bilateral walk, it is necessarily a convex combination of ˛  H ˇ and ˛  gH ˇ. (iv)  is invariant with respect to a mixed random walk iff  D c  C .1  c/   for some c 2 .0; 1/. Clearly, D ˛  G  ˇ satisfies this criterion. Iterating the preceding criterion, we obtain D

n   X n kD0

k

c k .1  c/nk nk    k ;

where   0 .0  / is interpreted as just .

(3.65)

3.4 Compact Semigroups

233

Let Mn be a mixed random walk beginning at x 2 S . The transition probabilities are given by Pmn .x; / D

n   X n kD0

k

c k .1  c/nk nk  ıx  k ; n D 1; 2; 3; : : : :

(3.66)

Lemma 3.7(ii) tells us that the Ces`aro averages of the terms (3.66), operating on a continuous f , converge to a P f .x/. For given x, f 7! P f .x/ is a continuous linear functional on C (S ). Furthermore, the constant function 1 is mapped onto itself. In terms of measures, this means that the Ces`aro averages of the measures (3.66) converge weakly to a probability measure x , say. We show that the limit is independent of x and invariant with respect to Pm and furthermore that any invariant measure is necessarily equal to . The proof of Lemma 3.5 shows that x is supported on K. Take any compact subset C of K c and approximate its indicator function from above by an f 2 C (S ) that is 0 on K. For any positive ", n fs 2 S jf .s/  "g decreases geometrically fast n since R the set is compact and disjoint from K. Thus, Pc f .x/ < 2" for large n and so f d x D 0 and x .C / D 0. Consequently x .K / D 0. Let f be an arbitrary continuous real-valued function on S and assume that P f is nonconstant on K, with maximum a and minimum b < a. Let s1 .s2 / be such that 2 P f .s1 / D a (P f .s2 / D b). Since P f D P f , we can actually choose s1 and s2 in K: Z Z 2 P f .s/ D P f .s/ D s .dt/.P f /.t/ D s .dt/.P f /.t/; K

where the last expression lies between the extreme values of P f on K. By continuity, there is an open set O 3 s2 such that P f .t/ < b C .a  b/=2 for t 2 O: We know that all elements of K are positive (Theorem 3.15). Indeed the proof of Proposition 3.18(ii) showed that, for x 2 K, the Ces`aro averages of P n .x; N.x// are bounded below by a positive constant ı 4 =2d 2 . We also noted that this inequality holds true for P n .y; N.x// for y in a neighborhood N 0 .x/ of x. In our context, this means that the Ces`aro averages of P n .; O/ are bounded below by a > 0 on a neighborhood N 0 of s2 . The following equation Z P nCk .s1 ; O/ D

P k .s1 ; dy/P n .y; O/

implies that the Ces`aro averages of P nCk .s1 ; O/ are bounded below by P k .s1 ; N 0 /

which is strictly positive for some k. (Recall that K is a communicating class for the mixed walk, see Lemma 3.4(iv).) Hence s1 .O/ > 0.

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3 Random Walks on Semigroups

Then PN PN f .s1 / D

Z s1 .dt/P f .t/ < a D P f .s1 /;

2

contradicting P D P . Thus, there is a common 0 such that Z P f .x/ D

Z f d x D

f d 0

8x 2 S:

0 is invariant: Z Z Z f d 0 D x Pm f D P Pm f .x/ D P f .x/ D f d x D f d 0 : Consider the expression (3.66) operating on a continuous f and integrated with respect to an invariant probability . Then Z lim

Z Z n 1X .dx/P k f .x/ D .dx/x .dy/f .y/ n kD1 Z Z D .dx/0 .dy/f .y/ Z D f d 0 :

On the other hand, (3.65) implies that the above expression equals Hence, the invariant probability measure is unique and equals .

R

f d . t u

Corollary 3.11. (i) The right random walk is ergodic on a minimal right ideal of the semigroup generated by . (ii) The left random walk is ergodic on a minimal left ideal of the semigroup generated by . (iii) The bilateral walk is ergodic on the minimal ideal of the semigroup generated by 2 . (iv) The mixed random walk is ergodic on the minimal ideal of the semigroup generated by . Proof. Note first that all the walks admit a unique invariant probability distribution on the ideal in question: For Pr , it is ıs  D ıx  G  ˇ on the minimal right ideal generated by s D .x; g; y/ 2 K, which is fxg  G  Y . For Pl , it is  ıs D ˛  G  ıy on the minimal left ideal generated by s D .x; g; y/ 2 K, which is X  G  fyg. For Pb , it is  ıs  D ˛  H  ˇ for s D .x; g; y/ in X  H  Y . For Pm , it is D ˛  G  ˇ for s 2 K D X  G  Y . Furthermore, the unique invariant probability measure is the weak Ces`aro limit of P k .s; / in all four cases. This enables us to deduce immediately that Z Z n Z 1X f .s/.ds/P k g.s/ D f d  gd ; n!1 n lim

kD1

(3.67)

3.4 Compact Semigroups

235

where  denotes the invariant probability measure for one of the random walks on the corresponding minimal ideal and f ,g are continuous functions on S . To prove ergodicity we need to verify (3.67) with f and g arbitrary measurable functions instead of continuous functions. It suffices to do it for measurable indicator functions f D 1A , g D 1B (see (3.62)).  is a regular measure so there R R are continuous positive f and g, bounded by 1, such that jf  1A jd  and jg  1B jd  are < ". Choose first B open and g  1B . Consider Z Z

Z .dx/P k .x; B/ 

j

f .x/.dx/P k .x; dy/g.y/j

A

Z Z Cj

Z .dx/P k .x; B/ 

j A

Z

f .x/.dx/P .x; B/  k

f .x/.dx/P k g.x/j

Z

Z Dj

f .x/.dx/P k .x; B/j

.1A .x/  f .x//.dx/P k .x; B/j C

.dx/f .x/.P k .x; B/  P k g.x// Z

Z "C

.dx/.P k .x; B/  P k g.x// D " C

.dx/.1B .x/  g.x//  2"

For B compact and g  1B , the calculation yields the same result. For an arbitrary Borel set B, we have n Z 1X lim inf .dx/P k .x; B/  .A/.C / n A kD1

for any compact C B. In a corresponding way lim sup

n Z 1X .dx/P k .x; B/  .A/.O/; n A kD1

where O B is open. Since .B/ can be arbitrarily well approximated from below and from above by .C /, C compact, and .O/, O open, respectively, the Ces`aro sum on the left is in fact convergent to the limit .A/.B/. Corollary 3.12. (i) Suppose that K is right simple. Let Zn be a right random walk on S starting at x. Then the sequence of empirical measures n1 1X ıZk n kD0

almost surely converges weakly to the invariant measure G  ˇ.

236

3 Random Walks on Semigroups

(ii) Suppose K is left simple. Let Sn be a left random walk on S starting at x. Then the sequence of empirical measures n1 1X ıSk n kD0

almost surely converges weakly to the invariant measure ˛  G . (iii) Suppose 2 generates all of S . Let Wn be a bilateral random walk on S starting at x. Then the sequence of empirical measures n1 1X ıWk n kD0

almost surely converges weakly to the invariant measure ˛  G  ˇ. (iv) Let Mn be the mixed random walk on S starting at x. Then the sequence of empirical measures n1 1X ı Mk n kD0

almost surely converges weakly to the invariant measure ˛  G  ˇ. Proof. We prove the assertions using a coupling argument. A walk beginning at an arbitrary x after a finite time almost coalesces with an ergodic process. The detailed argument will be given for the right random walk in Assertion (i), only. The ideas underlying the proofs for the other walks are quite similar. Let x 2 S and let Zn be the right random walk generated by  with starting point x. Zn D xX1 X2    Xn ; n D 1; 2; : : : ; where X1 ; X2 ; : : : are mutually independent -distributed random variables in S . Assertion (i) of Corollary 3.12 means that for arbitrary continuous f n1 1X f .Zn / n kD0

R almost surely converges with limit f d . Let Ui ; i D 1; 2; : : : N , be any open covering of K D G Y [since the right ideal structure was assumed to be trivial in Assertion (i)]. We may take the Ui to have a nonempty intersection with K, thus .Ui / > 0 for all i . D G  ˇ is the unique stationary probability distribution for a right random walk on S . By Corollary 3.11, the right random walk with initial distribution is ergodic. Let Y i be a right random walk on K generated by  with initial measure i the restriction of to Ui , appropriately normed to be a probability measure.

3.4 Compact Semigroups

237

R

Then Yni is ergodic in the sense that the sum in (3.64) converges to f d for all bounded measurable f Pi  almost surely. If not, then the set of exceptional trajectories would have positive measure with respect to P as well, contradicting the ergodicity of the right random walk with initial measure . Let f be a continuous function on S and take an " > 0. Then there exists a covering Ui ; i D 1; 2; : : : N of K such that (3.60) holds. Let O be their union [niD1 Ui . O K and from Proposition 2.6 we know that for some M and  < 1 and all x, n .x 1 O c /; n .O c /  Mn : (3.68) Let T be the time when the walk Zn first enters the open set O. Formally, T D inffn  0jZn 2 Og; where T is defined to be infinite if Zn never enters O. (3.68) shows that Ex T D

X

Px fT > ng

n

is finite. Hence, T is finite with probability 1. Let us now define the parallel right random walks Zni on K by letting Z0i be distributed according to i and i XT Cn ; n D 1; 2; : : : Zni D Zn1

In other words, increments of parallel walks are the same as those of the original process Zn from the (random) time T onwards. The Zni process is similar to the Yni process just discussed, apart from the numbering of the increments. By construction, ZT belongs to some Ui ; i D 1; 2; : : : N , U1 , say, and then f .ZT Cn / differs from f .Zn1 / by less than " for all n. Then T Cn1 1 X f .Zk / n kDT

differs by less than " from T Cn1 1 X f .Zk1 /: n kDT

R The latter sum was seen to converge almost surely to f d . Therefore repeating the argument for a decreasing sequence of values of ", we finally obtain that Px -almost surely Z n1 1X lim f .Zk / D f d : n kD0 t u

238

3 Random Walks on Semigroups

Remark 3.13. Theorem 3.12 is a special case of Corollary 3.12. The convergence of the Ces`aro sums in Theorem 3.12 are to hold Pe -a. s. We cannot use Corollary 3.11 directly since the equidistribution for arbitrary measurable f holds only P -almost surely and certainly not Pe -almost surely. Indeed, if  is deterministic or supported on a finite set of points, the statement of Theorem 3.12 does not hold for noncontinuous f . (Our stopping time T is identically 0 in Theorem 3.12.) Corollary 3.12 does not quite cover the case in Theorem 3.12, where the group generated by 2 is a proper subgroup H of G; however, the proof of Corollary 3.12 can easily be extended. The proof relies on the fact that the bilateral random walk confined to H (gH ) admits a unique invariant probability distribution H ( gH ). Note also that the mixed walk always can be used as a means of experimental determination of K and the idempotent measure D ˛  G  ˇ on K. In the general non-ergodic case, the empirical measures of Zn , Sn , and Wn converge to some (sample-dependent) stationary distribution. Corollary 3.13. Suppose S is a locally compact semigroup with a compact ideal I such that k .I / is strictly positive for some k. Then S admits a completely simple kernel K and the assertions of all the theorems in Sect. 3.4 are valid for the random walks generated by . Proof. Exactly as in Proposition 2.6, one shows, using the relation SIS I , the analog of (3.68) P n .x; I c /; n .I c /  Mn ; n D 1; 2; : : : ;

(3.69)

where P stands for any of the four transition probability operators considered. Asymptotic and ergodic properties of the random walks are thus entirely determined by the process on the compact subsemigroup I . Limit points of all sequences of measures are supported on subsets of I and hence on the compact minimal ideal K of I . Equation (3.68) is again satisfied for any open set O containing K. After a random almost surely finite time, trajectories of all our random walks enter the ideal I and hence any relatively open O \ I K. We leave the detailed verification to the reader. t u Spreadoutness and random walks with continuous components were useful in the study of random walks on locally compact groups. In a compact semigroup situation, such conditions affect the structure profoundly. Proposition 3.20. Suppose that the right random walk on the compact semigroup S has a nontrivial continuous component for every k 2 K. Let X  G  Y be the Rees–Suschkewitsch representation of the kernel K. Then X is finite. Proof. If k is an element of K, then kS has n -measure 1 for all n. Also Prn .; kS / D 1 on kS . Consequently, if T .; / denotes the continuous component, the set fsjT .s; kS / > 0g

3.4 Compact Semigroups

239

is an open set containing every element of kS . The compact kernel K is the union of these minimal right ideals kS . Since the minimal right ideals form an open covering of K, there is in fact only a finite number of minimal right ideals. t u The intersections of the minimal right ideals and the minimal left ideals are isomorphic groups (DkKk). If the left walk also satisfies the conditions of Proposition 3.20, then we can postulate Corollary 3.14. Corollary 3.14. If both unilateral walks have continuous components which are nontrivial at the points of K, then the subgroups kKk are open and K is a finite union of open groups. In Sect. 3.3, terminology for Markov random walks was developed, see the Definitions 3.9 and 3.10. We use the terminology here to present an equidistribution theorem on a compact semigroup S . Define a Markov random walk on S as in Definition 3.9 with C S and the function f the identity function on C . Then Sn is simply X1 X2    Xn . Assume further that Xn is ergodic on C and also that C intersects the kernel K of S . Proposition 3.21 is a special case drawn from the theory developed in [237]. Proposition 3.21. Suppose that the kernel of S is a compact group G. Then the set of essential elements for the Koutsky process .Xn ; Sn / is precisely C  G. If C  G is the only essential class for the Koutsky process, then Sn is ergodic in the sense that it satisfies (3.64) with respect to the Haar measure G lim

Z n 1X F .Sk / D F d G ; almost surely; F bounded measurable n kD1

Another example with applications to iterated function systems, see Sect. 4.2, is the following. We assume that the set C is finite. Let Xn be stationary on C , call its invariant distribution . Assume further that the transition matrix P for the X chain is strictly positive; i.e., for any c; c 0 2 C there is a positive probability for a transition from c to c 0 in one step. Our aim is to investigate the left Markov random walk Yn D f .Xn /f .Xn1 /    f .X2 /f .X1 / on a compact semigroup S that we assume generated by f .C /; i.e., SD

1 [

.f .C //n :

nD1

Suppose S is equipped with a kernel K D A  G that is left simple; i.e., there is a unique minimal left ideal, K itself. We also make another restrictive condition on S : sK is a minimal right ideal for every s 2 S (and not just for the elements of K).

240

3 Random Walks on Semigroups

The invariance semigroups ˘.c; c/ (see Definition 3.10) have the form f .c/S . This is an immediate consequence of the strong assumption made on the transition probability matrix P . Then, using the ideas behind the proof of Lemma 3.4, it is not difficult to show that the set of essential elements for the Koutsky process .Xn ; Yn / is C  K. In addition, the elements of C  K communicate. The transition probability operator U for the Koutsky process is equicontinuous and hence ergodic on the essential class C  K. The invariant measure is   , where D ˛  G is idempotent on K. The measure ˛ on the minimal right ideals is given by ˛.xK/ D fc 2 C jf .c/K D xKg: To complete Sect. 3.4, let us look at a case, where the transition probability matrix P for the Markov chain on C is extremely structured in contrast with the above situation, where all transitions were possible in one step. Take C to be the set f0; 1=4; 1=4; 1=6; 1=3; 1=2g and take f .c/ D exp.2 i c/ in the circle group T . The state space of the Markov random walk is thus (a subset of) the compact abelian group T . Let P be the following 6  6 matrix where the above ordering of the elements of C is preserved: 1 0 :2 :4 0 :4 0 0 B 0 0 1 0 0 0C C B C B B 1 0 0 0 0 0C C: B B 0 0 0 0 1 0C C B @ 0 0 0 0 0 1A 1 0 0 0 00 P is aperiodic and irreducible. Nevertheless, the invariance groups ˘.c; c/ are all reduced to the singleton exp.2 i 0/ D 1.

Section 3.4 Exercises Exercise 3.27. Consider the semigroup S of Example 3.5 (i) represented as points in the unit square. Suppose  charges only two particular points .a; a C d / and .a0 ; a0 C d /, where d < 13 . Show that the kernel of the semigroup generated by  is a Cantor-like compact subset of the diagonal of the unit square. Exercise 3.28. Let f be a continuous function on the compact semigroup S . Suppose f is 0 on the kernel K of S . Let Xn be a random walk on S . Prove that lim f .Xn / D 0

n!1

almost surely. Is it true that

1 X nD1

almost surely?

f .Xn / < 1

3.5 Completely Simple Semigroups

241

Exercise 3.29. Let S be the semigroup of d  d stochastic matrices. Suppose the support of  contains a transition matrix of an irreducible Markov chain. Show that the left random walk generated by  converges. Describe the asymptotic behavior of the right, bilateral, and mixed walks. Exercise 3.30. Let  be a probability measure on the semigroup of d  d doubly stochastic matrices. Suppose the support of k contains an all-positive matrix, for some k. Prove that the random walks generated by  converge. Exercise 3.31. (Continuation of Exercise 3.27) Consider matrices of the form 0 1 p 1p a @q 1  q b A ; 0 0 c where 0 < p; q < 1, a and b are real, and c D ˙1.

3.5 Completely Simple Semigroups The importance of the completely simple minimal ideal of a locally compact semigroup has been stressed throughout this book. For example, it was shown in Chap. 2 that any idempotent probability measure is supported on a completely simple subsemigroup, with a compact group factor. In Sect. 3.2 we showed, for a discrete semigroup S , that if a random walk admits recurrent points, these constitute the completely simple minimal ideal K of S . It is natural then to single out completely simple semigroup for special consideration. In Sect. 3.5, we assume from the start that the underlying state space is a completely simple, locally compact topological semigroup S . Recall that S is algebraically and topologically isomorphic to a Rees–Suschkewitsch product X G Y with multiplication .x; g; y/.x 0 ; g 0 ; y 0 / D .x; g. y; x 0 /g 0 ; y 0 /; where X and Y are locally compact spaces, G a locally compact group and the sandwich function  a continuous map from Y  X into G.  is not uniquely determined (see Proposition 1.10). One way of choosing the factors in the Rees–Suschekewitsch product is the following (see Theorem 1.1): Let e be an idempotent of S . Then eSe is a closed (thus locally compact) subgroup of S . Let X be the idempotents of Se and Y the idempotents of eS . With .y; x/ taken to be the product yx, X  eSe  Y is topologically and algebraically isomorphic to the completely simple semigroup S . (Notice that there is considerable freedom in the choice of the representation. e can in fact be chosen arbitrarily among the idempotents of S .) When dealing with completely simple subsemigroups of S , as D in Propositions and Corollaries below, we choose e 2 D, so that eDe eSe, E.De/ E.Se/ and E.eD/ E.eS / (where E./ denotes the idempotents of the set in question.) Thus, the Rees–Suschekewitsch factors of D are (chosen to be) subsets of the corresponding factors of S .

242

3 Random Walks on Semigroups

Let  be a probability measure on S generating a subsemigroup D: DD

1 [

Sn :

(3.70)

nD1

We will work mainly with the problem of recurrence/transience for our random walks generated by . This theory is developed for the unilateral walks only. Let us start out though by making some general observations on essentiality. Proposition 3.22. Let S be a completely simple locally compact semigroup and  a measure on S generating the closed subsemigroup D. Then either all or none of the elements of D are essential for the right (left) walk. If the elements of D are essential for one of the unilateral walks, they are also essential for the other unilateral walk. In that case, D is a completely simple subsemigroup of S . Proof. Suppose x is essential for the right random walk on D, i.e., xD is a minimal closed right ideal and x 2 yD D xD for all y 2 xD. In particular, let y D x 3 . Then there is a sequence dn in D such that x D lim x 3 dn D x 2 lim xdn . We have further x 2 D x 2 lim xdn x. In the group xSx, the equation x 2 D x 2 y has a unique solution, however. Thus, lim xdn x is necessarily equal to e, the identity of xSx. (Recall that D is closed.) Hence, e 2 D and also xD D eD D eD D xD. Let I be the set of all essential elements for the right random walk. Just as in the proof of Corollary 3.1 it is seen that I is a minimal ideal of D. I is a simple subsemigroup of S , containing idempotents. All idempotents of S are primitive and hence I is a completely simple subsemigroup of D and S (see Definitions 1.5 and 1.6). Also, I necessarily coincides with D. To see this, let d D .a; g; b/ 2 D. Since I is completely simple, a union of groups iIi;i 2 I , then dId I contains an idempotent f (D the identity of the group dSd) such that df D f d D d . Hence d 2 I. D being completely simple, the sets Dx, x 2 D, are also closed and minimal left ideals. In other words, the left random walk on D admits essential elements. Actually, all elements of D are essential since D is the union of minimal left ideals. t u Corollary 3.15. If D is completely simple, then all elements of D are essential for the bilateral and mixed walks. There is one essential class for the mixed walk and at most two essential classes for the bilateral walk. Proof. The assertions for the mixed random walk follow trivially from the fact that D is simple. The essential elements for the bilateral walk form a two-sided ideal of D. The verification of this fact can be made along the lines of the corresponding proof in Proposition 3.9. Thus, it remains to show that there are essential elements for the bilateral walk.

3.5 Completely Simple Semigroups

243

To this end, take e to be an idempotent of D. There is a sequence sn 2 Sn with sn converging to e. Then sn esn converges to eee D e. We can deduce that e is a return state for the bilateral walk. Let e ! h. Then, as in the proof of Proposition 3.9, hk D e implies e ! kh and j h D e implies e ! hj . For all s; t such that ths D e we have h ! sht. There is considerable freedom in the choice of s and t to satisfy ths D e. It is possible for example to take t D e and s 2 eDe. Making use of the explicit product representation (with G D eDe) we can then show that sht D e for that particular choice of s and t. We have thus proved that e ! h implies h ! e. Next, we sketch the proof of the fact that there are at most two essential classes for the bilateral walk. Write D in product form A  G  B D AGB. Consider the decomposition DD

1 [

Sn D

nD1

1 [ nD1

S2n [

1 [

S2nC1 :

(3.71)

nD0

The first term on the right-hand side is the subsemigroup generated by 2 . It contains all the squares d 2 of the elements of D and hence all the idempotents of D and consequently the range of . We can write it AHB where H is a normal subgroup of G. If H ¤ G, then G D H [ gH for a g 2 G n H . It remains to prove that the identity e of H (and G) communicates with all of AHB. In the calculation to follow the subscripted element, cn indicates an element of Sn . The elements of D are written in the product form AGB. Let sn  sn0 D agb  a0 g 0 b 0 be an element of S2n written as a product of two elements in Sn . Then the product is, in fact, unaltered if we expand it to agb  a2k ea2k  a0 g 0 b 0 ; where a2k is assumed to equal the idempotent a0 2 E.De/. This shows that sn sn0 is reachable from e by the bilateral walk in n C 2k steps. If lim a2k D a0 instead, we get the same result, e ! sn sn0 and, by continuity, e communicates with all limit points of sequences sn sn0 , i.e., with all elements of AHB. t u The starting point for the study of recurrence is the following analog of Theorem 3.5 Proposition 3.23. If x 2 D is recurrent for the right (left) random walk, then it is essential for the right (left) walk. The element x 2 D is recurrent for the right (left) random walk if and only if 1 X nD1

for all neighborhoods N.x/ of x.

n .N.x// D 1

(3.72)

244

3 Random Walks on Semigroups

For a proof we refer to the detailed treatment in Mukherjea and Tserpes [172], Chap. 2, Sect. 3. The primary difficulty in going from locally compact groups to locally compact completely simple semigroups is the fact that sets of the form K 1 K 0 (where K and K 0 are compact) are no longer compact. Propositions 3.22 and 3.23 enable us to draw the following conclusion. Theorem 3.17. The points of D are either all or none recurrent with respect to the unilateral walks. The right (left) random walk admits recurrent points if and only if 1 X

n .K/ D 1

(3.73)

nD1

for some compact K D. Under (3.73), D is a completely simple subsemigroup of S . The only assertion left to prove is the relationship between (3.73) and recurrence. A compactness argument shows that (3.73) holds for some compact K if and only if (3.72) holds for some x 2 D, i.e., if and only if the set of recurrent points is nonempty. P n Corollary 3.16. The right (left) random walk is recurrent if and only if the 1 nD1  is not a Radon measure. Remark 3.14. If the Rees–Suschkewitsch product happens to be direct, then the behavior of the random walks can essentially be read off from that of the group P factor. n Let 2 be the marginal distribution of the middle factor g. Then n  is a P n Radon measure if and only if n 2 is. We go on to characterize positive recurrence for the random walks on the completely simple semigroup. Proposition 3.24. The set of elements x satisfying lim sup n .N.x// > 0

(3.74)

for all neighborhoods N.x/ of x, is either empty or consists of all D. Proof. The points x satisfying (3.74) form a two-sided ideal D 0 of D: If N.xs/ is a neighborhood of xs (s 2 S ) we choose neighborhoods N.x/ and N.s/ of x and s, respectively, with N.x/N.s/ N.xs/. Then k .N.s// > 0 for some k and nCk .N.xs//  n .N.x//k .N.s//. Thus, xs satisfies (3.73) if x does. The left ideal property is proved similarly. If x satisfies (3.74), it is obviously recurrent and so D is completely simple (Propositions 3.22 and 3.23). It is easily seen that any bilateral ideal D 0 of D must necessarily coincide with D. t u

3.5 Completely Simple Semigroups

245

Remark 3.15. (i) In the same way as in Theorem 3.17, we see that (3.74) is satisfied for some x 2 D if and only if n does not converge to zero vaguely, i.e., lim sup n .K/ > 0 n!1

for some compact subset of K D. (ii) The right random walk has a nontrivial continuous component on D if and only if some power of  is nonsingular with respect to the measure 1  3 . For a proof see [97], Sect. 5. Note that the topology of the first factor A is necessarily discrete. Theorem 3.18. If the group factor G in the Rees–Suschkewitsch representation is compact then all elements of D are positive recurrent. If the right random walk has a nontrivial continuous component on D, then the elements of D are positive recurrent if and only if (3.74) is satisfied for some x 2 D. This is the case if and only if G is compact. Proof. If G is compact then (3.74), or rather the equivalent formulation in Remark 3.15(i), is certainly satisfied since nC2 .A0  G  B 0 / D .A0  G  B/n .A  G  B/.A  G  B 0 / D .A0  G  B/.A  G  B 0 / for compact subset A0 ; B 0 of A and B, respectively. Hence, the random walk is recurrent. Consider the measure   1   3 , where 1 .3 / is the marginal distribution of the first (third) component a (b) and is the normed Haar measure on G. (Formally, 1 .A0 / D .A0 G B/ for A0 A.) Clearly,  is the required probability measure invariant with respect to the right and left random walks. By Proposition 3.12, a positive recurrent point x satisfies the positivity condition lim sup n .x 1 N.x// > 0 n!1

for all neighborhoods N.x/ of x. Take N.x/ to be compact and let the lim sup above be  2ı > 0. The set x 1 N.x/ is noncompact in general but the Rees–Suschkewitsch product structure D D A  G  B will enable us to use almost the same arguments as in the group case. Write x D .a; g; b/ and take N.x/ to be A0  G 0  B 0 . Let A00 be a compact subset of A with 1 .A00 / D .A00  G  B/ > 1  ı. Then x 1 N D .a; g; b/1 .A0  G 0  B 0 / [ A00  .g.b; A00 //1 G 0  B 0 .A n A00 /  G  B: Since n ..A n A00 /  G  B D 1 .A n A00 / < ı, we conclude that the n -measure of the compact set A00  .g.b; A00 //1 G 0  B 0 is at least ı for infinitely many n. Thus, (3.74) is satisfied for all x 2 D.

246

3 Random Walks on Semigroups

Conversely, assume that (3.74) holds for the elements of D. Take an idempotent e 2 D. Then e 1 N.e/ contains a neighborhood N 0 .e/ of e. Thus, lim sup n .e 1 N.e// is strictly positive. In other words, e is a positive point for the right random walk and hence positive recurrent. If e has the Rees–Suschkewitsch representation (a; ..b; a//1 ; b) we see that ıa   3 is invariant with respect to the right random walk on eD D fagG B. By positive recurrence, this invariant measure is a probability measure. Thus, the group factor G has to be compact. Clearly all the elements of D are then positive recurrent. t u Remark 3.16. Unfortunately, the previous theorems do not carry over to the completely 0-simple case. Let  generate the multiplicative locally compact semigroup S D Œ0; 1/ D f0g [ .0; 1/ Then 0 is essential, but the nonzero elements are not, in general.  can be chosen to generate a null-recurrent random walk on the open interval (0; 1) – take  to be a distribution on a small interval around 1, such that its geometric mean is 1. Then 0 will be the only essential, positive recurrent point, while the other elements of S are inessential and null-recurrent.

Section 3.5 Exercises Exercise 3.32. Prove the assertion in Remark 3.14: If the Rees–Suschekewitsch product is direct, then the random walk on S is recurrent if and only if the random walk on the middle factor G is. Exercise 3.33. Let S be the semigroup of 2  2 matrices of the form 

 ka a ; kac ac

where k; a; and c are real numbers (see Example 1.2). S is a nonabelian, completely 0-simple semigroup. Construct a probability measure  in such a way as to make the nonzero elements of S null-recurrent. Of course, 0 is always positive recurrent. Exercise 3.34. Let S be the semigroup of 3  3 matrices of the form 0

1 B3 B B1 B @3 0

1 2 xC 3 C 2 C; yC 3 A 0 "

where x and y are real and " is ˙1. Show that S supports null-recurrent left and right random walks.

3.6 Notes and Comments

247

Exercise 3.35. Let S be the semigroup of 3  3 matrices of the form 0 p 1p @q 1  q 0 0

1 x yA ; "

where 0  p; q  1, x; y 2 < and " is ˙1. Show that one can construct a nullrecurrent right random walk on S . Does S support a recurrent left random walk as well?

3.6 Notes and Comments There is certainly no clear distinction between the study of convolution sequences on semigroups (the subject of Chap. 2) and random walks as in this chapter. After all, the distribution of the random walk Xn is the n-fold convolution of the generating measure . These very brief Notes and comments will, however, show a preference for work on path-wise and almost sure properties, such as communication structure, recurrence/transience, and ergodicity. Chapter 2 of the lecture notes by Mukherjea and Tserpes [172] opens up with a more detailed historical survey of the development of theory of random walks on groups and semigroups. The volume edited by Cohen et al. [44] is a survey of the study of random matrices. It also contains a large bibliography. The recent book by Woess [236] treats random walks on graphs and discrete groups. The recurrence/transience problem and its relation to the structure of the underlying space is the focus of Chap. 1. Guivarc’h in [80] studies random walks on large classes of classical groups, notably groups of matrices. He offers a deep survey of current work on limit theorems and also Gaussian laws (central limit theorems) on such groups.

Section 3.1 The book by Revuz [195] gives a solid introduction to Markov chains and random walks. The concept of continuous component was introduced Tuominen and Tweedie in [224]. They also saw the (essential) equivalence between their concept and that of spreadoutness (´etalement), a nonsingularity condition of great utility in the study of random walks on groups. The bilateral random walk seems to have been first studied by Larisse [131] (in discrete semigroups) and the mixed random walk by H¨ogn¨as [94] (in compact semigroups).

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3 Random Walks on Semigroups

Section 3.2 The pioneering paper by Furstenberg and Kesten [70] on random matrices deals with a semigroup structure but does not primarily view it in semigroup terms. The very influential and inspiring book by Grenander [77] insists on the importance of working with the underlying algebraic structure. Grenander’s student Martin-L¨of [138] highlights the interplay between algebra and probability theory in his very clearly written paper. Martin-L¨of deals exclusively with discrete semigroups as does Larisse [131], who develops and deepens the theory. Our treatment owes very much to these two authors. There is a deep connection between algebraic coding theory and semigroup theory. This has been recognized especially in France, where the school of Sch¨utzenberger has developed both theories in parallel. A particularly nice probabilistic example is provided by the study of densities of code words. Let A be an alphabet and A the set of all words on the alphabet A. A is a semigroup under concatenation of words, the empty word being the identity element. A code over A is a subset X of A such that any nonempty word in X  can be written uniquely as a product of words in X (see [11], Chap. 1). Given a probability distribution  on the alphabet A, it is easily extended to a probability distribution on longer words by the recursion .aw/ D .a/.w/, where a 2 A and w 2 A . Such a distribution is called a Bernoulli distribution on A . By definition, it is also a homomorphism from A into the unit interval. The Bernoulli distribution  is positive if it is positive on all letters of A. In the terminology of this book, we are studying an irreducible random walk on A . If L is a subset of A , then it has a density if the sequence .L\An / converges in the Ces`aro sense (where An consists of the n-letter words in If L is highly structured then it turns out that the density of L may be shown to exist. Let M be a well-founded monoid, i.e., a semigroup with identity admitting a completely simple minimal ideal with finite group factor. If  W A ! M is a homomorphism, then it may be shown that all sets of the form  1 .m/ do have a density. Furthermore, this density is nonzero if and only if m 2 K, the completely simple kernel of M (see [11], Theorem 2.4 of Chap. 6). As shown in [11], Chap. 6, the applications in coding theory emanate from the fact that certain codes X are naturally associated with particular monoids M . Hence if the completely simple kernel of such a monoid is known, then the inverse images of its elements, minimal ideals and maximal groups have densities which are expressible in terms of the Rees–Suschkewitsch product representation of the kernel (in fact, it is the order of the group factor that is the most important feature). Among other things, an expression for the density of X  , the interesting quantity in this connection, is obtained. The ergodicity results of Corollary 3.3 indicate ways of experimentally determining the idempotent measure ˛  G  ˇ. One of the reasons the mixed random walk was devised in [94] was precisely the wish to find a simple way to simulate the idempotent measure (see formula (3.28) and Corollary 3.12(iv)).

3.6 Notes and Comments

249

The quotient of G by the normal subgroup N generated by the sandwich function in Proposition 3.7 defines the maximal homomorphism from S to G, see [110, 131, 209]. The Markov random walks, cf. Proposition 3.8 and Definition 3.9, were introduced by Koutsky [130] and Cigler [40] in the finite group case. Schmetterer [206], Muthsam [174], and H¨ogn¨as [99] study Markov random walks on discrete semigroups and obtain structure theorems, which are not restated here. The group case and the compact semigroup case are, however, treated in some detail in Sects. 3 and 4, respectively. The main sources are [237] and [102,103]. The same or similar processes have been widely studied under different names, such as Markov walks (see [79]), semi-Markov processes and Markov renewal processes, e.g., in [123] and [2]. For Exercise 3.17 and further discussion on the continuity of convolution, see [138], Sect. 3. Exercise 3.18 is taken from [131], Part II. Exercise 3.25: Recurrence and transience of random walks on the semidirect product of a compact group with Z d or 0 such that for n > n0 , d X

.yn /ij > ı;

1  j  d;

i D1

so that for some m0 > 0 and n > n0 , ı  .zn /jk 

d X

.yn /ij .zn /jk

i D1



d X

.yn zn /i k  m0 ;

i D1

whenever 1  j  d and 1  k  d . This means the sequence .zn / is bounded so that there exists z in S such that it is a subsequential limit of the zn s and y D yz. Since y has no zero column, z also cannot have a zero column. Thus, we have shown that y 2 y .S  Jc /: (4.3) Notice that S  Jc is a semigroup and let u 2 y .S  Jc /. Then u D ys, s 2 S  Jc . Let Ns be an open set containing s. There exists a positive integer k such that 0 < P .Zk 2 Ns /   D P Zk 2 Ns ; Zn 2 Ny1 Ny infinitely often      P Zk 2 Ns ; XkC1 : : : Xn 2 Ns1 Ny1 Ny infinitely often     D P .Zk 2 Ns / P XkC1 : : : Xn 2 Ns1 Ny1 Ny infinitely often : This means that given any open set Nu containing u, Nu1 Ny ¤  if Ny is chosen so that Ny Ns  Nu for some open set Ns containing s. Again, by using a similar argument to that used above in establishing (4.3), it follows that there exists v in S such that y D u  v and v 2 S  Jc . Thus, we have u 2 y .S  Jc / ) y 2 u .S  Jc /: This means y .S  Jc / is a minimal right ideal of S Jc . Since K is an ideal (which can be easily verified), it follows that K  Jc D [ fy .S  Jc / W y 2 K  Jc g :

(4.4)

Now if I  K  Jc and I is an ideal of S  Jc , then for any y 2 K  Jc , yI is a right ideal of S  Jc and yI  y .S  Jc /. Since y .S  Jc / is a minimal right ideal containing y, it follows that y 2 y .S  Jc / D yI  I so that K  Jc is simple.

256

4 Random Matrices

It remains to show that K  Jc has at least one primitive idempotent element. To this end, let y 2 K  Jc . Since y 2 2 y .S  Jc / and y .S  Jc / is a minimal right ideal, y 2 y 2 .S  Jc /, so there exists z 2 y .S  Jc / such that y D y  z. Then for each positive integer m, y D yzm . Since y has no zero column, the sequence .zm / is a bounded sequence. Notice also that whenever a sequence yzn converges to some element z0 , the sequence .zn / is bounded (since y has no zero column), so z0 2 yS . This means yS is a closed subset of S . Thus, the closure of fzm W m  1g, which is a compact subsemigroup, contains an idempotent element e (see Theorem 1.6) and y D y  e. Now e has no zero column, since y has no zero column. Consequently, since z 2 yS , e 2 yS  Jc D y .S  Jc / ; which is now a right group. Now we show that every idempotent in K  Jc is primitive. Let e1 , e2 be any two idempotents in K  Jc such that e1 e2 D e2 e1 D e1 . Then e1 is an idempotent in the right group e2 .S  Jc /. This means e1 D e2 . This proves that every idempotent in K  Jc is primitive, and therefore, K  Jc is completely simple. t u Corollary 4.1. Suppose there exists y … J such that (4.1) holds for y. Then, K J , K as in Theorem 4.1, is a completely simple (minimal) ideal of S  J . Proof. Notice that if w 2 z .S  Jc / and z 2 K  J , then by Theorem 4.1, z 2 w .S  Jc /. This means w … Jr , since z … Jr . Thus, it follows from (4.4) that [ fz .S  Jc / W z 2 K  J g  K  J: By Theorem 4.1, for each z 2 K  J , z 2 z .S  Jc /. Then the preceding inclusion is an equality, and K  J is a union of right groups. Since K  J is an ideal of S  J , the corollary follows. t u Our next result shows that results similar to those for random walks with values in Rd also hold for random walks in the present context. Theorem 4.2. Let y … Jc . Then the following results are equivalent:   n Ny D 1 for every open set Ny containing y. nD1   (ii) P Zn 2 Ny infinitely often j Z0 D y D 1 for every open set Ny containing y. 1   P n y 1 Ny D 1 for every open set Ny containing y. (iii) nD1   (iv) P Zn 2 Ny infinitely often > 0 for every open set Ny containing y. (i)

1 P

If y … Jr , then results similar to the preceding also hold for .Wn / with y 1 Ny above replaced by Ny y 1 . If y … J , then each of the preceding four statements is also equivalent to each of the following:   (i) P Wn 2 Ny infinitely often j W0 D y D 1 for every open set Ny containing y. (ii) P Wn 2 Ny infinitely often > 0 for every open set Ny containing y.

4.2 Recurrent Random Walks in Nonnegative Matrices

(iii)

1 P

257

  n Ny y 1 D 1 for every open set Ny containing y.

nD1

Proof. (i) ) (ii). (i) implies (as in Theorem 4.1) that for every open set Ny containing y, we have   Pr Zn 2 Ny1 Ny infinitely often D 1: (4.5) Since y 2 Jcc , it can easily be verified that given an open set Ny containing y, there is always an open set N0y containing y such that 1 N0y  y 1 Ny : N0y

Then (ii) follows from (4.5). (ii) ) (iii). Follows by the Borel–Cantelli  lemma.    (iii) ) (i). Assume statement (iii). Let Nmy , cl NmC1;y  Nmy , be a countc able local  base  at y such that each is relatively compact. Since y 2 Jc , the set 1 y cl Nmy is compact for each m  1. Now utilizing this along with (iii) with Nmy (in place of Ny ), we see there exist zm such that the sequence yzm converges to y and for every open set Nzm containing zm , 1 X

n .Nzm / D 1:

(4.6)

nD1

Since y 2 Jcc , it follows that the zm s are bounded and as such there exists z in S such that z is a subsequential limit of the zm s and y D yz. It follows from (4.6) that for every open set Nz containing z, we also have 1 X

n .Nz / D 1:

(4.7)

nD1

Since the set of all points of S with the property (4.7) is an ideal of S , it follows that y D yz also has the property in (4.7). Thus (i) follows. (iv) ) (i) [) (ii)] follows by the Borel–Cantelli lemma. (ii) ) (iv). We use an argument similar to the one in (iii) ) (i). By (ii), there is a countable local base Nmy such that for m  1,      Pr Zn 2 y 1 cl Nmy infinitely often D 1:    Again by the compactness of y 1 cl Nmy , there exist zm as before such that yzm converges to y, zm s are bounded and Pr .Zn 2 Nzm infinitely often/ > 0 for every open set Nzm containing zm : If z is a subsequential limit of the zm s, then yz D y and Pr .Zn 2 Nz infinitely often/ > 0 for every open set Nz containing z:

258

4 Random Matrices

Now given an open set Ny containing y, there exist open subsets N1y containing y and N1z containing z such that N1y N1z  Ny . Let k be a positive integer such that  Pr Zk 2 N1y > 0. Then we have   Pr Zn 2 Ny infinitely often

   Pr Zk 2 N1y ; XkC1 : : : Xn 2 N1z infinitely often   D Pr Zk 2 N1y  Pr .Zn 2 N1z infinitely often/ > 0: This establishes (iv). The rest of Theorem 4.2 follows by similar dual arguments.

t u

Remark 4.1. We call an element x in S positive recurrent if lim sup n .Nx / > 0

n!1

for every open set Nx containing x. Suppose the set I of positive recurrent states is nonempty and I \ J c ¤ . Suppose also .J / D 0. Then it is easily verified that for each positive integer n, n .J / D 0, and as such, we can restrict  to J c (since J c is open) when we are considering only elements in S that have the property in (4.1). It then follows from Theorem 2.11, see also Proposition 4.1, that n is tight. By Theorem 2.7, S has a completely simple (minimal) ideal, which is precisely the set I . Thus, we have proven that whenever I \ J c ¤  and .J / D 0, I is the kernel of S , and by Theorem 4.1 and Corollary 4.1, I \ J c D K \ J c , I \ Jrc D K \ Jrc , and I \ Jcc D K \ Jcc ; where K is the set of all elements in S with the property in (4.1). Let us now formally define the following recurrent sets: ( ) 1 X n RD x2S W  .Nx / D 1 for every open set Nx containing x ; nD1

R.Z/ D fx 2 S W P .Zn 2 Nx infinitely often/ > 0 for every open set Nx containing xg ; R.W / D fx 2 S W P .Wn 2 Nx infinitely often/ > 0 for every open set Nx containing xg : Note that R  R.Z/ [ R.W /, R.Z/ is a left ideal of S , and R.W / is a right ideal of S , while R is a (two-sided) ideal of S . Theorems 4.1 and 4.2 did not deal with elements in the set J in the context of recurrence. In what follows, we show that under mild conditions, we can say more about the recurrent sets R.Z/ and R.W /. First, we need the two following technical lemmas.

4.2 Recurrent Random Walks in Nonnegative Matrices

259

Lemma 4.1. Let H be a (multiplicative) semigroup of d  d nonnegative matrices and J the semigroup of all d  d strictly positive matrices such that J1 H \ J ¤ , m.H  f0g/ \ Jrc ¤ : Write:    X m.H  f0g/ [ f0g and Y m.H  f0g/ \ Jrc D m H \ Jrc : As in Chap. 1, m.A/ D fx 2 A j rank.x/  rank.y/ for all y 2 Ag. Then X is a completely 0-simple subsemigroup if and only if Y is a completely simple subsemigroup. When 0 … H , the same result holds with obvious modifications. Proof. Let us assume Y is a completely simple subsemigroup. Write H1 H f0g. Observe that J1  H1 J1  J so that J1  m .H1 /  J1  m .H1 / \ J: Let y 2 m .H1 / \ J . Then y 2 Y , and by our assumption y  Y  y is a group. Let e be the identity of this group. Then e 2 J . Since e 2 J , e  Œm .H1 / \ Jr  e  J  Jrc : Then it follows that e  m .H1 /  e   D e Œm .H1 / \ Jr  e [ e m .H1 / \ Jrc e D e  Y  e. yYy/; which is a group. Since e 2 J , e Œm .H1 / \ Jr   Jrc : Therefore, e  m .H1 / D eY , which is a minimal right ideal of Y . This means eY [ f0g is a nonzero minimal right ideal of X . Now let us consider m .H1 /  e. Let x 2 m .H1 /. Then there exists z in m .H1 / such that .eze/.exe/ D e: Consequently, Œm .H1 / e xe  Œm .H1 / eze exe D m .H1 / eI

260

4 Random Matrices

then m .H1 / e [ f0g is a nonzero minimal left ideal of X . Write: A e  m .H1 / [ f0g, L m .H1 / e [ f0g: Then L  A  L [ A, and L  A is an ideal of X . It is actually a nonzero minimal ideal of X , since if I  L  A and I is a nonzero ideal of X , then for any w 2 A, w ¤ f0g, Lw is a nonzero minimal left ideal of X and Lw D I  Lw  I: Thus, L  A is a completely 0-simple subsemigroup (see Proposition 1.11). We now show that L  A D X . Let x 2 m .H1 /. Since e 2 J , xex ¤ 0; also, x and xex have the same (minimal) rank, and thus as linear transformations from Rd into itself, they have the same range space. Since xex 2 L  A (e is in L  A), there exists an idempotent e0 in L  A such that e0  xex D xex: Now let u 2 Rd . Then there exists v 2 Rd such that x.u/ D Œxex.v/. Thus Œe0  x .u/ D Œe0  .xex/ .v/ D Œxex.v/ D x.u/: This means x D e0 x 2 L  A. This proves that x D L  A, and X is a completely 0-simple subsemigroup. Now suppose X is a completely 0-simple subsemigroup. Let y 2 Y . Then by Proposition 1.11, yXy  f0g is a group containing y. Since y … Jr , every element of this group is in Jrc \ m .H1 / Y so yXy  f0g D y ŒyXy  f0g y  y  Y  y; which is a group containing y. We show that Y is simple. If I is an ideal of Y , then for any z 2 Y , I  zI z. Since I \ zY z ¤ , z 2 zY z, which is a group, it follows that I  zY z. Thus, Y is simple. It is easily verified that for y 2 Y , Y  y is a minimal left ideal and yY is a minimal right ideal. If L0  Y  y and L0 is a left ideal of Y , then for z  y 2 L0 and z 2 Y , yzy 2 L0 \ yYy. y 2 yYy  L0 . Hence Y  y  L0 . Also, if R0  yY and R0 is a right ideal of Y , then for y  z 2 R0 and z 2 Y , we have yzy 2 R0 \ yYy. Then y 2 yYy  R0 and yY  R0 . Hence, Y is completely simple. t u  d To consider recurrence for affine maps f on RC , f .x/ D Ax C B with A and B as in (4.8), we need Lemma 4.2.

4.2 Recurrent Random Walks in Nonnegative Matrices

261

Lemma 4.2. Let H be a (multiplicative) semigroup of .d C1/.d C1/ nonnegative matrices of the form   AB ; (4.8) 0 1 where A is a d  d matrix, B is a d  1 vector, 0 .0; : : : ; 0/ is a 1  d all-zero vector. Let M be the set of all such matrices such that  

AB 2 M if and only if max min Aij ; Bi > 0; 0 1 1j d for each i , 1  i  d . Suppose H \ M ¤  and m.H / \ Jrc ¤ , where Jr is the set of all .d C 1/  .d C 1/ nonnegative matrices with at least one zero row. Then m.H / is a completely simple (minimal) ideal of H if and only if    m.H / \ Jrc D m H \ Jrc is a completely simple (minimal) ideal of H \ Jrc . Proof. The proof follows the same lines as that of Lemma 4.1. The only point we need to note here is that   M  H  fw g  M  M ; where w is the special element (the .d C 1/  .d C 1/ matrix), where the only nonzero element is 1 and it is in the last row and the last column. t u Now we can present a version of Theorem 4.2 even for elements in J , but with the additional assumptions: .J / D 0 and S \ J ¤ :

(4.9)

Theorem 4.3. Let , S , Zn , and Wn be defined as before. (Here, we are considering general d  d nonnegative matrices as in Theorems 4.1 and 4.2.) Assume (4.9). Suppose R.Z/ is nonempty. Then it must either be f0g (in case 0 2 S ) or R.Z/ D R.W / D m .S  f0g/ [ f0g: The same is true if R.W / is assumed nonempty instead of R.Z/. Proof. Recall that R is defined by ( RD y2S W

1 X nD1

) n .N.y// D 1 for every open set N.y/ containing y :

262

4 Random Matrices

Then R  R.Z/ [ R.W /. Suppose R ¤  and R contains a nonzero element of S . If x ¤ 0 and x 2 R, then since R is an ideal for y 2 J yx 2 R \ Jrc ;

xy 2 R \ Jcc :

Therefore, by Theorem 4.1, yx 2 R.W / and xy 2 R.Z/. By Theorem 4.1, R \ Jrc is a completely simple (minimal) ideal of S \ Jrc , and the same is true for R \ Jcc , which is a completely simple (minimal) ideal of S \ Jcc . Thus   R \ Jrc D m S \ Jrc ;

  R \ Jcc D m S \ Jcc :

(4.10)

By Lemma 4.1, m .S  f0g/ [ f0g is a completely 0-simple ideal of S . We claim that (

  R \ Jcc  cl R \ Jrc ;   R \ Jrc  cl R \ Jcc :

(4.11)

Let us prove only the first statement in (4.11). The proof of the second statement is similar. Let z 2 R \ Jcc . Then by Theorem 4.2, z 2 R.Z/. Let V be an open set containing z such that 0 … V and V has compact closure. Suppose V \ J  R D :

(4.12)

Then for any x 2 J , x 1 V is compact and x 1 V \ R D . This means 1 X

  n x 1 V < 1:

(4.13)

nD1

Let W be an open set containing z such that W  V . Then for each x 2 J and any y 2 W , given any open set N.y/ containing y, there exist open sets N.x/ containing x and Nx .y/ containing y such that N.x/1 Nx .y/  x 1 N.y/:

(4.14)

It follows from (4.13) and (4.14) that given any compact set A  J , 1 X

  n A1 W < 1:

nD1

Notice that z 2 R.Z/ and lim n .J / D 1;

n!1

(4.15)

4.2 Recurrent Random Walks in Nonnegative Matrices

263

where J is an open ideal of S . The reason is the following: Z     n J c x 1 .dx/ nC1 J c D 

n

and therefore,



J

c



1 X

Jc



   n  Jc   Jc ;   n J c < 1;

nD1

since some k  1,  .J / > 0. Thus, there is a compact set A  J such that k

P .Zk 2 A; Zn 2 W infinitely often/ > 0:

(4.16)

However, (4.16) implies that 0 < P .Zk 2 A; Zk  XkC1 : : : Xn 2 W infinitely often/    P XkC1 : : : Xn 2 A1 W infinitely often   D P Zn 2 A1 W infinitely often ; which contradicts (4.15). Thus, (4.12) is impossible, and there exist elements y1 2 J , y2 2 R such that y1  y2 2 V : Since 0 … V , y2 ¤ 0 and y1  y2 2 Jrc \ R. This establishes (4.11). Notice that the preceding proof also demonstrates   R.Z/  cl R \ Jrc  R.W /; and similarly,

(4.17)

  R.W /  cl R \ Jcc  R.Z/:

(4.18)   c Therefore, R.Z/ D R.W /. Since x 2 m .S  f0g/, J x  m S \ Jr , it is clear that   m S \ Jrc D m .S  f0g/ \ Jrc  m .S  f0g/ ; Therefore,      R.Z/  cl R \ Jrc D cl m S \ Jrc  m .S  f0g/ [ f0g:

(4.19)

Now we prove the final part of Theorem 4.3, namely, R.Z/ D m .S  f0g/ [ f0g D R.W /:

(4.20)

264

4 Random Matrices

Let x 2 m .S  f0g/. Then S x ¤ f0g. Since m .S  f0g/ [ f0g is completely 0-simple, xS x  f0g is a group with an identity e and S x D Se. Notice that J .R  f0g/ J  J \ R  R.Z/ \ R.W /: Let y 2 R.Z/\J . Then ye 2 R\Jrc . By (4.11), ye 2 R.Z/ since R.Z/ is closed and R \ Jcc  R.Z/ by Theorem 4.2. Since R.Z/ is a left ideal, S  ye  R.Z/. Since x 2 S x and S x D Se D Sye  R.Z/, x 2 R.Z/. Thus, R.Z/ D m .S  f0g/ [ f0g ŒD R.W /:

t u

To examine the situation for affine maps, let us now consider (as in (4.8)) the .d C 1/  .d C 1/ random nonnegative matrix   An Bn ; Xn 0 1

(4.21)

where X0 ; X1 ; : : : are i.i.d. with distribution . Then   A0 A1 : : : An Cn Zn X0 X1 : : : Xn D ; 0 1 where Cn D B0 C A0 B1 C    C A0 A1 : : : An1 Bn . Notice that each Cn is a d  1 vector and for 1  i  d .CnC1 /i  .Cn /i then lim .Cn /i exists but is possibly infinite. We also have n!1

  An An1 : : : A0 Dn ; Wn Xn Xn1 : : : X0 D 0 1 where Dn D An An1 : : : A1 B0 C An : : : A2 B1 C    C An Bn1 C Bn : Notice that for each n  0, P .Cn 2 B/ D P .Dn 2 B/: Furthermore if R.Z/ ¤ , then there exists u > 0 such that P ..Cn /i  u for 1  i  d eventually/ > 0:

(4.22)

4.2 Recurrent Random Walks in Nonnegative Matrices

265

This implies that there exists ı > 0 and a positive integer N such that ˚  n  N ) n x 2 S W xi;d C1  u for 1  i  d > ı:

(4.23)

Note that R.W / ¤  does not immediately lead to an inequality of the form (4.23), since, unlike Cn s, Dn s cannot be assumed to be an increasing sequence. However, if Wn s are assumed to have a positive recurrent state, that is, when lim sup P .Dn 2 B/ > 0

n!1

for some bounded open set B, we must then also have by (4.22), lim sup P .Cn 2 B/ > 0:

n!1

(4.24)

This implies an inequality of the form (4.23). At this time, it is not known exactly when R.W / ¤  implies R.Z/ ¤ . However, when the rank of the matrices in m.S /, where S is the closed multiplicative semigroup generated by S is greater than 1, then R ¤  implies (and therefore R.W / ¤  also implies) that R.Z/ ¤  if S contains at least one matrix of the form   AB ; 0 1 where A is strictly positive. The reason is if we take   C D 0 1 in R, then C ¤ 0, and consequently if A is strictly positive, then    C D AB 2 R \ Jcc D R.Z/ \ Jcc : 0 1 0 1 In what follows, we present a number of interesting results for affine maps, namely, that under very mild conditions, R.Z/ ¤  (in fact an assumption even weaker than this) implies the tightness of the sequence .n / and therefore the existence of a -invariant probability measure  where n 1X k  ; n!1 n

 D .weak/ lim

kD1

and S./ D m.S /. These results will also help us identify the sets R.Z/ and R.W /. Theorem 4.4. Let  be a Borel probability measure on .d C1/.d C1/ nonnegative matrices of the form (4.8). Let S be the closed (with usual topology) multiplicative semigroup of such matrices generated by S . Suppose the following three conditions hold:

266

4 Random Matrices

(i) There exist q > 0 and a subset Bz with z > 0, defined by ˚  Bz D x 2 S W xi;d C1  z for 1  i  d C 1 ; such that for some subsequence .nk / of positive integers nk .Bz / > q, k  1; (ii) The set J0 defined by ˚  J0 D x 2 S W for each i; 1  i  d; xi;d C1 > 0 is nonempty; and (iii) .J / D 0. Then the sequence .n / is tight. Proof. We defer the proof to Sect. 4.3, Theorem 4.13. Theorem 4.5. Let X0 ; X1 ; : : : be a sequence of .d C 1/  .d C 1/ i.i.d. (random) nonnegative matrices of the form given in (4.8). Suppose  is the distribution of X0 and S is the closed (with usual topology) multiplicative semigroup generated by S./, the support of . Let Jc (respectively, Jr ) be the set of all matrices in S with at least one zero column (respectively, zero row) and J D Jc [ Jr . Let us also define the set I by



I D x 2 S W max min xij ; xi;d C1 > 0 for each i; 1  i  d : 1j d

(4.25)

Assume .J / D 0; and there exists in S a matrix

I ¤

(4.26)

  AB 0 1

for which B ¤ 0. (Without this last assumption, the situation is exactly as in Theorem 4.3.) Let Zn X0 X1 : : : Xn and Wn Xn Xn1 : : : X0 : Suppose R.Z/ ¤ . Then every point in m.S / (matrices in S with the minimal rank) is positive recurrent with respect to both .Zn / and .Wn /, and R.Z/ D m.S /  R.W /: Moreover, the sequence .n / is tight, and there is a unique -invariant probability measure  on S such that   D   D  and S./ D m.S /, which is a completely simple subsemigroup of S .

4.2 Recurrent Random Walks in Nonnegative Matrices

267

Proof. Suppose R.Z/ ¤ . Suppose also   00 2 S: 01 Let us call this special element w. Notice that R.Z/ is a left ideal of S , and w  R.Z/ D fwg: This means that if R.Z/ ¤ , then w 2 R.Z/. Let   AB 2 S; 0 1

B ¤ 0:

Call this element w1 . Then w1 w ¤ w and w1 w 2 R.Z/. If x 2 I and y 2 S with y ¤ w, then xy 2 Jrc . Thus, R.Z/ \ Jrc is a nonempty set containing I  .w1 w/. Notice that when w 2 S , w 2 m.S /, and w1 w 2 m.S /. In other words, m.S / always contains an element other than w. Since I Œm.S /  fwg  m.S / \ Jrc ; it follows from Theorem 4.1 that     m S \ Jrc D m.S / \ Jrc D R.W / \ Jrc : Let us now show that R.Z/  m.S /. Let x 2 R.Z/  m.S / if possible. Since m.S / is closed, there is an open set V containing x such that V is compact and V \ m.S / D . Exactly as in the proof of Theorem 4.3, we then have V \ IR ¤ :

(4.27)

(To establish this, we need to observe only that n .J / D 0 for all n  1. We can verify that I is a nonempty open ideal of S  J , and as such, lim n .I / D 1.) n!1

Then it follows from (4.27) that there exists z1 2 I , z2 2 R such that z1 z2 2 V . Since V \ m.S / D , z2 must be different from w. This means z1 z2 2 Jrc \ R \ V : In other words,     R.Z/  m.S /  cl R \ Jrc D cl m.S / \ Jrc  m.S /; which is a contradiction. This proves R.Z/  m.S /:

(4.28)

268

4 Random Matrices

By Lemma 4.2, m.S / is a completely simple subsemigroup of S . Now we claim that the rank of the matrices in m.S / is 1. Suppose this rank is greater than 1. In this case when this rank is greater than 1, we can verify that I  m.S /  I  m.S / \ I:

(4.29)

Thus, m.S / \ I ¤ . Let x 2 m.S / \ I . Since m.S / is a completely simple subsemigroup, x 2 xm.S /x and xm.S /x is a group with an identity. Let e be this identity. By (4.29), e 2 e  m.S /  e D x  m.S /  x  I \ m.S / and since e is an idempotent, it must then take the form   A0 0 ; 0 1 where A0 is a strictly positive d  d matrix. The reason is the following: If for some i , 1  i  d , ei;d C1 > 0, then for some other j , 1  j  d , ej;d C1 D 0 means that ej i > 0 (since e is in I ). Since e D e 2 , ej;d C1  ej i ei;d C1 > 0: Then, either the .d C 1/th column of e is strictly positive, in which case e must have rank 1 (since e is an idempotent), or else the first d elements on the .d C 1/th column of e are all zeros. We assumed e has rank greater than 1 and e 2 I ; it follows that e must have the preceding form. Since there exists y 2 xm.S /x such that xy D e, it is clear that x must have the form   C 0 : 0 1 Thus, every matrix in m.S / \ I has this form where C ¤ 0. But we assumed that there is an element in S of the form   AB ; B ¤ 0: 0 1 Notice that



AB 0 1

    C 0 AC B D y 0 , say; 0 1 0 1

which is an element of m.S / (and where AC cannot be zero), if   C 0 2 m.S /: 0 1

4.2 Recurrent Random Walks in Nonnegative Matrices

Notice that taking

we see that

269

  A1 B1 in I; 0 1 

   A1 B1 A1 B1  y0  2 m.S / \ I 0 1 0 1

and this element is not of the form   C 0 ; 0 1 since the matrix B in y 0 is nonzero. This contradicts the form of the elements in m.S /. Thus, we have proven that the rank of the matrices in m.S / is 1 and consequently every matrix in m.S / has the form   0B : 0 1 Thus for x, y 2 m.S /, xy D x, and consequently since R.Z/ is a left ideal, it follows from (4.28) that R.Z/ D m.S /. Now as we pointed out earlier in (4.23), R.Z/ ¤  implies that Condition (i) in Theorem 4.4 holds. Also notice that the element w1 w (considered earlier in the proof ) ¤ w, w1 w 2 m.S /, and I  w1 w  I  I \ m.S /: Since the matrices in m.S / have the form   0B ; 0 1 it is clear that matrices in I \ m.S / have the same form   0B 0 1 with B strictly positive. Consequently, Condition (ii) in Theorem 4.4 holds also. It follows then that .n / is tight. Thus, the weak limit n 1X k  n!1 n

 D lim

kD1

270

4 Random Matrices

exists, and we also have: u  D   D  and S./ D m.S /: t u

(See Theorem 2.7.)

Remark 4.2. Under the assumptions of Theorem 4.5, for any bounded, real, contin d  d uous function f on RC and any u in RC , Z n 1X f .Wk .u// D f d n!1 n lim

kD1

almost surely, where  D  ıu . (The proof is deferred to Sect. 4.4, where we present a law of large numbers for the more general mixed random walks.) Now we present conditions when R.Z/ ¤  actually implies R.Z/ D R.W / D m.S /. Theorem 4.6. Let Zn , Wn , S ,  be as in Theorem 4.5. Let us replace the sets I , Jc , and Jr in Theorem 4.5 by the sets I  , Jc and Jr , respectively, which are defined as follows:  

AB I D 2 S W A is strictly positive ; 0 1  

AB  Jc D 2 S W A has at least one zero column . Jc /; 0 1  

AB Jr D 2 S W A has at least one zero row . Jr /: 0 1 Suppose there exists

  AB 2 S with B ¤ 0 and 0 1 I  ¤ ;

   Jc [ Jr D 0:

(4.30)

Then R.Z/ ¤  ) R.Z/ D R.W / D m.S /: Proof. Suppose R.Z/ ¤ . Then as in the proof of Theorem 4.5, R.Z/ \ Jrc is a nonempty subset of m.S /, and thus, using Theorem 4.2, we have: R.W / \ m.S / ¤ : It also follows that m.S / is a completely simple subsemigroup of S .

4.2 Recurrent Random Walks in Nonnegative Matrices



271

 

Since I  is an open ideal of S  Jc [ Jr , it follows from arguments used earlier that   lim n I  D 1: (4.31) n!1

If the rank of the matrices in m.S / is more than 1, there does not exist an   AB 0 1 in S for which A D 0. Then we can easily verify that I   m.S /  I   I  \ m.S /: Using this inequality instead of (4.29), it follows easily as in the proof of Theorem 4.5 that the rank of the matrices in m.S / must be 1 and also R.Z/ D m.S /  R.W /:

(4.32)

Now let x 2 R.W /, x … m.S /. Again following an argument similar to the one used earlier and using (4.31), we can prove that for any open set V containing x such that V \ m.S / D ; we must have

V \ RI  ¤ :

Thus, there exist y 2 R, z 2 I  such that yz 2 V . Since V \ m.S / D ; y does not have the form

  0B : 0 1

Therefore, as is easily verified, yz 2 Jcc \ R (since R is an ideal y 2 R and z 2 I  ). It follows that   V \ R \ Jcc is nonempty. This means that     x 2 cl R \ Jcc D cl R.Z/ \ Jcc  R.Z/: This contradicts (4.32).

t u

272

4 Random Matrices

Let us now consider the sets R.Z/ and R.W / in the special case when  is a probability measure on the set of nonnegative 2  2 matrices of the form   ab : 01 As before, let X0 ; X1 ; : : : be a sequence of i.i.d. nonnegative random matrices of the form   ab : 01 Let Zn X0 X1 : : : Xn ; Wn X n : : : X 1 X 0 : Write: Xn

  n n : 0 1

Let  be the distribution of X0 and S the closed (with usual topology) multiplicative semigroup generated by S./, the support of . The case when  

a0 S Wa0 01 can easily be taken care of, since S can then be regarded as a closed multiplicative semigroup of Œ0; 1/. When S

 

0b Wb0 ; 01

it is then immediate that S D S./ D R.Z/ D R.W /; since then for x, y 2 S , x  y D x. Let us now consider the case when  

ab S\ W a > 0; b > 0 ¤ : 01

(4.33)

Let Jr (respectively, Jc ) as usual be the set of those matrices in S that have at least one zero row (respectively, one zero column). Let Jc [ Jr J . Suppose first that .J / > 0: (4.34)

4.2 Recurrent Random Walks in Nonnegative Matrices

273

Then we observe that J D

 

0b 2S jb0 ; 01 1 X

n .J c / < 1:

nD1

If K is compact, K  J and .K/ > 0, then since K  S D K, it follows that for all positive integers n, n .K/  .K/ > 0: It also follows that R D R.Z/ D R.W / D J and every point in J is positive recurrent. Now suppose .J / D 0: If J D , then

(4.35)

 

ab S W a > 0; b  0 : 01

In this case, m.S / D S and by Theorem 4.2, R D R.Z/ D R.W /: If R ¤ , m.S / D S is a completely simple semigroup that must be a group (since S is cancellative). In this case, the inverse of 

ab 01



can exist only when b D 0, and then it is   1=a 0 : 0 1 In other words, since S is closed, S must be a singleton (the identity matrix). Let us then assume (4.33), and also, let .J / D 0 and J ¤ : In this case, m.S / D

 

0b 2S Wb0 ; 01

(4.36)

274

4 Random Matrices

and by Theorem 4.5, R.Z/ ¤  ) R.Z/ D m.S /:

(4.37)

Suppose now that (4.33) and (4.36) hold and also R ¤ :

(4.38)

Since by Theorem 4.1, R \ Jcc D R.Z/ \ Jcc ; it is clear from (4.37) that R  m.S /. Since R is an ideal, for x, y in m.S /, x  y D x; If m.S / ¤

R D m.S /:  

00 ; 01

then R \ Jrc ¤ . By Theorem 4.1, R \ Jrc D R.W / \ Jrc ; so that

 

00 m.S /  R.W /  m.S /  : 01

(4.39)

Note that by assumption (4.33), there exist a > 0, b > 0 such that   ab 2 S; 01 Since

     ab 00 0b D ; 01 01 01

it is clear that m.S / ¤

 

00 01

so that R.W / ¤ . We now claim that under (4.33), (4.36), and (4.38), R D R.W / D m.S /: To prove (4.40), we need to show only that when   00 2 S; 01

(4.40)

4.2 Recurrent Random Walks in Nonnegative Matrices

275

then it must be in R.W /. Let us write:   00 w ; 01

  an bn wn : 0 1

First suppose w is not an isolated point in S . Then there exist wn 2 S such that .an ; bn / ¤ .0; 0 and wn ! w as n ! 1. Now there exists b > 0 such that   0b 2 m.S /: 01 Since

     an bn 0b 0 an b C bn D ; 0 1 01 0 1

and also since .an ; bn / ¤ .0; 0/, an b C bn ¤ 0, and therefore by (4.39), for each n  1,   0 an b C bn 2 R.W /: 0 1 Since R.W / is closed and an b C bn ! 0, it follows that w 2 R.W /. Now suppose w is an isolated point in S . Now w  S D fwg; therefore, there is a positive integer k such that k .fwg/ > 0 and n .fwg/  k .fwg/ for n  k: It follows that w 2 R.W /. This proves (4.40). Recall that     Xn n n : 0 1 Let us now mention that the assumptions P .0 D 0/ D 0; 1  E .log 0 / < 0;

(4.41) (4.42)

E log max f0 ; 1g < 1;

(4.43)

imply R.Z/ ¤ ;

R.Z/ D m.S / and .n / is tight:

(4.44)

Also, under (4.41), 0 < E .log 0 /  1 ) R.Z/ D R.W / D ;

(4.45)

and E .log 0 / > 1 and R.Z/ ¤  ) E log max f0 ; 1g < 1:

(4.46)

276

4 Random Matrices

The proof of (4.45) is immediate by the law of large numbers, since E .log 0 / > 0 implies  P lim 0 1 : : : n D 1 D 1: n!1

Proofs of assertions (4.44) and (4.46) are deferred until Sect. 4.3, where they are part of Theorem 4.17. It is also relevant to point out (and very easy to show) that if E .log o / D 0; P .o D 1/ < 1; and P .0  "/ D 1 for some " > 0; then R.Z/ D . In the rest of Sect. 4.2, we look briefly into the connection of the set R.W / with what has been called an attractor in the literature.  d  d Let F be a family of affine maps from RC into RC such that each f 2 F can be represented as      f .x/ Af Bf x D : 1 0 1 1  d where x 2 RC , and the nonnegative .d C 1/  .d C 1/ matrix on the right has the form of (4.8). Notice that for f and g in F ,        f ı g.x/ Af Bf Ag Bg x D : 1 0 1 0 1 1 This means that if .Xn /, n  0, is a sequence of i.i.d. F -valued random variables with common distribution  and Wn D Xn Xn1 : : : X0 , then the sequence .Xn / can be considered a sequence of i.i.d. matrices of the form of (4.8) and the Wn s as products of such matrices.  d Let us now define the attractor A .u/, u 2 RC , as the set n  d A .u/ D y 2 RC W for every open Ny containing y;    P Wn  u 2 Ny infinitely often > 0 ; where Wn  u D Wn .u/ and Wn is the affine map Xn Xn1 : : : X0 . Using matrix notation, we can then write    d   y A .u/ D y 2 RC W for every open N y  containing y  D ; 1  

    u P Wn  2 N y infinitely often > 0 ; 1 where Wn is the product of Xn : : : X0 , Xi s now regarded as nonnegative .d C 1/  .d C 1/ matrices, as mentioned.

4.2 Recurrent Random Walks in Nonnegative Matrices

277

The next result is useful in identifying the attractor as the support of a -invariant probability measure, as we show in Theorem 4.7. Lemma 4.3. Let .Xn / be a sequence of d  d i.i.d. nonnegative matrices with com d mon distribution . Then for x 2 RC , x > 0 (that is, every entry of x is positive), A .x/ D R.W /  x . fC  x W C 2 R.W /g/ ; where R.W / is the recurrent set as defined earlier. Proof. Let x > 0, C 2 R.W /, and y D C x. Notice that given an open set N.y/ containing y, there exists an open set N.C / containing C such that N.C /x  N.y/. This means y 2 A .x/. For the converse, let y 2 A .x/. Let Nn .y/, 1  n < 1, be a countable open (and relatively compact) local base at y such that cl .NnC1 .y//  Nn .y/ 8n: For each n, define the set U .Nn .y// D fD W D is a d  d nonnegative matrix and Dx 2 Nn .y/g : Then U .Nn .y// is an open set; since x > 0, it can be verified that U .Nn .y// is relatively compact. Now notice that for each positive integer m, Pr .Wn 2 cl .U .Nm .y/// infinitely often/ > 0: Since U .Nm .y// is relatively compact, there exists for each m a matrix Cm in R.W / such that Cm 2 cl .U .Nm .y/// : Since for each m, CmC1 x  cl .Nm .y//  N1 .y/; it follows that the Cm s are bounded (since x > 0). If C0 is a limit point of these Cm s, then C0 2 R.W /, since R.W / is a closed set, and C0 x D y. t u Theorem 4.7 identifies the attractor as the support of a -invariant probability measure. Theorem 4.7. Let , S , .Zn /, .Wn /, I , Jc , Jr be as in Theorem 4.5. Let  be the -invariant probability measure in Theorem 4.5. Consider the hypothesis of this  d theorem. Then for u 2 RC , u > 0, the attractor A .u/, defined by n    d A .u/ D y 2 RC W for every open set N y  containing y  ;      P Wn u 2 N y  infinitely often is positive ;

278

4 Random Matrices

where y D

  y ; 1

u D

  u ; 1

n o  d turns out to be given by the set A y 2 RC W y  2 S./  u , except for a set with d -dimensional Lebesgue measure zero. With the hypothesis of Theorem 4.6, the attractor A .u/ is actually equal to the set A, which is the support of the -invariant probability measure  ıu , where ıu is the unit mass at u.  d Proof. Note that by Lemma 4.3, for u 2 RC , u > 0, fy  W y 2 A .u/g D fx  u W x 2 R.W /g : Under the hypothesis of Theorem 4.5, R.W / \ Jrc  m.S / D S  R.W /; and under the hypotheses of Theorem 4.6, R.W / D S , where n 1X k  ; n!1 n

 D lim

kD1

as in Theorem 4.5. The proof therefore follows immediately, since for x 2 R.W / \ Jr , x  u 2 fy  W yi D 0 for some i; 1  i  d g M; and

n

o  d y 2 RC W y  2 M

has d dimensional Lebesgue measure zero.

t u

Now we use the formula given in Lemma 4.3 and Theorems 4.5–4.7 to describe completely what is wellknown as a Sierpinski gasket. We also describe one of its unbounded versions. Even though the examples below are simple, they well illustrate the concepts. Sierpinski gasket. Consider the three functions on the plane given by f .x; y/ D

    x y xC1 y x yC1 ; , g.x; y/ D ; and h.x; y/ D ; : 2 2 2 2 2 2

4.2 Recurrent Random Walks in Nonnegative Matrices

279

Note that the semigroup (with respect to the composition of maps) generated by these three functions can be identified with the multiplicative semigroup generated by matrices (called by the same names, for convenience) given by 0 1 0 1 1=2 0 0 1=2 0 1=2 f D @ 0 1=2 0A ; g D @ 0 1=2 0 A ; 0 0 1 0 0 1 0 1 1=2 0 0 h D @ 0 1=2 1=2A : 0 0 1 Take any (this is quite interesting, though the logic is very simple) distribution  such that S./ D ff; g; hg. Then S , the smallest closed (with usual topology for matrices) multiplicative semigroup containing S./ is a compact semigroup. It is easy to see that m.S / consists of all rank 1 matrices in S . Since R.W / D m.S /, where S is a compact semigroup (see Chap. 3), it follows by Lemma 4.3 that for any point .x; y/, the attractor set A .x; y/ is given by 1 1 0 00a A A .x; y/ D @.a; b/ W @0 0 b A belongs to m.S /A : 001 0

Recall that m.S / is the set of all those matrices in S that have minimal rank (in this case, rank 1). Now compute m.S / to identify the set A as the Sierpinski gasket with vertices at .0; 0/, .0; 1/, and .1; 0/. Since f , g, h are substochastic, it is clear that .a; b/ 2 A ) fa; bg  Œ0; 1: Also notice that for any positive integer m, 0 1 0 1 a a=2m f m  @b A D @b=2m A ; 1 1 and

0 1 0 1 a a=2m C .1=2 C    C 1=2m/ A g m  @b A D @ b=2m 1 1

0 1 0 1 a a=2m hm  @b A D @b=2m C .1=2 C    C 1=2m /A : 1 1

Since m.S /  m.S / D m.S /, by the simplicity of m.S / as an ideal, m.S / maps any point .x; y; 1/ into the point .a; b; 1/, where .a; b/ is on or within the triangle 4 bounded by the lines a D 0, b D 0, and a C b D 1. Observe that h maps the points .x; y; 1/, where .x; y/ 2 4, to the points .a; b; 1/, where .a; b/ is on or within the triangle with vertices at .0; 1=2/, .1=2; 1=2/, and .0; 1/. g maps the same

280

4 Random Matrices

Fig. 4.1

set of points to the points .a; b; 1/, where .a; b/ is on or within the triangle with vertices at .1=2; 1=2/, .1=2; 0/, and .1; 0/. f maps those points to points .a; b; 1/, where .a; b/ is on or within the triangle with vertices at .0; 0/, .0; 1=2/, and .1=2; 0/. Since m.S / D m.S /m.S /m.S /, it follows that m.S / maps the points in R2  .1/ to the points .a; b; 1/, where .a; b/ is on or inside the triangle 4 but outside the interior of the triangle with vertices at .0; 1=2/, .1=2; 1=2/, and .1=2; 0/. Repeating this argument and noting that m.S / D Œm.S /m for every positive integer m, the attractor set is easily seen to be the known Sierpinski gasket (see Fig. 4.1) and its graph is the closure of the graph of E0 [E1 [E2 : : : , where E0 D .0; 0/. For n  0, EnC1 D f.a=2m ; b=2m C .1=2 C    C 1=2m // W m  1 and .a; b/ 2 En g [ f.a=2m C .1=2 C    C 1=2m/ ; b=2m / W m  1 and .a; b/ 2 En g [ f.a=2m ; b=2m / W m  1 and .a; b/ 2 En g : These formulas describe the attractor completely. Unbounded attractor for an expansive iterated function system. Consider the functions .f; g; h; k/ from the plane into itself, where the functions f , g, and h are the same as in the example of the Sierpinski gasket and functions k is given by k.x; y/ D .2x; y/. Take a probability distribution  such that S./ D ff; g; h; kg and such that the average contractivity (discussed in Example 4.7, condition (4.187) or any other condition that implies the tightness of .n /. As before, we identify these functions with the corresponding 3  3 matrices and let S be the smallest closed matrix semigroup (with usual topology and multiplication) containing S./. It is then immediate from the example of the Sierpinski gasket that the set m.S / of all matrices in S with minimal rank consists of those matrices in S with rank 1, and this set is given by 80 9 1 m  0 and < 0 0 2m  a = m.S / D @0 0 b A W .a; b/ 2 the Sierpinski triangle : : ; 00 1 of the previous example

4.2 Recurrent Random Walks in Nonnegative Matrices

281

Fig. 4.2

The attractor set A .x; y/ is the set (graphed in Fig. 4.2), the graph actually extending to infinity, and given by 8 9 m  0 and < = A .x; y/ D .2m  a; b/ W .a; b/ 2 the Sierpinski triangle : t u : ; of the previous example Other relevant results for attractors for general metric spaces are outlined in the problems.

Section 4.2 Exercises Exercise 4.1. Verify the inclusion (4.29). Exercise 4.2. Why is the set R.Z/, in Theorem 4.5, contained in the set R.W /? Exercise 4.3. Why does the condition R.Z/ ¤ ; imply in Theorem 4.6 that R.W / \ m.S / ¤ ;? Explain in details. Exercise 4.4. The statement S D S./ D R.Z/ D R.W / appears just preceding (4.33). Prove it. Exercise 4.5. Prove that in the presence of the condition (4.33) and (4.34) the fol1 P lowing inequality holds: n .J c / < 1. nD1

Exercise 4.6. Prove that under (4.33) and (4.34), the following is true: R D R.Z/ D R.W / D J , and also, every point in J is positive recurrent with respect to .Zn / as well as .Wn /. Exercise 4.7. Let X1 ; X2 ; : : : be a sequence of independent random variables with values in the interval Œ0; 1 such that P .Xi 2 B/ D i .B/ for any Borel set

282

4 Random Matrices

B  Œ0; 1, where 1 ; 2 ; : : : is a sequence of probability measures on the Borel subsets of Œ0; 1. Write: Xk;n D XkC1 : : : Xn for k < n. Show that Yk D lim Xk;n n!1 exists almost surely, and so does Y D lim Yk . Show also that either Y D 0 alk!1

most surely or Y D 1 almost surely, and that Y D 1 almost surely iff

1 P

Œ1  Xn 

nD1

converges almost surely. Exercise 4.8. Describe the attractor set A .x; y/ determined by the functions 

x y ; ; f .x; y/ D 2 2

g.x; y/ D

 x y C1 ; ; 2 2

 h.x; y/ D

 xC1 yC1 ; : 2 2

Exercise 4.9. previous exercise replacing the function f in it by the function  Do the y f0 .x; y/ D xC1 . ; 2 2 Exercise 4.10. Let S be a complete metric space with metric d and ffi W 1  i  ng be a finite family of contraction maps from S into itself (that is, there exists 0 < r < 1 such that d .fi .x/; fi .y//  rd.x; y/ for any x, y in S , 1  i  n). Show that there is a unique compact set K  S , called the attractor of the contractn S ing system ffi W 1  i  ng such that K D fi .K/. i D1

(Hint: Define for A, B in the family F of nonempty compact subsets of S the metric by .A; B/ D inf fk > 0 W A  Nk .B/; B  Nk .A/g ; where for C 2 F , Nk .C / D fy 2 S W d.y; x/ < k for some x in C g : Show that .F ; / is a complete metric space. This is called the Hausdorff metric. n S Next, define ˚ W F ! F by ˚.A/ D fi .A/. Then ˚ is a contraction with a i D1

unique fixed point K.) Exercise 4.11. Let .S; d / be a metric space and ˇ > 0. Define Fˇ by Fˇ D ff W S ! R j for all x; y 2 S; jf .x/  f .y/j  d.x; y/ and jf .x/j  ˇg : Consider the set P1 .S / of all regular (with respect to compact sets from inside) probability measures on the Borel subsets of S . For 1 , 2 in P1 .S /, define ˇ

ˇZ Z ˇ ˇ ˇ ˇ dˇ .1 ; 2 / D sup ˇ hd1  hd2 ˇ W h 2 Fˇ

4.2 Recurrent Random Walks in Nonnegative Matrices

283

Show that dˇ is a metric on P1 .S /. This metric is known as the Monge– Kantorovich metric. Exercise 4.12. In the previous exercise, show that if . in P1 .S /, R n / is a sequence R  2 P1 .S / and ˇ > 0, then n !  weakly iff hdn ! hd for each bounded real Lipschitz function h on S (that is, jh.x/  h.y/j  d.x; y/ for x, y in S and jh.x/j  M for all x in S for some M > 0) iff dˇ .n ; / ! 0.   Exercise 4.13. In Exercise 4.11, show that a Cauchy sequence in P1 .S /; dˇ , when S is complete, is tight. Thus, using Exercise 4.12 and the fact that B1 .S / D  f j  is a nonnegative regular measure, .S / not exceeding  1g is weak -compact, it follows that every subsequence of a Cauchy sequence in P1 .S /; dˇ must have a weakly convergent subsequence, which must also converge in the metric dˇ . This   means that P1 .S /; dˇ is complete for any ˇ > 0, when S is complete. Exercise 4.14. Let S be a complete separable metric space, and fi , 1  i  n, be n contraction maps from S into itself such that d .fi .x/; fi .y//  rd.x; y/, 0 < r < 1, 1  i  n, for all x, y in S . Let p1 ; p2 ; : : : ; pn be nonnegative numbers n P such that pi D 1. Then show that there is a unique invariant probability measure i D1

 on the Borel subsets of S such that for all Borel subsets A  S , .A/ D

n X

  pi  fi1 .A/ :

i D1

Show also that the support of  is the compact set K of Exercise 4.10. (Hint: Let fi .xi / D xi , 1  i  n, s D max d .xi ; x1 /, t D .1Cr/sC1 . Show 1r 1i n

that if u  t, then for each i , 1  i  n, fi .Bu .x1 //  Bu1 .x1 /, where Bu .x1 / D fy 2 S W d .y; x1 /  ug : Define ˚ W P .S / ! P .S / by ˚./.A/ D

n X

  pi  fi1 .A/ :

i D1

Show that for ˇ > t, for 1 , 2 in Bt , dˇ .˚ .1 / ; ˚ .2 //  r dˇ .1 ; 2 / :/

Exercise 4.15. Let S be a complete metric space and f1 ; f2 ; : : : ; fn be n contraction maps from S into itself such that d .fi .x/; fi .y//  rd.x; y/ for some r, 0 < r < 1, for all x, y 2 S , 1  i  n. Let K be the attractor given as in

284

4 Random Matrices

Exercise 4.10. Let X1 ; X2 ; : : : be i.i.d. random variables with values in ff1 ; f2 ; : : : ; n P fn g such that P .X1 D fi / D pi , 1  i  n, pi D 1. Let x0 2 S and i D1

xn D Xn Xn1 : : : X1 .x0 /. Show that

K D fx 2 S j P .Xn Xn1 : : : X1 .x0 / 2 N.x/ i.o./ D 1 for each open set N.x/ containing xg :

4.3 Tightness of Products of I.I.D. Random Matrices: Weak Convergence Consider the discrete time Markov process .n / on the state space Rd , d finite, where the transition rule is governed by a d  d random matrix Y with real entries, so that nC1 D YnC1 n ; n  0; (4.47) where Y1 ; Y2 ; : : : are an i.i.d. sequence of d  d random matrices (which are copies of Y ). We are interested in the limiting distribution of n as n ! 1, when it exists. Let  be the distribution of Y and ˇ that of 0 , so that  is a probability measure on d  d real matrices and ˇ on Rd . Let S be the closed (in usual matrix topology) multiplicative semigroup generated by the support S of ; in other words, ! 1 [ n S D cl S : nD1

Assuming the Yi s are independent of 0 , the distribution ˇn of n is given by ˇn D n ˇ;

n  1:

(4.48)

Here for any probability measure  on Rd , the convolution product   is defined by Z  .B/ D fA W Ay 2 Bg.dy/: Note that for 1 , 2 in P .S /, 1 .2 / D .1 2 / ; and the map .; / !   is continuous with respect to the weak topology. It is also clear that if ˇ D the unit mass at the zero vector, then for each n  1, ˇn is also (no matter what  is). In this context, Theorem 4.8 is interesting. Theorem 4.8. Suppose S contains only matrices with all entries and  nonnegative d at least one matrix with all entries strictly positive. Let ˇ 2 P RC such that

4.3 Tightness of Products of I.I.D. Random Matrices: Weak Convergence

285

is is not the unit mass at the zero vector. Then in (4.48), the sequence .ˇn / is tight if and only if the sequence .n / is tight. Proof. The “if” part is immediate since ˇn fA  x j A 2 B; x 2 C g  n .B/  ˇ.C /: For the only if part, let us suppose the sequence ˇn D n ˇ is tight. Let Jk be the set defined by

1 Jk D x D .x1 ; x2 ; : : : ; xd / W xi > ; i D 1; 2; : : : ; d : k By the theorem hypotheses, there is a positive integer m such that m .the strictly positive matrices in S / > 0:  d Notice that if A is a strictly positive d  d matrix, x 2 RC and x ¤ 0, then .Ax/i > 0 for i D 1; 2; : : : ; d . This means there exits ı > 0 such that n o  d m ˇ x 2 RC W xi > 0 for i D 1; 2; : : : ; d D 2ı:

(4.49)

Then there is a positive integer k such that   m ˇ Jk > ı:

(4.50)

 d Notice that if C is a compact subset of RC , then the set C J 1 k defined by ˚  C J 1 k D A 2 S W A  x 2 C for some x 2 Jk is relatively compact. Let us now suppose the sequence .n / is not tight. Then we have   D 1  2u < 1: sup lim inf n C J 1 k n!1 d C .RC / ;C compact

(4.51)

 d It follows from (4.50) and (4.51) that for any compact subset C  RC , Z   nCm

ˇ.C / D n C x 1 m ˇ.dx/       .1  u/m ˇ Jk C m ˇ Jkc        m ˇ Jk C m ˇ Jkc  u  m ˇ Jk  1  uı < 1; for infinitely many n. This contradicts the assumption that .n ˇ/ is a tight sequence. t u

286

4 Random Matrices

Let us now give a convergence theorem for the sequence .ˇn / in (4.48). We write ı0 to denote the unit mass at the zero vector in Rd as well as the unit mass at the zero matrix, that is, the matrix with all entries zeros. (Its meaning should be clear from the context.) Theorem 4.9. Let S consist of only d  d nonnegative matrices (that is, matri d ces with nonnegative entries). Let ˇ 2 P RC and suppose .n / is tight. If S contains the zero matrix, then n converges weakly to ı0 , and consequently ˇn converges weakly to ı0 . When S does not contain the zero matrix and S contains a strictly positive matrix, then S is necessarily compact and the sequence n converges weakly to some  2 P .S / such .J / > 0, where J is the set of all strictly positive matrices in S . When ˇ ¤ ı0 , the sequence ˇn converges weakly to some  d  2 P RC such that o n  d  x 2 RC W xi > 0 for 1  i  d > 0: If S contains a matrix with rank 1 instead of a strictly positive matrix, it remains true that the sequence n converges weakly. Proof. Suppose .n / is tight. By Theorem 2.7 (Chap. 2), the sequence n 1X k  n kD1

converges weakly to some  in P .S / and S , the support of , is the completely simple minimal ideal of S with a compact group factor. If 0 2 S , then S D f0g and  D ı0 . It follows from Theorem 2.7(iii) that n converges weakly to ı0 . Now suppose 0 … S and S contains a strictly positive matrix. Let  be as just defined. Then for n  1, n  n D :

(4.52) ˚  Since J  J  S  J A1 A2 A3 W A1 and A3 belong to J and A2 2 S , it follows from (4.52) that .J /  n .J /.S /n .J /;

n1

so that .J / > 0. Thus, S \ J ¤ ;. Let x 2 S \ J . Since S is completely simple, the set xS x is a compact group (see Theorems 1.7 and 2.7(i)). Let e be the identity of this group. Since x 2 J and 0 … S , it follows (by direct multiplication of matrices) that e 2 xS x  J:

4.3 Tightness of Products of I.I.D. Random Matrices: Weak Convergence

287

Since eSe D eS e D xS x is a compact group of strictly positive matrices, it is a singleton set (see Corollary 1.8). Thus, there exists M > 0 such that if A 2 S and 1  i  d , 1  j  d , then ei i Aij ejj  .eAe/ij  M: Since e 2 J , it follows that S is compact. It follows from Theorem 2.7 that n converges weakly to  (since eSe is a singleton set). To prove the second part, observe that when 0 … S and S contains a rank 1 matrix, then S (the minimal ideal of S ) consists of all rank 1 matrices in S (see Proposition 1.9). Then eSe, a compact group of nonnegative rank 1 matrices, is necessarily a singleton. The convergence of n then follows from Theorem 2.7. t u Theorem 4.10 (essentially due to Kesten and Spitzer [124]) deals with the problem of weak convergence of .n /. Consider the following condition: .J / D 0;

(4.53)

where J D Jc [ Jr , Jc (respectively, Jr ) is the set of all d  d matrices in S with at least one column (respectively, row) containing only zeros. Since (4.53) implies Z   C1 .J c / D n J c x 1 .dx/  n .J c /  .J c / ; Jc

it follows from (4.53) that for n  1, n .J / D 0:

(4.54)

Also if S contains only nonnegative matrices and S \ J ¤ ;, where J is the set of all strictly positive matrices, then since J is an ideal of J c , it follows from (4.53), (4.54), and Proposition 2.7 that n .J / " 1:

(4.55)

Note that J is an open set in S and thus, for some positive integer m  1, m .J / > 0. Theorem 4.10. Let  be a probability measure on nonnegative d  d matrices and S the closed multiplicative semigroup generated by S . Suppose S \ J ¤ ; and (4.53) holds. Then the following statements are equivalent: (i) n !  2 P .S / weakly and S \ J c ¤ ;. (ii) For each u > 0, there exists ku > 0 such that for each positive integer n, n .Bu / > 1  u, where



Bu D A 2 S W min Aij  ku ; i;j

288

4 Random Matrices

and there exists v>0 and a subsequence .nt / such that for t  1, nt .Av / > v, where

Av D B 2 S W max Bij > v : i;j

n

n

(iii) . / is tight and  does not converge weakly to ı0 , the unit mass at the zero matrix. (iv) S is compact and also the second condition in (ii) holds. Proof. We prove only that (ii) ) (iii). The rest of the proof can easily be completed following arguments in Theorem 4.9. Assume that (ii) holds. Write:

1 Jk D A 2 S W Aij  for all i; j : k

Let " > 0. From (4.55), there exists a positive integer N and an integer m  1 such that   N Jm > 1  ": (4.56) Let .Xi / be i.i.d. matrices in S with distribution . Write: Yn X 1 X 2 : : : X n : Notice that for any positive integer s, ˚

Y2N Cs 2 Bu; YN 2 Jm ; XN CsC1 : : : X2N Cs 2 Jm ˚    XN C1 : : : XN Cs 2 Jm1 Bu Jm1



and   ˚  Jm1 Bu Jm1 D C W ACB 2 Bu for some A; B in Jm

2  C W max Cij  ku m : i;j

It follows that



s C W max Cij  ku m2  1  2"  u: i;j

Remark 4.3. Under the conditions of Theorem 4.10, we can easily show that the following statements are also equivalent: (i) (ii) (iii) (iv)

.n / is tight and 0 … S ; n converges weakly to some  2 P .S / and .J / D 1; S is compact and 0 … S ; m.S / is bounded and m.S / ¤ f0g, where m.S / D fA 2 S W rank A  rank B for all B in S g:

4.3 Tightness of Products of I.I.D. Random Matrices: Weak Convergence

289

Note that the condition (iv) implies that 0 … S and as such, J \ m.S /  .J \ S /m.S /.J \ S /: It is also easily shown that (iv) implies m.S / is a compact ideal of S and it has a completely simple kernel (see Theorem 1.6), which must be itself (see Proposition 1.9). Thus if x 2 J \ m.S /, then xm.S /x, contained in J , is a compact group containing an identity e, and eSe D e m.S /e is compact (in fact, a singleton). As in Theorem 4.9, it follows immediately that S is compact. It is clear from the preceding discussion that the question of when the sequence .n / is tight is fundamental in the context of weak convergence. Theorem 2.12 characterizes this tightness property. As we show below, we can be more specific in the context of matrices. First we need a general lemma (which can be easily proven following the proof of Theorem 2.12). Lemma 4.4. Let .n / be a sequence of probability measures on a locally compact Hausdorff second-countable topological semigroup. Suppose there is a closed subset F such that lim n .F / D 0. If for every open set W  F , lim n .W / D 1, n!1

n!1

then the sequence .n / is tight. We now present Theorem 4.11 characterizing the property of tightness of products of i.i.d. random matrices. Theorem 4.11. Let  be a (Borel) probability measure on d  d real matrices with usual topology and support S . Let S be the closed multiplicative semigroup generated by S . Let m.S / be the set defined by m.S / D fx 2 S W rank x  rank y for all y 2 S g: Let a be the rank of the matrices in m.S /. Then the following results hold: (i) Suppose a D d . Then the sequence of convolution powers .n / is tight if and only if S is a compact group. (ii) Suppose a D 0. Then the sequence .n / is tight if and only if n converges weakly to ı0 , the unit mass at the zero matrix. (iii) Suppose 0 < a < d . Then the sequence .n / is tight if and only if the following two conditions hold: a) There is a compact group G of invertible a  a matrices and an invertible d  d matrix y such that for any x in S , the matrix y 1 xy can be uniquely represented in the form y 1 xy D



 A BD ; DC D

(4.57)

290

4 Random Matrices

where D is an element of G , C is an a  .d  a/ matrix, B is a .d  a/  a matrix, and A is a .d  a/  .d  a/ matrix; b) For any open set V containing the set of matrices given by  ˇ

BDC BD ˇˇ there exists A such that the matrix M D ; DC D ˇ in (4.57) is an element of y 1 Sy   lim n yVy 1 D 1: n!1

  In the only if case, the set M coincides with the set m y 1 Sy . Therefore when the sequence .n / is tight, .n / converges weakly if and only if there does not exist a proper normal subgroup H of G0 , where G0 is the compact group with identity e0 of d  d matrices given by 



0 0 WD2G 0D

with G as in Condition a), such that     E e0  y 1 Sy E y 1 Sy  e0  H and

  e0 y 1 S./y e0  gH

for some g 2 G0  H . (Here, E.W / stands for the set of all idempotent elements in the set W .) Proof. Let us prove (i). The “if” part is obvious. For the “only if” part, notice that from Theorem 2.7, n 1X k  converges weakly to a probability measure ; n kD1

  D   D  D  : Here, the support S of  is the minimal (completely simple) ideal of S . If e is an idempotent matrix in S , then e  S  e is a compact group. Since matrices in S are assumed to be of full rank and S is an ideal of S , it follows that e is the usual identity matrix and S D S  e  S D e  S  e: Let us prove (ii). The “if” part is obvious. For the “only if” part, notice that when 0 2 S , the support S , the minimal ideal of S , must be the singleton f0g, so that  D ı0 . Then the weak convergence of n to  follows from Theorem 2.7. For the “if” part of (iii), suppose that Conditions (a) and (b) hold. If for each positive integer k, k .m.S // D 0;

4.3 Tightness of Products of I.I.D. Random Matrices: Weak Convergence

291

then it follows from Lemma 4.4 that the sequence .n / is tight. Note that condition (b) is used here. Let us then suppose that there is a positive integer k such that k .m.S // > 0: Since m.S / is an ideal of S , it follows from Chap. 2 that lim n .m.S // D 1:

n!1

(4.58)

By Condition (a), there is an invertible matrix y such that for each x in S , y 1 xy has representation given in (4.57). Let us define the set S1 and the measure  as follows: ˚ 

S1 D y 1 xy j x 2 S ;  1 (4.59)  y By D .B/; B  S: It is easily verified that for n  1,   n y 1 By D n .B/ and also

m .S1 / D y 1  m.S /  y:

It follows from (4.48) that lim n .m .S1 // D 1:

n!1

(4.60)

it is also clear that the sequence .n / is tight if and only if the sequence .n / is tight. Let " > 0. Then it follows from (4.59) that there is a positive integer N and a compact subset A  m .S1 / such that N .A / >

p 3 1  ":

(4.61)

By Condition (a), every matrix in S1 has the form in (4.57); that is, it looks like 

 A BD ; DC D

(4.62)

where D is an element of the compact group G of a  a matrices. Note that the matrix D itself has rank a. Thus if the matrix in (4.62) belongs to m .S1 /, then it has rank a and it must look like   BDC BD ; (4.63) DC D

292

4 Random Matrices

where D 2 G . In fact every matrix in m .S1 / has the form (4.63). Now we claim that the set A  m .S1 /  A fx1  x2  x3 W x1 ; x3 2 A and x2 2 m .S1 /g is a compact subset of S1 . To prove this claim, let us consider   B1 D1 C1 B1 D1 2A; D1 C1 D1

  B2 D2 C2 B2 D2 2A D2 C2 D2

  BDC BD 2 m .S1 / : DC D

and

Notice that 

where

   B1 D1 C1 B1 D1 BDC BD B2 D2 C2 B2 D2 D1 C1 D1 D2 C2 D2 DC D      B1 D1 C1 B1 D1 BD C2 BD D D1 C1 D1 D  C2 D    B1 D  C2 B1 D  D ; D  C2 D 

(4.64)

D  D D .CB2 C Ia / D2 2 G ;

Ia the a  a identity matrix, and D  D D1 .C1 B C Ia / D  2 G : Now observe that the sets ˇ 

ˇ BDC BD ˇ Bˇ 2 A for some D 2 G and for some C DC D and



ˇ 

ˇ BDC BD ˇ Cˇ 2 A for some D 2 G and for some B DC D

are both compact subsets in the usual topology of .d  a/  a and a  .d  a/ matrices, respectively, since A is a compact subset of d  d matrices. The reason is the following: For every matrix B in the first set, entries in the matrix B  D ( B1 , say), which is the block in the upper right-hand corner of the matrix   BDC BD ; DC D

4.3 Tightness of Products of I.I.D. Random Matrices: Weak Convergence

293

which belongs to A , must be bounded, since A is a compact set. Since the matrix D belongs to the compact group G of matrices with full rank, it is clear that entries in all such matrices B D B1 D 1 must be bounded. A similar argument applies to the second set. This observation along with the form of the product in (4.64) implies that the set A  m .S1 /  A is a compact subset of S1 . This establishes our claim and consequently it follows from (4.60) and (4.61) that lim nC2N .A  m .S1 /  A /

n!1

 lim N .A /  n .m .S1 //  N .A / n!1

> 1  ": It follows that the sequence .n / and therefore the sequence .n / is tight. Let us now prove the “only if” part in (iii). Assume the sequence .n / is tight. Then the sequence n 1X k  n kD1

converges weakly to some probability measure  such that S , the support of , is the (completely simple) minimal ideal of S , and consequently S D m.S /. Now we must exploit the algebraic structure of m.S /. Note that m.S / is a completely simple subsemigroup, and thus, it has at least some idempotent e. Then e has rank a, and there is an invertible d  d matrix y such that y

1

  0 0 ; ey D 0 Ia

(4.65)

where Ia is the a  a identity matrix. Let x 2 S . Write y 1 xy as follows: y

1



 AB xy D ; C D

(4.66)

where D is a  a, B is .d  a/  a, A is .d  a/  .d  a/, and C is a  .d  a/. Now since m.S / is completely simple and the support of an idempotent probability measure , the set eSe D e m.S /e is a compact group (see Theorem 2.7). Observe that       AB 0 0 0 0 0 0 1 y .exe/y D D : C D 0 Ia 0D 0 Ia Let ˇ

  ˇ A B 1 ˇ G D D ˇ there exist A; B; C as in (4.66) such that 2 y Sy : C D

294

4 Random Matrices

Then it is clear that G is a compact group of a a matrices. Now recall from Chap. 1 that m.S / D E.m.S /e/  Œe m.S /e  E.e m.S //: (4.67) where E.W / denotes the set of all idempotent elements in the set W . A typical element in y 1 Œm.S /ey has the form    y 1 .xe/y D y 1 xy y 1 ey D



AB C D

    0B 0 0 D : 0D 0 Ia

If this element is idempotent, then        0B 0B 0B 0 BD D D 0D 0D 0D 0 D2 so D must be Ia . Thus, elements in y 1  E.m.S /e/y   0B : 0 Ia

have the form

Similarly, elements in have the form

y 1 E.e m.S //y  0 0 : C Ia



Then it follows from (4.67) that a typical element in y 1 m.S /y has the form       0B 0 0 0 0 BDC BD D ; 0 Ia 0D DC D C Ia

(4.68)

where D belongs to the compact group G . Consider an arbitrary element in y 1 Sy. Let it be y 1 zy, which looks like   A1 B1 ; C1 D1

D1 2 G ;

expressed as in (4.66). Then notice that y 1 .ze/y 2 y 1 m.S /yI

(4.69)

4.3 Tightness of Products of I.I.D. Random Matrices: Weak Convergence

therefore,

295

     A1 B1 0 0 0 B1 D C1 D1 0 D1 0 Ia

must have the same form as in (4.68), so that B1 D BD1 for some .d  a/  a matrix B. Notice that this B is unique when B1 and D1 are given, since D1 2 G and G is a group. Similarly, y 1 .ez/y 2 y 1 m.S /y: Therefore,

     A1 B1 0 0 0 0 D C1 D1 C1 D1 0 Ia

must have the form in (4.68). This means that C1 D D1 C for some unique a  (d-a) matrix C . Thus, it follows from (4.69) that every matrix in y 1 Sy has the form   A1 BD1 ; (4.70) D1 C D1 where D1 is an element of the compact group G of a  a matrices and (4.70) has the same form described in (4.66). The proof of (iii) is complete once we show that whenever the matrix   A BD ; DC D with the same form as in (4.70) is in y 1 Sy, then the matrix   BDC BD DC D must also be an element of y 1 Sy and therefore in y 1 m.S /y. To prove this part, let us take x in S such that   A BD 1 y xy D DC D with the same form as in (4.70). Notice that eSe D e m.S /e is a group and as such, .exe/1 , the inverse of exe in this group, belongs to S . If   z y 1 x.exe/1 x y; 

then zD

A BD DC D

     0 0 A BD BDC BD D : 0 D 1 DC D DC D

296

4 Random Matrices

Thus, we have proven that the set m.S / consisting of all matrices in S with the minimal rank coincides with the set yM y 1 . Since n 1X k  n kD1

converges weakly to the probability measure  and the support of  is m.S /, the Condition (b) now follows immediately from Theorem 2.7. t u Next we consider the case of 2  2 real matrices. In this case, as we show in Theorem 4.12, we can be more specific. Theorem 4.12. Let  be a (Borel) probability measure on 22 real matrices and S the closed (with respect to the usual topology) multiplicative semigroup generated by S./, the support of . Suppose the sequence .n / is tight and the rank of the matrices in m.S / is 1. (When this rank is 0 or 2, exactly what happens is clear from Theorem 4.9.) Suppose also that m.S / does not contain a group of the form f1; 1g. Then either there is a common right nonzero eigenvector for every matrix in S with common eigenvalue 1 or there is a common left nonzero eigenvector for every matrix in S with common eigenvalue 1. In particular, there is an invertible 2  2 matrix y such that in case of the first possibility y

1

 

a0 Sy  W a; c scalars : c1

In case of the second possibility, y

1

 

ab Sy  W a; b scalars : 01

Suppose now that m.S / contains a group of the form f1; 1g. Then either m.S / consists of exactly eight elements or there is a common right eigenvector for every matrix in S with eigenvalue 1 or 1, or there is a common left eigenvector for every matrix in S with eigenvalue 1 or 1. In case of the last two possibilities, there is an invertible 2  2 matrix y such that in case of the second possibility,  

a0 y 1 Sy  W a; c scalars and b D ˙1 ; cb and in case of the third possibility, y

1

 

ac Sy  W a; c scalars and b D ˙1 : 0b

4.3 Tightness of Products of I.I.D. Random Matrices: Weak Convergence

297

Proof. Suppose .n / is tight and the rank of the matrices in m.S / is 1. Then as in the proof of Theorem 4.11, m.S / is a completely simple subsemigroup of S with a compact group factor. Let e D e 2 be a fixed idempotent element of m.S /. Since rank.e/ D 1, there is an invertible 2  2 matrix y such that   00 y 1 ey D : 01 Consider the set

y 1 .m.S /e/y:

Since the rank of the matrices in y 1 m.S /y is 1, a typical element in the set y 1 .m.S /e/y has the form      a1 b1 a1 b2 00 0 a1 b2 D : a2 b1 a2 b2 0 a2 b2 01 If this is idempotent, then a2 b2 D 1, so the set   E y 1 m.S /ey D y 1 .E.m.S /e//y; where E.W / is the set of idempotent elements in W , consists of elements of the form   0a : 01 Similarly, the set

  E y 1 e m.S /y D y 1 E.e m.S //y 

 00 : b1

consists of elements of the form

Now suppose E.m.S /e/ ¤ e and E.e m.S // ¤ e:

(4.71)

Then there exist elements x1 , x2 in m.S / such that x1 e D .x1 e/2 , ex2 D .ex2 /2 , and   0a y 1 .x1 e/ y D ; a ¤ 0; 01   00 1 y .ex2 / y D ; b ¤ 0: b1 Notice that y

1



00 .ex2 x1 e/ y D b1

where ab C 1 ¤ 1, since ab ¤ 0.

    0a 0 0 D ; 01 0 ab C 1

(4.72)

298

4 Random Matrices

  00 But since y .e m.S /e/y is a compact group of matrices of the form , 0c where c is a nonzero scalar, it is clear that in (4.72), ab C 1 2 f1; 1g. Since ab ¤ 0, ab C 1 D 1 or ab D 2. It follows that if (4.71) does occur, then we must have 1



E y and

1



E y





.e m.S //y D

1

  

0 0 00 ; 2=a 1 01

(4.73)

   

0a 00 .m.S /e/y D ; 01 01 

for some a ¤ 0. It is also clear that when m.S / does not contain a group of the form f1; 1g, the element ab C 1 must be 1, so that ab D 0. This means that (4.71) cannot occur in this case. Observe that the set y 1 .e m.S /e/y, as a compact group must be either    

 

00 00 0 0 or ; : 01 01 0 1 As a consequence, since m.S / is completely simple, it follows easily that every element in m.S / is idempotent when y 1 .e m.S /e/y is a singleton. When this set is not a singleton, m.S / D m.S /, so that every element in m.S / is either an idempotent or the negative of an idempotent. Since we have the set equality m.S / D E.m.S /e/  e m.S /e  E.e m.S //; it follows that when (4.73) occurs, y 1 m.S /y consists of exactly the following eight elements; that is, there is a ¤ 0 such that y

1

      00 0 0 0a m.S /y D ; ; ; 01 0 1 01       0 a 0 0 0 0 ; ; ; 0 1 2=a 1 2=a 1    

2 a 2 a ; : 2=a 1 2=a 1

(4.74)

Now we look at the situation when m.S / does not contain a group of the form f1; 1g. In this case, (4.71) does not occur, and then we have either Se D m.S /e D E.m.S /e/ D feg;

(4.75)

4.3 Tightness of Products of I.I.D. Random Matrices: Weak Convergence

299

or eS D e m.S / D E.e m.S // D feg:

(4.76)

Suppose (4.75) holds. Then for any s 2 S , let y 1 sy D Then, y 1 .se/y D

  ab : cd

     ab 00 00 D cd 01 01

in which case, b D 0 and d D 1. This means that when (4.75) occurs y 1 Sy 

 

a0 W a; c scalars : c1

Similarly when (4.76) occurs, y 1 Sy 

 

ab W a; b scalars : 01

Finally, m.S / may contain a group of the form f1; 1g and (4.71) also does not occur. In this case, as is clear from the preceding, we must have either Se D m.S /  e D fe; eg or eS D e  m.S / D fe; eg: The rest of the details are clear and left to the reader.

t u

In Theorem 4.10, we studied weak convergence of the sequence .n / in nonnegative matrices. We now present a sufficient condition for weak convergence in real matrices. First, we need Lemma 4.5. Lemma 4.5. Suppose x D x 2 is a d  d real matrix with no zero row. Suppose there are two rank 1 d  d real matrices y and z such that xy D x and xz D x. Then we must have: y D y 2 , z D z2 , and yz D y. Proof. Notice that x has rank 1. Write x as     x D xij D ai xj ; where each ai is nonzero (since x has no zero row) and not all the xj s are zero. Write y as     y D yij D bi yj :

300

4 Random Matrices

Since xy D x, for 1  i , j  d we have ai xj D

d X

ai xk bk yj :

(4.77)

kD1

Since x D x 2 and x ¤ 0, d X

ai xi D 1:

(4.78)

i D1

It follows from (4.77) that d X

ai xi D

i D1

d X

!

d X



ai yi

i D1

! bk xk :

(4.79)

kD1

Canceling ai and then multiplying by bj in (4.77), we have bj xj D bj yj 

d X

bk xk

kD1

0

so that d X

bj xj D @

j D1

d X

1 bj yj A

j D1

d X

! bk xk :

(4.80)

kD1

It follows from (4.78)–(4.80) that d X

bk xk ¤ 0 and

d X

bj yj D 1:

(4.81)

j D1

kD1

This means y D y 2 . Similarly, we write z as     z D zij D ci zj : Since xz D x, it follows as before that z D z2 . It follows from (4.77) that yj D 0 ) xj D 0I also xj =yj D

d X kD1

bk xk :

(4.82)

4.3 Tightness of Products of I.I.D. Random Matrices: Weak Convergence

301

Let us now prove that yz D y. First notice that .yz/ij D bi zj 

d X

ck yk :

(4.83)

xj D 0, whenever zj D 0I

(4.84)

kD1

As in (4.82), we also have

and, xj =zj D

d X

ck xk ;

kD1

when zj ¤ 0. Suppose yj D 0. Then xj D 0 by (4.82). Since xz D x, d X

ai xj D ai zj 

bk xk :

kD1

Since by (4.81), d X

bk xk ¤ 0;

kD1

we have zj D 0. Then .yz/ij D 0 by (4.83). Thus when yj D 0,   .yz/ij D yij D bi yj : Now suppose yj ¤ 0. By (4.77) and (4.81), xj D yj 

d X

bk xk ¤ 0:

(4.85)

kD1

Therefore by (4.84), zj ¤ 0 and 0 ¤ xj D zj 

d X

ck xk :

(4.86)

kD1

It also follows from (4.77) that d X sD1

cs xs D

d X sD1

! cs ys



d X kD1

! bk xk :

(4.87)

302

4 Random Matrices

By (4.85)–(4.87), yj D zj 

d X

cs ys :

(4.88)

sD1

By (4.83) and (4.88), we have .yz/ij D bi yj D yij : This proves that yz D y.

t u

In Theorem 4.10, the strictly positive matrices played an important role. A similar role is played by the set of matrices of rank 1 in Theorem 4.13, which provides a sufficient condition for the weak convergence of .n / in real matrices. Theorem 4.13. Let  be a probability measure on d  d real matrices and let S be the closed multiplicative semigroup generated by S . Let I be the set of all matrices in S with rank 1. Suppose for every open set G  I , lim n .G/ D 1:

n!1

(4.89)

(Note that this condition holds when S does not contain the zero matrix and for some positive integer m, m .I / > 0, since then I is an ideal of S .( Suppose for some x .x1 ; x2 ; : : : ; xd /, not all xi s zero, x  S D x or S  x T D x T . (In other words, suppose x is a common left (or respectively, right) eigenvector for every matrix in S with common eigenvalue 1.) Then there is a probability measure  on S such that n !  weakly as n ! 1 and S  I . Proof. First, we show that .n / is tight. Let us assume that xS D x and x ¤ 0. Since xS D x, we have x  S D x so that S does not contain the zero matrix. If m .I / D 0 for every m  1, then since I is a closed subset of S , it follows from (4.89) and Lemma 4.4 that .n / is tight. Suppose now that for some positive integer m, m .I / > 0. It follows from Proposition 2.7 that lim n .I / D 1:

n!1

(4.90)

Since xS D x for the nonzero vector x, we can form a rank 1 d  d idempotent matrix x0 such that x is its first row; then x0 S D x0 . It follows from Lemma 4.5 that y 2 I and z 2 I ) y  z D y: (4.91) Let p > 0. It follows from (4.90) that there exists q > 0 and a positive integer N such that p n  N ) n .I / > 1  q > 1  p: (4.92)

4.3 Tightness of Products of I.I.D. Random Matrices: Weak Convergence

303

Let K be a compact subset of I such that N .K/ > 1  q. Let n  N . Then Z   nCN .K/ D N Ky 1 n .dy/ Z    N Ky 1 n .dy/ I

 N .K/n .I / > .1  q/2 > 1  p: (Notice that for y 2 I , Ky 1  K, since K  I and K  y D K by (4.91).) This proves that .n / is tight. By Theorem 2.7, n 1X k  n kD1

converges weakly to some  2 P .S /, where S is a complete simple (minimal) ideal of S with a compact group factor. Thus, S  I . Since a compact group of rank 1 d d matrices is either a singleton or a group of the form f1; 1g and since xS D x, it is clear that the group factor of S is a singleton. (If A is a rank 1 d  d matrix in the compact group factor of S , then if the i th row of A is .bi a1 ; bi a2 ; : : : ; bi ad /, the matrix AnC1 D ˛ n  A, where ˛ D b1 a1 C b2 a2 C    C bd ad , and consequently, j˛j must be 1 so that A2 D A or A.) It follows from Theorem 2.7 that n !  weakly. t u Let us now present some examples illustrating the above theory. Example 4.1. We show that Theorem 4.12 cannot be extended to d  d matrices with d > 2. Consider the semigroup S of 3  3 strictly positive matrices given by S D fe0 ; e1 ; e2 ; e3 g, where 0 1 0 1 1=2 1=6 1=3 3=4 1=12 1=6 e0 D @1=2 1=6 1=3A ; e1 D @3=4 1=12 1=6A ; 1=2 1=6 1=3 3=4 1=12 1=6 0 1 0 1 1=2 1=6 1=3 3=4 1=12 1=6 e2 D @ 1 1=3 2=3A ; e3 D @3=2 1=6 1=3 A : 1=4 1=12 1=6 3=8 1=24 1=12 Then the multiplication table for S is given by  e0 e1 e2 e3

e0 e0 e0 e2 e2

e1 e1 e1 e3 e3

e2 e0 e0 e2 e2

e3 e1 e1 e3 e3

304

4 Random Matrices

Consider the probability measure  on S given by  .ei / D

1 ; 4

i D 0; 1; 2; 3:

Then by Theorem 4.10, n converges weakly to some probability measure on S . However, unlike Theorem 4.12, there does not exist a nonzero common eigenvector for every matrix in S with common eigenvalue 1. Suppose there exists a nonzero vector x D .x1 ; x2 ; x3 / such that xei D x for i D 0; 1; 2; 3. Then we can choose a ¤ 0, b ¤ 0, and c ¤ 0 such that ax1 C bx2 C cx3 D 1 and 0 1 ax1 bx2 cx3 x0 D @ax1 bx2 cx3 A ax1 bx2 cx3 is an idempotent matrix and x0 ei D x0 , i D 0; 1; 2; 3. It follows from Lemma 4.5 that for y, z in S , yz D y. But this contradicts the multiplication table for S . Similarly, it can be shown that there is no nonzero x such that ei x T D x T ;

i D 0; 1; 2; 3:

Example 4.2. This example was considered in [124]. Consider the probability measure  such that S D fA; Bg, where A and B are the 2  2 matrices given by AD

  1p ; 0q

BD

  r0 ; s1

where p > 0, q > 0, r > 0, and s > 0. Assume 0 < q < 1, 0 < r < 1, and .1  q/.1  r/ D ps. Consider the vector  x D 1;

p 1q

 :

Then xA D x and xB D x. It follows that x  S D x, so S must be compact and S does not contain the zero matrix. Notice that AB is strictly positive, and it follows by Theorem 4.10 that n converges weakly to some probability measure  and the strictly positive matrices in S have  measure 1. It is relevant to point out that S \ J ¤ ;, since lim An 2 J and S D m.S /:

n!1

Example 4.3. This example illustrates Theorem 4.13. Consider the probability measure  such that S D fA; B; C g, where A, B, C are the 2  2 matrices given by   1p AD ; 0q

  r0 BD ; s1



 0 0 C D ; .1  q/=p 1

4.3 Tightness of Products of I.I.D. Random Matrices: Weak Convergence

305

where q ¤ 1, p, q, r, s are nonzero real numbers such that ps D .1  q/.1  r/. It is then easily verified that the vector  x D 1;

p 1q



satisfies the equations xA D x, xB D x, and xC D x. It follows by Theorem 4.13 then n converges weakly to some , where  (the rank 1 matrices in S ) D 1. Example 4.4. This example illustrates Theorem 4.13 when S is a noncompact semigroup of 2  2 nonnegative matrices. Consider the sets K1 and K2 defined by K1 D and

 

1m W m; n are nonnegative integers 0 n

 

1m W m is a nonnegative integer : K2 D 0 0

Consider a probability measure  on S0 D K1 [ K2 , which is a semigroup with respect to matrix multiplication such that    

11 10 ; : S0  S  01 00 Notice that if x D .1; 0/, then      1m 1 1 D 0 n 0 0 so that S0 x T D x T . Then it follows from Theorem 4.13 that n converges weakly to some probability measure  such that S D K2 . Now we appeal to Theorem 2.11 for an interesting result on tightness for nonnegative matrices. Proposition 4.1. Let  be a probability measure on the Borel subsets of d  d (0 < d < 1) nonnegative matrices. Suppose S is the closed multiplicative semigroup generated by S and S  J c ; that is, no matrix in S has a zero row or a zero column. If n .K/ does not converge to zero as n ! 1 for some compact set K, then the sequence .n / is tight. Proof. The proof follows immediately from Theorem 2.11 and the remark immediately preceding Theorem 2.8. t u As we saw in the preceding examples, we do not often have the situation when S  J c , and as such, the next result, due to Kesten and Spitzer [124], is interesting. The proof, not given in [124], is taken from Mukherjea [149].

306

4 Random Matrices

Theorem 4.14. Let  and S be as in Proposition 4.1. Instead of assuming S  J c , suppose .J / D 0 and S \ J ¤ ;; where J is the set of all d  d strictly positive matrices in S and J is the set of all d  d matrices in S that have at least one zero row or one zero column. Then either n converges to zero in the weak sense (or vaguely) in the nonzero matrices of S or the sequence .n / is tight. Proof. We prove slightly more than is necessary. Suppose there exist q1 > 0, q2 > 0, " > 0 and a subsequence .nk / of positive integers such that for k  1, nk .B .q1 ; q2 // > ";

(4.93)

where



B .q1 ; q2 / D A 2 S W max Aij  q2 and for each s, min Ai s  q1 : i

i;j

We show that if (4.93) and the assumptions .J / D 0;

S \J ¤ ;

(4.94)

hold, then the sequence .n / is tight. Step 1. There is a positive number q0 , a positive integer N , and a compact subset W  J such that for large k, nk CN .W / > q0 :

(4.95)

To prove (4.95), let " be as in (4.93). It follows from (4.94), (4.54), and (4.55) that there is a positive integer N and v > 0 such that   " N Jv > 1  ; 4 where



1 Jv D A 2 S W  Aij  v for every i; j : v

It follows from (4.93) and (4.96) that for nk > N , Z D

" < nk .B .q1 ; q2 //   nk N x 1 B .q1 ; q2 / N .dx/

  "  nk N Jv1 B .q1 ; q2 / C : 4

(4.96)

4.3 Tightness of Products of I.I.D. Random Matrices: Weak Convergence

307

It is easily verified that the product set   Jv Jv1 B .q1 ; q2 / Jv is contained in a compact set W  J . Notice that Jv1 B .q1 ; q2 /

q2  max Aij  q1 v  F .q1 ; q2 ; v/ D A 2 S W i;j vd and there exist a > 0, b > 0 such that ˚  Jv F .q1 ; q2 ; v/ Jv  A 2 S W a  Aij  b for all i; j : It follows that for nk > N ,       nk CN .W /  N Jv nk N Jv1 B .q1 ; q2 / N Jv >

.1  "=4/2  3" : 4

This proves (4.95). Step 2. If there exists q > 0, a compact set W  J , and a subsequence .mk / of positive integers such that k  1;

mk .W / > q;

(4.97)

then .mk / is tight, and moreover given " > 0, there are positive integers n0 and k0 and a compact set W"  J such that for k  k0 , mk Cn0 .W" / > 1  ":

(4.98)

To prove this, write for u > 0, 1  i  d , 8 9 d < X 1= Bu D A 2 S W u  Aij  : : u; j D1

Then we have

  sup lim sup n x 1 Bu D 1: u>0 n!1 x2S

(4.99)

To see this, notice that the set W in (4.97) is contained in some Bu and the set Bu Jv1 , Jv as in Step 1 is contained in Bw for some w > 0. Now let the sup

308

4 Random Matrices

in (4.99) be the number b. Suppose b < 1. Because of (4.97), b > 0. Let b < c < 1 and c.1 C c/=2 < b. Let u > 0. There exists a positive integer N , v > 0, and 0 < " < c=4 such that   N S  Jv < ";

  sup N z1 Bu < c  ": z2S

Let E D Bu Jv1 . Then E  Bw for some w > 0. For n > N and x 2 S , n





x

1



Z

  N y 1 x 1 Bu nN .dy/ Z Z C D x 1 E Sx 1 E h  1   c  i nN  " 1  nN x 1 E x E C  .c  "/ 2 c C c2 c nN  1  c < b; x E C    2 2 2

Bu D

which is a contradiction. This establishes (4.99). It follows from (4.97) and (4.99) that given 0 < " < q, there exist u > 0 and elements xn in S such that for all sufficiently large n and each k  1, mk .W / > q;

  " n xn1 Bu > 1  : 2

Thus, for sufficiently large k, 1 xm Bu \ W ¤ ; k

so that xmk 2 Bu W 1 Tu , say. We can now verify easily that there exist a" > 0, b" > 0 such that Tu1 Bu  L"  D" ;

(4.100)

where sets L" and D" are defined by

L" D A 2 S W max Aij  a" and min Aij  b" i;j

i;j

and

D" D A 2 S W min Aij  b" : i;j

It follows that given " > 0, there exists k0 such that for k  k0 , mk .D" /  mk .L" / > 1  ":

(4.101)

4.3 Tightness of Products of I.I.D. Random Matrices: Weak Convergence

309

Now we use the proof of (ii) ) (iii) in Theorem 4.10. Let the Xi s and Yi s be as in that proof. Let k  k0 . Then, for mk > 2N with N as before, we have  ˚ 1  2"  Pr Y2N C.mk 2N / 2 D" ; YN 2 Jv ; Xmk N C1    Xmk 2 Jv ˚    Pr YN 2 Jv ; XN C1    Xmk N 2 Jv1 D" Jv1 ;  Xmk N C1    Xmk 2 Jv     mk 2N Jv1 D" Jv1 : It follows as in the proof of Theorem 4.10 that the sequence .mk / is tight. It is easily verified that Jv  L"  Jv  J (4.102) so that (4.98) follows immediately from the tightness of .mk / and (4.102). This completes Step 2. Step 3. It follows from Steps 1 and 2 that when (4.93) occurs, the sequence .n / has a weak limit point in P .S / that is not the unit mass at the zero matrix. Thus, there is a subsequence .pk / of positive integers such that pk ! 0 ¤ ı0 weakly. Notice that for any positive integer j , given " > 0, taking compact sets E1  S and E2  S  J such that pk .E1 / > 1  "; we have

j .E2 / > 1  "

Z   1  " < pk .E1 / D pk j E1 x 1 j .dx/ Z   pk j E1 x 1 j .dx/ C "  E2

so that

  pk j E1 E21  1  2":

  Since E2  S  J , the set E1 E21 is compact, and consequently pk j is a tight sequence. Following the proof of Lemma 2.2, we observe that there is a further subsequence .tk /  .pk / such that for each j  1, tk j ! j 2 P .S /; weakly. We can also assume   that (4.98) holds with .tk / replacing .mk /. Now we show that the sequence tk is tight. Since for j  1, j j D j j D 0 :

310

4 Random Matrices

Then for k  1, we have tk tk D tk tk D 0 : Since (4.98) and (4.101) hold with .tk / replacing .mk /, it follows from (4.102) that given " > 0, there exists a compact set F  J and a positive integer N such that for k  1, tk C2N .F / > 1  "; 0 2N .F / > 1  ": (Note that 0  D  0 and because of (4.101), 0 .L" /  1  ".) Thus for k  1, Z   1  " < 2N 0 .F / D tk x 1 F tk C2N .dx/ Z Z   C  tk F 1 F C ": D F

SF

Using a similar argument, it follows that for k  1,   tk F 1 F \ FF 1 > 1  6":

(4.103)

Note that F 1F \ FF 1 is compact and contained in S  J . Thus, it follows that the sequence tk is tight and every weak limit point of tk is in P .S / and has zero mass for the set J . Nowfollowing the proof of Lemma 2.2, one observes that  if  is a weak limit point of tk , then  2 P .S /;

  D ;

.S  J / D 1

(4.104)

and there is a subsequence .sk /  .tk / such that skC1  sk ! 1 as k ! 1 and

skC1 sk !  weakly:

(4.105)

We show that .J / D 1. Write



rk D skC1  sk and Ev D A 2 S W max Aij > v : i;j

By (4.104) and (4.105), there are positive u, v, w such that for large k, rk .Ev / > 1  "; and

  N Ju > 1  "

  2N Jw > 1  "

4.3 Tightness of Products of I.I.D. Random Matrices: Weak Convergence

311

for some positive integer N . It is easily seen that Jw1 Ev  Ez for some z > 0 and for k large,   rk 2N Jw1 Ev > 1  2": Since the closure of the set Ju  Ez  Ju  J , it follows that given " > 0, there is a closed set D  J such that for large k, rk .D/ > .1  "/2 .1  2"/: It follows from (4.105) that .J / D 1. Now (4.93) gives (4.97) and by (4.97), there exists x 2 J such that for any open set U containing x, n .U / 6! 0: (4.106) Notice that if y 2 S  J such that x … Sy, then there exist open sets Ux , Uy such that x 2 Ux  S  J; y 2 Uy  S  J , and Ux Uy1 D ;: A similar result holds when x … yS . Following the proof of Theorem 2.11, it follows that if (4.106) holds for x 2 J , then x 2 S x and x 2 xS I

(4.107)

y 2 S x \ J c ) S x D SyI y 2 xS \ J c ) xS D yS:

(4.108) (4.109)

Let S1 S  f0g. Then xS1 x is a subsemigroup of J . Let y 2 xS1 x. Then yx 2 xS \ J c and xy 2 S x \ J c : It follows from (4.108) and (4.109) that y .xS1 x/ D xS1 x D .xS1 x/ y so that xS1 x is a group contained in J . Define the set K0 by   ˚  K0 D y 2 J W n Uy 6! 0 as n ! 1 for every open Uy containing y : Then K0 is a nonempty ideal of S \ J c . Since it is clear from (4.108) and (4.109) that for each y 2 K0 , S1 y is a minimal left ideal of K0 and yS1 is a minimal right ideal of K0 , it follows from Propositions 1.7 and 1.9 that the set m .K0 / of matrices with minimal rank from K0 is the completely simple minimal ideal of K0 . By (4.107), y 2 K ) y 2 yS1 \ S1 y: (4.110) Since the union of all minimal right ideals of K0 is the minimal ideal of K0 (see the proof of Proposition 1.7), it follows from (4.110) that m .K0 / D K0 .

312

4 Random Matrices

 To complete  the proof, consider the probability measure  in (4.105). Note that  J \ S D 1, so if x 2 J \S and Ux is an open set containing x,  .Ux / > 0. It follows from (4.105) that n .Ux / 6! 0 as n ! 1: It follows that J \S  K0 so  .K0 / D 1. Note that   D   and  D  , so . /n D n  and . /n .K0 /  n  ..S \ J c / K0 /  n .S \ J c /  .K0 / D 1 8n  1: This allows us to restrict probability measures  and   to the completely simple semigroup K0 (which is a closed subsemigroup of S ) with relative topology. Note that the support of   D   restricted to K0 generate the closed semigroup cl .S  S / \ K0 , which is an ideal of K0 since cl .S  S / D cl .S  S / : Since K0 is simple, the support of   restricted to K0 generates K0 . Notice that if x 2 J \ S   K0 ; then x 2 2 K0 and for any open set U containing x 2 , if Ux is an open set containing x such that ux  Ux  U , then . /n .U /  n .Ux /  .Ux / 6! 0 as n ! 1. It follows from Theorem 2.10 that the group factor of K0 is compact. Since K0  J , the group factor of K0 must be a singleton. (See Corollary 1.8.) Then Theorem 2.7(iii) implies n  D . /n converges weakly as n ! 1. Thus given " > 0, there exist a compact set E1  K0  J and a compact set E2  J such that for n  1, n  .E1 / > 1  " and  .E2 / > 1  ": Thus for n  1,

Z

  n E1 x 1 .dx/    n E1 E21 C ":

1"
0 and a subset Bz with z  1, defined by ˚  Bz D x 2 S W xi;d C1  z for 1  i  d C 1 ; such that for some subsequence .nk / of positive integers nk .Bz / > q;

k  1:

(ii) The set J0 defined by ˚  J0 D x 2 S W for each i; 1  i  d; xi;d C1 > 0 is nonempty. (iii) .J / D 0, where J is the set of all matrices in S that have at least one zero row or zero column. Then the sequence .n / is tight. Proof. Notice that Condition (iii) implies that n .J / D 0 for n  1 (see (4.53) and (4.54)). It is also easily verified that J0 is a nonempty open ideal of Jrc , where Jr D fx 2 J j x has at least one zero rowg. It follows as in (4.55) that   lim n J0 D 1:

n!1

(4.112)

314

4 Random Matrices

Define for each u > 0, the set Hu by Hu D

8 < :

x 2 S W min xi;d C1  u; min 1i d

d X

1i d

Then J0 \

Jcc

D

xj i

j D1

1 [

9 1=  u; max xij  : u; 1i;j d C1

H1:

mD1

(4.113)

m

For each u  1, define the set Bu by ˚  Bu D x 2 S W xi;d C1  u for 1  i  d C 1 : Notice that for any x 2 S ,

Bu x 1  Bu I

then, for k  1, n  1, nCk .Bu /  n .Bu / so that lim n .Bu / exists. n!1 We claim that b sup lim n .Bu / D 1:

(4.114)

u1 n!1

By Condition (i) of Theorem 4.15, b > 0. To prove (4.114), we proceed as we did in establishing (4.99). Let b < 1. Choose c such that r b 1  .c=2  q1 / ; : M .Bs / < c  q1 It is easily verified that Ht 1 Bs  Bs0 ; Consequently,

s0 D ds=t:

Bs y 1 \ Ht ¤ ; ) y 2 Ht 1 Bs  Bs0 :

(4.115)

4.3 Tightness of Products of I.I.D. Random Matrices: Weak Convergence

315

Let n > M . Then Z   n .Bs / D M Bs y 1 nM .dy/ Z Z D C Bs0

SBs0

h   c  i  q1 1  nM Bs0  .c  q1 / nM Bs0 C 2   c c nM D  q1 C   Bs0 : 2 2 It follows that for any s  1, lim n .Bs / 

n!1

c2 c C < b; 2 2

contradicting the definition of b. Thus b D 1, and given " > 0, there exists u  1 such that for all n  1, n .Bu / > 1  ": (4.116) Choose the positive integers N and 0 > 0 such that   N Hu0 > 1  ":

(4.117)

Now we can easily verify that the set Bu Hu1 is relatively compact, and also for 0 n > N, Z   n 1  " <  .Bu / D nN Bu y 1 N .dy/   C ":  nN Bu Hu1 0 t u

It follows that .n / is tight. Theorem 4.16. Let , S , and J be as in Theorem 4.15. Assume (i) .J / D 0. (ii) H ¤ ;, where H is given by





H D x 2 S W for each i; 1  i  d; max xi;d C1; min xij 1j d

>0 :

(iii) The sequence .n / is tight. Then the sequence .n / converges weakly to a probability measure , where S D m.S / ( the set of all matrices in S with the minimal rank) and .H / D 1.

316

4 Random Matrices

Proof. Notice that for n  1, n .J / D 0. H is easily verified to be a nonempty open ideal of S  J . It follows as in (4.55) that n .H / " 1 as n ! 1: Assume that w … S , where

  00 wD : 01

Notice that since w … S , H  m.S /  H  H \ m.S /:

(4.118)

By Condition (iii), the sequence n 1X k  n kD1

converges weakly to a probability measure  on S such that  D   D   and S D m.S / (see Theorem 2.7 and Proposition 1.9). Since for each positive integer n,  D n  n ; it follows from (4.118) that .H \ m.S //  n .H /.m.S //n .H / ! 1 as n ! 1. It follows that .H / D 1. Note that S is a completely simple subsemigroup (Theorem 2.7), and for x 2 H \ m.S /, the set xS x D xm.S /x is a group containing x. Let e be the identity of this group. By (4.118), e 2 xm.S /x  H \ m.S /:

(4.119)

Suppose that the rank of the matrices in m.S / is greater than 1. Then e has rank greater than 1, and as such, e, a nonnegative idempotent matrix, cannot have a strictly positive column (see Theorem 1.11). Now if for some i , 1  i  d , ei;d C1 > 0, then since there exists j , 1  j  d , such that ej;d C1 D 0, it follows that ej i > 0 since e 2 H (see the definition of H ) and ej;d C1 D .e  e/j;d C1  ej i ei;d C1 > 0; which is impossible. Thus, the first d elements in the .d C 1/th column of e are all zeros. Since e 2 H , it follows that e has the form   A0 ; 0 1

(4.120)

4.3 Tightness of Products of I.I.D. Random Matrices: Weak Convergence

317

where A is a strictly positive d  d matrix. Since e D e 2 , A D A2 ; therefore by Theorem 1.11, the matrix has rank 1, and consequently e has rank 2. Now notice that if x 2 S and   BC ; C ¤ 0: xD 0 1 Then exe is given by exe D

  ABA AC ; 0 1

where AC ¤ 0, since A is strictly positive and C ¤ 0. Since exe 2 eSe D e m.S /e D xm.S /x; which is a group with identity e, it follows that there exists y in this group such that   DE yD 0 1 and .exe/y D e. Then .exe/y D

  ABAD ABAE C AC ; 0 1

where ABAE C AC ¤ 0, since AC ¤ 0. This contradicts the representation of e as given in (4.120). Thus, every element in S has the form   B0 ; B ¤ 0; (4.121) 0 1 since we have assume the element w … S . Now e m.S /e D eS e, the group factor of S , the support of an idempotent probability measure, must be a compact group. It is a compact group of strictly positive matrices, since if A is a strictly positive d  d matrix and B is a nonzero nonnegative d  d matrix, then ABA is a strictly positive d  d matrix. By Corollary 1.8, the set eS e must be a singleton. By Theorem 2.7, it follows that the sequence .n / converges weakly to . If the rank of the matrices in m.S / is 1, then for x, y 2 m.S /, xy D x; as a result, the set e m.S /e is again a singleton. As before it follows that n !  weakly. If we assume that the element w 2 S , then the rank of the matrices in m.S / is 1, and therefore, as we just argued, the sequence n converges weakly to . t u It is clear from Theorem 4.12, if  is a probability measure on 2  2 nonnegative matrices, then .n / is tight if and only if there exists either a common left eigenvector x .x1 ; x2 / with x1 > 0; x2 > 0

318

4 Random Matrices

for every matrix in S with common eigenvalue 1 or a common right eigenvector x .x1 ; x2 / with x1 > 0;

x2 > 0

for every matrix in S with common eigenvalue 1. Notice that xS D x, where x1 > 0, x2 > 0, implies xS D x and S is compact, where S D cl

1 [

! Sn

:

nD1

No such result holds for general 2  2 real matrices, though Theorem 4.17 is a result in that direction for real matrices. Let .Xn / be a sequence of 2  2 i.i.d. real matrices such that the distribution of X1 is . Write   yn zn ; n  1; Xn D 0 wn where wn D ˙1. According to Theorem 4.12, this is the most general situation, in the context of tightness for 22 real matrices if we ignore one exceptional situation. If Pr .y1 D 0/ > 0, then the sequence .n / is tight. The reason is the following: Let ! 1 [ n S D cl S nD1

and I be the set defined by  

0a I D x2S WxD for b D 1 or 1 and some real a : 0b Notice that I is a closed ideal of S and if .I / D Pr .X1 2 I / > 0; then by Proposition 2.7, lim n .I / D 1:

n!1

Therefore given " > 0, there is a positive integer N and a positive number r such that  

0a 1  " < N x 2 S W x D ; jaj  r and b D ˙1 : 0b Let us call the preceding set on the right B. Note that if b D ˙1, d D ˙1, then      0a 0c 0 ad D 0b 0d 0 bd

4.3 Tightness of Products of I.I.D. Random Matrices: Weak Convergence

319

so it is clear that B  I  B. This implies kCN .B/  N .B/k .I /: The tightness of .n / follows from this. Let us now assume 9 Pr .y1 D 0/ D 0; = 1  E .log jy1 j/ < 0; ; E log max fjz1 j ; 1g < 1:

(4.122)

We claim that .n / is tight by (4.122). To prove this claim, notice that   y1 y2 : : : yn z1 ˙ y1 z2 ˙    ˙ y1 y2 : : : yn1 zn X1 X2 : : : Xn D 0 w1 w2 : : : wn     n n , say: 0 n It follows from (4.122) and the law of large numbers that for any ˇ > 0, Pr

n X

! log jyi j < ˇ eventually D 1;

i D1

so that for any " > 0, Pr .jy1 y2 : : : yn j < " eventually/ D 1: Let us now write ˚n D log max fjzn j ; 1g and let F .x/ D Pr .˚n  x/: Then for any A > 0, it follows from (4.122) that A

1 X

Pr .˚n > An/

nD1

DA

1 X

Z

1

Œ1  F .An/ 

Œ1  F .x/ .dx/ D E .˚1 / < 1: 0

nD1

Using the Borel–Cantelli lemma, 

 ˚n Pr  A eventually D 1: n

(4.123)

320

4 Random Matrices

This means lim sup

n!1

˚n D 0 almost surely: n

Thus with probability 1, lim sup

n!1

 lim sup e

p n jy1 y2 : : : yn1 zn j

1 n ˚n

n!1

 exp 1=n

n1 X

! log jyi j

i D1

D e E logjyi j < 1; by (4.122). Thus, Pr .n converges as n ! 1/ D 1. This and (4.123) immediately imply the tightness of the sequence .n /. Now notice that if we have 0 < E .log jy1 j/  1;

(4.124)

then by the law of large numbers,  Pr

lim jy1 y2 : : : yn j D 1 D 1;

n!1

so that .n / cannot be tight. Let us now suppose that E jlog yi j > 1; E log max fjz1 j ; 1g D 1:

(4.125)

Then for any A > 0, with F and ˚n as before, ACA

1 X

Œ1  F .An/

nD1

Z

1

Œ1  F .x/ dx D E .˚1 / D 1

 0

so that for any A > 0,  ˚n > A infinitely often D 1: Pr n 

This implies lim sup

n!1

˚n D 1 almost surely n

(4.126)

4.3 Tightness of Products of I.I.D. Random Matrices: Weak Convergence

321

and consequently from (4.125) and 4.126, p lim sup n jzn y1 y2 : : : yn1 j n!1  1  lim sup e n ˚n  e E logjy1 j D 1 almost surely: n!1

This means that given any A > 0, Pr .jzn y1 y2 : : : yn1 j D jn  n1 j > A infintely often/ D 1: If .n / is tight, then given 0 < " < 1=2, there exists A > 0 such that Pr .jn j  A/ > 1  ";

n  1;

so that for n > 1, Pr .jn  n1 j  2A/ > 1  2" > 0; which is a contradiction. Thus, we have proven Theorem 4.17. Theorem 4.17. Let .Xn / be a sequence of 2  2 i.i.d. real matrices with common distribution . Suppose   yn zn Xn D ; 0 wn

n  1;

where wn D ˙1 almost surely. Then the sequence .n / is tight if one of the following two conditions hold: (i) Pr .y1 D 0/ > 0; (ii) Pr .y1 D 0/ D 0, 1  E log jy1 j < 0 and E log max fjz1 j ; 1g < 1. Also if E log jy1 j > 1 and the sequence .n / is tight, then we must have: E log max fjz1 j ; 1g < 1: The rest of Sect. 4.3 is devoted to connections between tightness for the sequence of products of i.i.d. random matrices with distribution  and the existence of -invariant probability measures. It is of course known to us (see Theorem 2.7) that if .n / is tight, then the sequence n 1X k  n kD1

converges weakly to a probability measure  such that   D   D . It is the converse of this result that we are more interested in. Let us first look at the following general lemma, Lemma 4.6. This is a result of Guivarc’h and Raugi [84], generalizing an earlier result of Furstenberg [69].

322

4 Random Matrices

Lemma 4.6. Let .Xn / be a sequence of i.i.d. random variables taking values in a topological semigroup S with distribution  2 P .S /. Let Y be a locally compact second-countable space such that there is a continuous map .s; y/ ! s  y from S  Y ! Y . Suppose that  2 P .Y / such that   D . Then for almost all ! (in the sample space), there is a probability measure ! 2 P .Y / such that the two sequences of probability measures given by .X1 .!/X2 .!/ : : : Xn .!/s/ and .X1 .!/X2 .!/ : : : Xn .!// both converge weakly to ! as n ! 1 for almost all s in S with respect to the probability measure 1 X 1  n : n 2 nD1

Here for any s in S , s.B/ D fy 2 Y W s  y 2 Bg for B  Y: Moreover if f is any bounded Borel measurable real function on Y , then 

Z

Z f d D E

f d! :

Proof. Let f be a bounded continuous function on Y . Define the function ˚ on S by Z ˚.s/ D

f .sy/.dy/:

(4.127)

Write: Zn X1 X2 : : : Xn : Notice that E .˚ .ZnC1 / j X1 ; X2 ; : : : ; Xn / Z D ˚ .Zn  s/ .ds/  Z Z D f .Zn sy/ .dy/ .ds/ Z D f .Zn y/  .dy/ Z D f .Zn y/ .dy/ D ˚ .Zn /; so that .˚ .Zn // is a martingale. Since ˚ is bounded, ˚ .Zn / converges almost surely to some random variable W .˚/ and Z E.W .˚// D E .˚ .Z1 // D

f d:

(4.128)

4.3 Tightness of Products of I.I.D. Random Matrices: Weak Convergence

323

Now for any positive integers m, n, E j˚ .ZmCn /  ˚ .Zm /j2 D E Œ˚ .ZmCn /2 C E Œ˚ .Zm /2  2E Œ˚ .ZmCn / ˚ .Zm / D E Œ˚ .ZmCn /2  E Œ˚ .Zm /2 ; since E .˚ .ZmCn / ˚ .Zm // D E ŒE .˚ .ZmCn / ˚ .Zm / j ˚ .Zm //

 Z n D E ˚ .Zm / ˚ .Zm s/  .ds/



Z Z n f .Zm sy/ .dy/  .ds/ D E ˚ .Zm / 

Z n D E ˚ .Zm / f .Zm y/  .dy/ 

Z D E ˚ .Zm / f .Zm y/ .dy/ D E Œ˚ .Zm /2 : Thus for any positive integer k, k X

E j˚ .ZmCn /  ˚ .Zm /j2

mD1

D

k h X

E .˚ .ZmCn //2  E .˚ .Zm //2

i

(4.129)

mD1

 2n  kf k; after canceling appropriate terms. Notice that for any real measurable function h E Œh .ZkCm / D E .E .h .ZkCm / j Zk //

Z  DE h .Zk s/ m .ds/ so that for n  1, we have 1 X

Z E

mD1 1 X

D

mD1

 j˚ .Zm s/  ˚ .Zm /j2 n .ds/

E j˚ .ZmCn /  ˚ .Zm /j2  2nkf k:

324

4 Random Matrices

This means

1 X

j˚ .Zm s/  ˚ .Zm /j2 < 1;

(4.130)

mD1

almost everywhere with respect to the measure ! 1 X 1 n  ; P 2n nD1 where P is the probability measure. It follows from (4.129) and (4.130) that almost everywhere lim ˚ .Zm s/ D lim ˚ .Zm / I.f /; m!1

m!1

say, exists. Since the space of continuous functions with compact support on Y is a separable metric space in the usual sup norm, if .fi / is dense in Cc .Y /, then Z fi .Zm sy/ d.y/ D I .fi / ; i  1, almost everywhere lim (4.131) m!1

with respect to the measure ! 1 X 1 n P  : 22 nD1 Define m;s .!/ 2 P .Y / by  m;s .!/.B/ D  .Zm .!/s/1 B : It follows from (4.131) that almost everywhere, the weak limit of m;s .!/ is independent of s, and if we call this limit .!/, then from (4.130), we have Z lim

m!1

Z fi dm;s .!/ D

fi d.!/;

i  1:

By (4.128), for i  1, Z E



Z

fi d.!/ D

fi d:

Then for any bounded Borel function f on Y , we have Z E

 f .y/d.!/.y/ D

Z f .y/d.y/;

which implies that the measure .!/ must be almost surely a probability measure. t u

4.3 Tightness of Products of I.I.D. Random Matrices: Weak Convergence

325

Proposition 4.2 is due to P. Bougerol. Proposition 4.2. Let .Xn / be a sequence of d  d i.i.d. random real matrices with distribution . Suppose ! 1 [ S D cl Sn : nD1

  Suppose also that  2 P Rd ,   D , and for any proper subspace V of Rd , .V / < 1. Then the sequence .n / is tight. Moreover, if there is no nontrivial proper subspace V of Rd such that x  V  V for each x in S , then S is compact.   Proof. Consider the given  2 P Rd . Then   D . By Lemma 4.6, if Zn X1 X2 : : : Xn ; then the sequence of probability measures .Zn .!// weakly converges, almost surely, to a random probability measure .!/ as n ! 1. Suppose for some !, .w/ lim Zn .!/ D .!/; n!1

but sup fkZn .!/k W n  1g D 1: Since every element in the sequence kZn .!/k1 Zn .!/ has norm 1, there is a subsequence .nk / such that    1 lim Znk .!/ D 1 and lim Znk .!/ Znk .!/ D H.!/;

k!1

k!1

where H.!/ is a nonzero matrix with norm 1. Let u 2 Rd such that H.!/u ¤ 0. Then since  1   lim Znk .!/ Znk .!/  u D kH.!/uk

k!1

is positive,

  lim Znk .!/  u D 1:

k!1

Thus for any continuous f with compact support on Rd , Z

Z f .y/d.!/.y/ D lim

k!1

Z

D lim

k!1

D lim

k!1

Z

  f .y/d Znk .!/ .y/   f Znk .!/y d.y/   IK! .y/  f Znk .!/y d.y/;

326

where

4 Random Matrices

n o K! D y 2 Rd W H.!/y D 0 :

This implies  .K! / D 1. But this contradicts our assumption on , since K! is a proper subspace of Rd . Thus, it follows that almost surely the sequence .Zn .!// is bounded. Then the sequence .n / must be tight. 1 P 1 n Now by Lemma 4.6, for almost all s in S with respect to 2n  , nD1

.w/ lim Zn .!/s D .!/; n!1

(4.132)

almost surely. Since .Zn .!// is bounded almost surely, (4.132) holds for every s 2 S almost surely. If ksk < 1 for some s 2 S , then the zero matrix is in S , and it follows from (4.132) that .!/, almost surely, is the unit mass at the zero element in Rd , which contradicts  D E..!//, (see (4.128)). To prove the last part of Proposition 4.2, assume there is no proper subspace V of Rd such that for each x 2 S , x  V  V . Suppose S is not compact. Then there exists sn 2 S such that lim ksn k D 1; n!1

but

lim ksn k1 sn D s0 ;

n!1

where s0 is a nonzero matrix. Then almost surely .!/ D Z.!/ssn ; for n  1, where Z.!/ is a limit point of the sequence .Zn .!//. Suppose y 2 Rd such that Z.!/  ss0 y ¤ 0. Then lim kZ.!/ssn yk D 1:

n!1

Then if f is a continuous function with compact support in Rd , Z Z f .y/.!/.dy/ D f .Z.!/ssn y/ .dy/ Z IV .y/f .Z.!/ssn y/ .dy/; D lim n!1

where V D fy j Z.!/ss0 y D 0g : It follows that .V / D 1 and V D Rd . If y0 2 Rd is such that s0 y0 ¤ 0, then fss0 y0 j s 2 S g

4.3 Tightness of Products of I.I.D. Random Matrices: Weak Convergence

327

spans a subspace W of Rd such that W  fy j Z.!/y D 0g: Since sW  W for s 2 S , W D Rd and consequently Z.!/ must be the zero matrix. This contradicts that S does not contain the zero matrix. t u We conclude Sect. 4.3 by considering the preceding problem in nonnegative matrices. Proposition 4.3. Let  be a probability measure on d  d nonnegative matrices and let ! 1 [ n S D cl S : nD1

 d Suppose there exists a probability measure  on RC such that: (i) .f0g/ < 1  d and (ii)   D , where 0 is the zero vector in RC . If S contains a strictly positive matrix, then the sequence .n / is tight and  D ˇ , where n  d 1X k :  and  2 P RC n!1 n

ˇ D .w/ lim

kD1

Proof. Suppose 0 < b .f0g/ < 1: Consider the probability measure 0 D

.  bı0 / : .1  b/

Then 0 .f0g/ D 0 and  0 D 0 . Thus, we may assume .f0g/ D 0. Because of this, given ı > 0, there exists a positive integer N such that  .FN / > 1  ı; where

 d 1 < xi < N : FN D x 2 RC W for some i; 1  i  d; N Since n  D  for n  1, Z

  n FN x 1 .dx/    n FN FN1 C ı

 .FN / D

(4.133)

328

4 Random Matrices

so that for n  1,

  n FN FN1 > 1  2ı:

(4.134)

It follows that given ı > 0, there exists u > 0 such that for each positive integer n, n .H.u// > 1  ı;

(4.135)



where

H.u/ D A 2 S W min Aij  u ; i;j

since

  Fn Fn1  H n2 ;

n  1:

The strictly positive matrices form an open subset of S , and thus, there exist positive integers k, m and a positive number t such that   k Jm D t; where

(4.136)

1 : Jm D A 2 S W min Aij > i;j m

Let s be any positive integer and let .Xn / be a sequence of i.i.d. matrices with distribution . Write X1 X2 : : : Xn Zn : Choose ı > 0 such that ı < t 2 . For this ı, let u be such that (4.135) holds. Then we have   t 2  ı  Pr Z2kCs 2 H.u/; Zk 2 Jm ; XkCsC1 : : : X2kCs 2 Jm

   Pr Zk 2 Jm ; XkC1 : : : XkCs 2 Jm1 H.u/Jm1 ; XkCsC1 : : : X2kCs 2 Jm   D t 2  s Jm1 H.u/Jm1 ; where Jm1 H.u/Jm1  ˚ D A 2 S W BAC 2 H.u/ for some B and C in Jm ; which is easily verified to be contained in the compact set

A 2 S W max Aij  u  m i;j

2

:

It follows that the sequence .n / is tight. The rest of the proof is left to the reader.u t

4.3 Tightness of Products of I.I.D. Random Matrices: Weak Convergence

329

Our last result connecting  invariance to tightness of .n / is given in Proposition 4.4. Proposition 4.4. Let  be a probability measure on nonnegative d  d matrices and ! 1 [ n S D cl S : nD1

Suppose  is a probability measure on d  d nonnegative matrices such that 

 D . (We do not assume that S  S .) Consider the following conditions: (i)  .Jr / D 0, where Jr is the set of all matrices that have at least one zero row; (ii) S S  S ; (iii) S  S . (Note that (iii) ) (ii). But (ii) in general is weaker than (iii).) If (i) holds, the sequence .n / is tight. If (i) and (ii) hold, then the solution  of   D  must be an idempotent probability measure. If (i) and (iii) hold, then both 0 and , where n 1 X m  ; n!1 n mD1

0 D .w/ lim

being idempotent probability measures, have product representations (see Theorem 2.8) that can be described as follows: Let e be an idempotent matrix of minimal rank in S . Then there exist ˛, ˇ, , 1 in P .S / such that 0 D ˛ ˇ and  D ˛ ˇ 1 ; where Sˇ D eSe is a finite group and ˇ is the uniform distribution on it, S1  S  the set of all idempotent matrices in e  S , and S˛  the set of all idempotent matrices in S  e. Proof. Suppose (i) holds. Given " > 0, there is a compact set K  Jrc such that .K/ > 1  ". Then n  D  implies Z

  n Kx 1 .dx/ C "

.K/  K

  so that n KK 1 > 1  " for n  1. Since K  Jrc and K is compact, it is easily verified that KK 1 is a compact set. This implies that .n / is tight and n 1 X m 0 D .w/ lim  n!1 n mD1

330

4 Random Matrices

exists. Since   0 D  0  D 0 D  0 0 ; we have 0  D  and S0 is the completely simple (minimal) ideal of S . Now suppose (i) and (ii) hold. Then by (ii),   S  S 0  S  S S 0  S 0 : Let us write:  0 1 . Then it easily follows that

1 0 D 1 D 0 1 ; : S 1  S 0

(4.137)

Then (4.137) implies S1  S1  S0  S0  S0 S  S 0  S0  S0  S   S 0 D S 1 : It follows that S1 is an ideal of S0 . Therefore, since S0 is simple, S1 D S0 . Using (4.137) and Proposition 2.4, for any Borel set B  S and x 2 S0 , we have Z     1 Bx 1 D 0 Bx 1 y 1 1 .dy/ Z     D 0 Bx 1 1 .dy/ D 0 Bx 1 : Consequently, Z   1 .B/ D 1 Bx 1 0 .dx/ Z   D 0 Bx 1 0 .dx/ D 0 .B/; which means  0 D 1 D 0 . Thus, 0  D ;

 0 D 0 and 0 D 0 0 :

This implies  D 0  D . 0 / .0 / D  .0 / D  : Thus, we have proven that under (i) and (ii), the solution  of   D  must be an idempotent probability measure. We leave the rest of the proof to the reader. t u

4.3 Tightness of Products of I.I.D. Random Matrices: Weak Convergence

331

Section 4.3 Exercises Exercise 4.16. Establish (4.49). Exercise 4.17. Explain why the set C J 1 k in the proof of Theorem 4.8 is relatively compact. Exercise 4.18. Let  be a probability measure on d  d real matrices such that m fthe zero matrixg is positive for some positive integer m. Show that n converges weakly to the unit mass at the zero matrix. Does this mean f0g > 0? If not, give an example. Exercise 4.19. Let .Xn / be a sequence of i.i.d. nonnegative random variables such that either P .X1 D 0/ > 0 holds or both conditions P .X1 D 0/ D 0 and E .log X1 / < 0 hold. Show that the sequence X1 X2 : : : Xn must then converge to zero almost surely. Exercise 4.20. Let  be a probability measure on d  d nonnegative matrices and S be the closed multiplicative semigroup generated by S./. Define the sets Ac and Ar as follows: Let D D f1; 2; : : : ; d g; Ac  D; Ar  D such that Ac D fi 2 D j if x 2 m.S /, then the i th column of x is a zero columng; and Ar D fj 2 D j if x 2 m.S /, then the j th column of x is a zero rowg: Let Ac \Ar D Acr , Ac nAr D Ac0 , Ak nAc D Aro , B D D .Ac [ Ar /. Suppose that the zero matrix is not in S . Show that then Ac [ Ar is a proper subset of D, and if .n / is tight, then every element in S has the following form: Aco

B

Aco B 0 <M Acr 0 0 Aro 0 0

Acr Aro 0 0

Specifically, there exists M > 0 such that if x 2 S , then the B  B block of x has each of its entry bounded above by M . (This result is taken from [154].)

332

4 Random Matrices

(Hint: Let  D .w/ lim

1 n!1 n

n P

k . Then S./ D m.S / is completely simple

kD1

with a compact group factor, and as such, for any x, y in m.S /, xSy D xm.S /y is compact. Also, note that for any x in m.S / and s 2 S , xs 2 m.S /, and thus, for i 2 Ac , .xs/ui D 0 for all u in D.) Exercise 4.21. Let .Xn / be a sequence of d  d random i.i.d stochastic matrices such that the products Xn Xn1 : : : X1 converge in distribution to a random stochastic matrix Z with identical rows. Let  and , respectively, be the distributions of X1 and Z (that is, .B/ D P .X1 2 B/ and .B/ D P .Z 2 B/, for any Borel set B of d  d stochastic matrices). Show that if ˇ is a probability measure on d  d stochastic matrices such that ˇ  D ˇ, then ˇ must be . Exercise 4.22. Let .Xn / be as in Exercise 4.21. Suppose that P (X1 is strictly positive) > 0. Then show that the sequence Xn Xn1 : : : X1 converges almost surely to a d  d random stochastic matrix with identical rows. This result is due to Chamayou and Letac. (Hint: Let s be a d  1 positive vector and let sn D Xn Xn1 : : : X1 s. Write: vn D max .sn /i , un D min .sn / and Mn D min .Xn /ij . Note that almost surely, 1 P

i

i

i;j

Mk D 1 (using the fact that P .M1 > ı/ > 0 for some ı > 0). Show that

kD1

un1 C Mn .vn1  un1 /  un  vn  vn1  Mn .vn1  un1 / : This leads to lim vn  un D 0 almost surely.) n!1

Exercise 4.23. Suppose that d (> 1) committee members are evaluating an applicant for a teaching position at a university. Let 0 be a d  1 vector, .0 /i > 0 for each i , representing their initial evaluations. The members then start discussing the applicant in a number of subsequent meetings to influence each other, to revise their scores from the previous meeting, and to reach a consensus. These result in a sequence of i.i.d. d  d stochastic matrices .Xn / such that n D Xn n1 , n  1, where .n /i is the score that the i th member arrives at after the nth meeting. If P .X1 > 0/ > 0, show that  D lim n exists, and i is independent of i . n!1

Exercise 4.24. Let X1 ; X2 ; : : : be a sequence of i.i.d. 2  2 random nonnegative matrices such that X1 X2 : : : Xn Yn , X1 has distribution ,   An Bn ; Xn D 0 1

  Zn Wn Yn D ; 0 1

and  is not the unit mass at the identity matrix. Show that .n / is tight iff the following two conditions hold: (i) Zn converges to zero in probability, and (ii) Wn converges almost surely.

4.4 Invariant Measures for Random Walks in Nonnegative Matrices: Laws of Large Numbers 333

Exercise 4.25. Show that in the previous exercises, when P .A1 D 0/ D 0 and 1 < E .log A1 / < 0, then the sequence Wn is almost surely convergent iff E .log sup f1; B1 g/ < 1. Show also that when P .A1 D 0/ > 0, then the sequence Zn converges to 0 almost surely. (See [78].) Exercise 4.26. Consider i.i.d. 2  2 random nonnegative matrices .Xn / given by   An Bn ; Xn D Cn Dn such that the distribution of X1 is . Let S be, as usual, the closed semigroup generated by S./. Let Yn D X1 X2 : : : Xn , n  1. Suppose that .n / is tight. Show that one of the following possibilities must then occur: (i) S is compact;

  10 (ii) Yn converges in distribution to the unit mass at a matrix which is either 00     00 00 or or ; 01 00 (iii) Either P .An D 1; Bn D 0/ D 1 or P .An D 1; Cn D 0/ D 1 or P .Cn D 0; Dn D 1/ D 1 or P .Bn D 0; Dn D 1/ D 1. (This is taken from [154].) (Hint: Use Exercises 4.20 and 4.24.)

4.4 Invariant Measures for Random Walks in Nonnegative Matrices: Laws of Large Numbers Mixed random walks in the general context are discussed in Chap. 3. Here, we study mixed random walks with values in nonnegative matrices. First, we consider the existence and uniqueness of invariant measures for such walks, then we apply these results to obtain laws of large numbers for such walks. The main results in this section are Theorems 4.19–4.23. Let .Yn / be a sequence of i.i.d. random variables such that for each n  1 and for some ˛, 0  ˛  1, P .Yn D 1/ D ˛ and P .Yn D 0/ D 1  ˛: Consider a mixed walk .Mn /, where MnC1 D Mn XnC1 , if YnC1 D 1I D ZnC1  Mn , if YnC1 D 0; for n  0, where .Xi / and .Zi /

(4.138)

334

4 Random Matrices

are two i.i.d. sequences of d  d random nonnegative matrices. The Xi s, Yi s, and Zi s are independent of one another with M0 X0 . Let 1 be the distribution of Xi , i  0 and 2 be the distribution of Zi , i  1. Let us assume the following conditions: (i) S1 D S2 ; (ii) 1 .J / D 0 D 2 .J /, where J is the set of all d  d matrices with at least one zero row or one zero column; (iii) S \ J ¤ ;, where ! 1 [ S D cl Sn1 nD1

and J is the set of all d  d strictly positive matrices; (iv) n1 and n2 are both tight. We are considering the usual topology for matrices. It follows from Theorem 4.10 that under the preceding conditions, the (multiplicative) semigroup S defined in Condition (iii) is compact when S does not contain the zero matrix; however, in any case, there are probability measure ˇ1 and ˇ2 such that n1 ! ˇ1 and n2 ! ˇ2

(4.139)

weakly as n ! 1. Let K be the kernel (that is, the smallest ideal) of S . Then we know (Theorem 2.7) that K is closed and it is the support of both ˇ1 and ˇ2 . Let P n .x; A/ be the n-step transition probability function for the mixed random walk .Mn /. Then it is clear that for x in S , 9 P .x; A/ D ˛ıx 1 .A/ C .1  ˛/2 ıx .A/ and for n > 1; = n   P : (4.140) n k

ıx k1 .A/ ˛ .1  ˛/nk nk P n .x; A/ D ; 2 k kD0

Let us define the Markov operator T by Z Tf .x/ D f .y/P .x; dy/; where f 2 C.S / the space of all real continuous functions on S . Then for n  1, ! “ n X n k n ˛ .1  ˛ nk f .yxz/nk .dy/k1 .d z/: (4.141) T f .x/ D 2 k kD0

Now suppose 0 … S . Since then S is compact, n1 ! ˇ1 weakly and n2 ! ˇ2 weakly, it follows that given " > 0, there exists a positive integer N such that for n > N , for each y 2 S , we have ˇZ ˇ Z ˇ ˇ ˇ f .yxz/n .d z/  f .yxz/ˇ1 .d z/ˇ < " 1 ˇ ˇ

4.4 Invariant Measures for Random Walks in Nonnegative Matrices: Laws of Large Numbers 335

and for each z 2 S , ˇZ ˇ Z ˇ ˇ ˇ f .yxz/n .dy/  f .yxz/ˇ2 .dy/ˇ < ": 2 ˇ ˇ We can also choose a positive integer N2 such that for n > N2 , ! nN X n ˛ k .1  ˛/nk > 1  ": k kDN

Then it follows from (4.141) that for each x in S and f 2 C.S /, lim T n f .x/ D T f .x/;

(4.142)

n!1

where “ T f .x/ D

f .yxz/ˇ2 .dy/ˇ1 .d z/ Z

D

f .w/ˇ2 ıx ˇ1 .d w/:

Let us write x ˇ2 ıx ˇ1 . Then Sx D Sˇ2  x  Sˇ1 D K  x  K for each x in S . Since KxK  K, it is clear that KxK D K. Now for f 2 C.S /, Z Z f .y/x .dy/ D lim f .y/P n .x; dy/: (4.143) n!1

Notice that for f 2 C.S /,   Tf D T Tf ;

T f D T .Tf /:

(4.144)

Then for f 2 C.S /, Z Z

Z f .y/x .dy/ D

 f .z/P .y; d z/ x .dy/:

(4.145)

Since a probability measure on a compact (or a complete separable) metric space is always regular, the probability measure  defined by Z .A/ D

P .y; A/x .dy/

is regular. Therefore  D x , since for f 2 C.S /, Z Z

Z f d D

 f .z/P .y; d z/ x .dy/ D

Z f dx :

336

4 Random Matrices

Thus for any x 2 S and any Borel set A  S , Z x .A/ D

P .y; A/x .dy/:

(4.146)

Now let us show that for any x and y in K, x D y . To prove this, it is enough to show that for any f 2 C.S /, T f is a constant on K. Let a D b C ı, ı > 0, a D max T f .x/ D T f .x1 / ;

x1 2 K;

b D min T f .x/ D T f .x2 / ;

x2 2 K:

x2K

and x2K

Then Z

2

a D T f .x1 / D T f .x1 / D Z Z C D N .x2 /

T f .y/x1 .dy/ ;

KN .x2 /

where N .x2 / is an open set containing x2 such that for y 2 N .x2 /, T f .y/
0. This means T f .x/ (and therefore the measure x ) is the same for all x 2 K. Now using the second relationship in (4.144), for x 2 S , we have Z T f .x/ D

T f .y/x .dy/ Z

D

T f .y/x .dy/; K

since K is the support of x . It follows that T f .x/ is the same for all x 2 S , and consequently for any x and y in S , x D y .

4.4 Invariant Measures for Random Walks in Nonnegative Matrices: Laws of Large Numbers 337

Now let  be any invariant probability measure, so that for any Borel set A  S , Z .A/ D

P .y; A/.dy/:

(4.147)

It follows from (4.147) that for any f in C.S / and any positive integer n, Z Z

Z f d D

 f .x/P n .y; d z/ .dy/

and by (4.143), taking n to infinity, we have Z Z

Z f d D

 f .z/y .d z/ .dy/

Z D

f dy ;

since for all y 2 S , y is the same. It follows that  D y . Thus, we have proven Theorem 4.18, when 0 … S . Theorem 4.18. Let

1 .Xn /1 nD0 and .Zn /nD1

be two sequences of i.i.d. random d  d nonnegative matrices such that the distribution of each Xn is 1 and that of each Zn is 2 . Let S1 be the support of 1 and S the closed (multiplicative) semigroup generated by S1 with usual matrix topology. Let us assume Conditions (i)–(iv) given at the beginning of this section. Let .Yn /1 nD1 be a sequence of i.i.d. random variables such that for each n  1, P .Yn D 1/ D ˛;

P .Yn D 0/ D 1  ˛;

where 0 < ˛ < 1. Assume the Xi s, Yi s and Zi s are all independent of one another. Consider the mixed walk .Mn /1 nD1 as defined in (4.138). Then the mixed walk has a unique invariant probability measure  given by  D ˇ2 ıx ˇ1 ; where x 2 S and ni ! ˇi (i D 1; 2) weakly as n ! 1.

t u

Theorem 4.20 treats the case when 0 2 S . But first we have the following theorem. Theorem 4.19. Let  be a probability measure on a locally compact Hausdorff second-countable topological semigroup S such that S D cl

1 [ nD1

! Sn

:

338

4 Random Matrices

Suppose .n / is tight and in this case, the weak limit ˇ, given by n 1X k  ; n!1 n

ˇ D .w/ lim

kD1

exists. Consider the S -valued mixed random walk .Sn / induced by an i.i.d. sequence of S -valued random variables .Xi / with distribution  as introduced in (4.138), with Zi D Xi almost surely for each i . The transition probability function P .x; B/ of this random walk is given by     P .x; B/ D ˛ x 1 B C .1  ˛/ Bx 1 ;

(4.148)

where 0 < ˛ < 1;

Bx 1 D fy 2 S W yx 2 Bg and x 1 B D fy 2 S W xy 2 Bg:

Let  be an invariant probability measure on S for .Sn /, so that for B  S , Z .B/ D

P .x; B/.dx/:

(4.149)

Then  satisfies the following conditions: (i) S , the support of , is the smallest ideal of S , so that S D Sˇ and (ii) ˇ D ˇ  D  ˇ and n converges weakly to ˇ. Proof. Let us prove (i) first. From (4.148) and (4.149),  D ˛  C .1  ˛/ :

(4.150)

This implies that supports of   and   are both contained in S . Therefore, S  S [ S  S  S ;

(4.151)

so that S  S [ S  S  S and S is an ideal of S . Since Sˇ is the smallest ideal of S , Sˇ  S :

(4.152)

To prove the converse inclusion, let us observe that by repeated applications of (4.150), for n  1, we have ! n X n D .1  ˛/nk ˛ k nk  k : k kD0

(4.153)

4.4 Invariant Measures for Random Walks in Nonnegative Matrices: Laws of Large Numbers 339

Let " > 0 and V be any open set containing Sˇ . Since .n / is tight, there is a compact set E  S such that for n  1, n .E/ > 1  ":

(4.154)

Now notice that E  S  Sˇ  E  Sˇ  V; since Sˇ is an ideal of S . Let A be a compact subset of S so that .A/ > 1  " and W be an open set containing Sˇ so that E  A  W  E  V: Since

n 1X k  ; n!1 n

ˇ D .w/ lim

kD1

W is open and W  Sˇ , there exists a positive integer N such that N .W / > 1  ": Then for integers k and s greater than N , k  s .V /  k .E/.A/N .W /sN .E/

(4.155)

> .1  "/ : 4

Now for a sufficiently large n and n > 2N C 2, nN X1 kDN C1

! n k ˛ .1  ˛/nk > 1  ": k

(4.156)

It follows from (4.153), (4.155), and (4.156) that .V / > .1  "/6 : Since " > 0 is arbitrary, this proves S  Sˇ : Assertions (4.152) and (4.157) establish (i). To prove (ii), from (4.150) we have, for n  1,  n D ˛ nC1 C .1  ˛/ . n /:

(4.157)

340

4 Random Matrices

Taking average and then weak limit as n tends to infinity, we then have  ˇ D ˛ ˇ C .1  ˛/ . ˇ/ or  ˇ D  . ˇ/:

(4.158)

Similarly for n  1, we can have n  D ˛ .n /  C .1  ˛/nC1  and as above ˇ  D .ˇ / :

(4.159)

Now (4.158) gives  ˇ D n . ˇ/ and consequently  ˇ D ˇ . ˇ/. Similarly from (4.159), ˇ  D .ˇ / ˇ: Therefore,  ˇ D ˇ  D ˇ  ˇ:

(4.160)

Since ˇ D ˇ ˇ, for B  S ,     ˇ Bx 1 D ˇ Bx 1 y 1 ;

(4.161)

whenever x, y are in Sˇ (see Proposition 2.4). Since Sˇ D S , it follows from (4.160) and (4.161) that for x 2 Sˇ ,   ˇ  Bx 1 D

Z

    ˇ Bx 1 y 1 .dy/ D ˇ Bx 1 :

Therefore, Z ˇ.B/ D ˇ ˇ.B/ D Z D

  ˇ Bx 1 ˇ.dx/

  ˇ  Bx 1 ˇ.dx/ D ˇ  ˇ.B/:

Thus, we have ˇ D ˇ  D  ˇ:

(4.162)

4.4 Invariant Measures for Random Walks in Nonnegative Matrices: Laws of Large Numbers 341

Now we prove the final assertion: n weakly converges to ˇ as n ! 1. We first observe that S D Sˇ is the completely simple (minimal) ideal of S with a compact group factor (see Theorem 2.2). Thus, we can write S D X  G  Y;

Y  X  G;

where G is a compact group. If A  X and B  Y are compact subsets such that .AGB/ > 1  "; then for n  3, n .A  G  B/  .AGB/n2 .XGY /  .AGB/  .1  "/2 ; which means that .n / is tight. Consider the compact semigroup  fn W n  1g ;

where the semigroup operation is convolution and the bar denotes weak closure. Such a semigroup always has an idempotent element, say, 0 , so that for some subsequence .nk / of positive integers w

n k ! 0 D 0 0 :

(4.163)

It follows from (4.162) and (4.163) that ˇ 0 D 0 ˇ D ˇ:

(4.164)

Since from (4.150), we have nk D ˛  nk 1 C .1  ˛/ nk ; it is clear that 0  .1  ˛/ 0 ;

(4.165)

0  ˛0 :

(4.166)

and similarly The assertions (4.165) and (4.166) imply S  S 0 [ S 0 S   S 0 and therefore S0 is an ideal of S , and consequently S ˇ  S 0 :

342

4 Random Matrices

Since S0 is completely simple, it follows that Sˇ D S0 . From (4.164), for B  S and x 2 Sˇ , Z     ˇ Bx 1 D 0 Bx 1 y 1 ˇ.dy/ Z     D 0 Bx 1 ˇ.dy/ D 0 Bx 1 Since 0 D 0 0 , Z 0 .B/ D Z D

  0 Bx 1 0 .dx/   ˇ Bx 1 0 .dx/

D ˇ 0 .B/ D ˇ.B/: Thus, 0 D ˇ. Since S D Sˇ D S0 , it follows immediately that the sequence n converges weakly to 0 D ˇ. t u Let us now state Theorem 4.20, the first part of which is contained in Theorem 4.18, and present the proof of its converse part. Theorem 4.20. Let .Xi /1 i D1 be a sequence of d  d i.i.d. random nonnegative matrices with distribution . Let .Yi /1 i Di be an i.i.d. sequence, independent of the Xi s such that P .Yi D 1/ D ˛;

P .Yi D 0/ D 1  ˛;

0 < ˛ < 1:

Define the mixed random walk .Sn / by SnC1 D Sn XnC1 , if YnC1 D 1I D XnC1 Sn , if YnC1 D 0: Let S be the smallest closed multiplicative semigroup generated by S . Suppose  satisfies the following conditions: (i) .n / is tight; (ii) .J / D 0 and S \ J ¤ ;, where J and J are as in Theorem 4.18. Then the mixed random walk has a unique invariant probability measure ; and the support of  is the smallest ideal of S with .J / D 1, when S does not contain the zero matrix, and  D ı0 , when S contains the zero matrix. Conversely under (ii), the existence of an invariant probability measure (such that it has no mass on the zero matrix) for the mixed random walk .Sn / implies the tightness of the sequence .n /.

4.4 Invariant Measures for Random Walks in Nonnegative Matrices: Laws of Large Numbers 343

The proof of the converse part of Theorem 4.20, not given earlier, follows. If  is an invariant probability measure for .Sn /, then as we have seen in (4.153), that for n  1,  must satisfy: ! n X n .1  ˛/nk ˛ k nk  k : (4.167) D k kD0

Notice that if x and z are strictly positive matrices and y is a nonzero nonnegative matrix, then the product xyz is a strictly positive matrix and as such, nk  k .J /  nk .J /.S  f0g/k .J /:

(4.168)

It follows from (ii) in the theorem that lim n .J / D 1, (see (4.55)). It follows n!1

from (4.156), (4.167), and (4.168) that .J / D 1. This observation is crucial in establishing the tightness of .n / as we will observe. Let " > 0. Let A be a compact set such that A  J and .A/ > 1  ". Let ı > 0 such that ı < min fxi i W 1  i  d and x 2 Ag : Let s 2 A and x 2 S and y 2 S . Then if 1  i  d and 1  `  d , xsy 2 A )

X

xij sjj yj `  .xsy/i `  M

j

for some M > 0. This means xsy 2 A and s 2 A ) ı

X

xij yj ` D ı.xy/i `  M

j

so there is a compact set B such that IA .xsy/IA .s/  IB .xy/: From (4.167), for n > 1 1  " < .A/ • n nk k D .1  ˛/ IA .xsy/nk .dx/.ds/k .dy/ ˛ k kD0 ! “ n X n .1  ˛/nk ˛ k "C IB .xy/nk .dx/k .dy/ k n X

!

kD0

D " C n .B/: This establishes that .n / is tight. Thus, we have completely proved Theorem 4.20. t u

344

4 Random Matrices

Now we consider mixed random walks taking values in a set of affine maps  d from RC into itself. Such affine maps can be identified with .d C 1/  .d C 1/ nonnegative matrices of the form   xy ; (4.169) 0 1 where x is a d  d matrix, y is a d  1 vector, 0 .0; 0; : : : ; 0/ is a 1  d vector. For a mixed random walk on such matrices, (ii) with S \ J ¤ ; in Theorem 4.20 becomes meaningless. However if we replace (ii) with the assumption that S contains a rank 1 matrix, then the conclusion of Theorem 4.20 holds. The reason is the special multiplication property for such matrices given by      0z xy 0z D 01 0 1 01

(4.170)

so that a rank 1 matrix of the form in (4.169) multiplied by another matrix of the form in (4.169) (not necessarily rank 1) remains unchanged. Now we need to use Theorem 4.19. Note that in Theorem 4.19, S D Sˇ D the kernel of S , which, in this particular situation, is the set of all rank 1 matrices of S (see Proposition 1.9), and moreover ˇ D ˇ  D  ˇ, which now implies that for any Borel set B  S , Z     ˇ B \ Sˇ D ˇ.B/ D ˇ w1 B .d w/ Z   ˇ w1 B .d w/ D S

D  .B \ S /;

since

w1 B D fu 2 S j w  u 2 Bg D fu 2 S j w 2 Bg;

where w has rank 1 and consequently w  u equals w. Thus  D ˇ and the invariant probability measure is unique. When S contains no rank 1 matrix, then (ii) in Theorem 4.20 can be replaced by (ii)0 .J / D 0 and S \ I ¤ ;, where J is as in Theorem 4.20 and I as in (4.25) ro reach the same conclusion. The proof is similar (see (4.120), (4.121), and Theorem 4.10) and omitted. Thus, we can state Theorem 4.21. Theorem 4.21. Consider the mixed random walk .Sn /, defined as in Theorem 4.20, with the corresponding i.i.d. sequence .Xi / taking values in .d C 1/  .d C 1/ nonnegative matrices in the form of (4.169). Let , S , and J be as in Theorem 4.20. Suppose .n / is tight and one of the following assumptions hold: (i) S contains a matrix with rank 1; (ii) .J / D 0 and S \ I ¤ ;, where I is as in (4.25).

4.4 Invariant Measures for Random Walks in Nonnegative Matrices: Laws of Large Numbers 345

Then the mixed random walk has a unique invariant probability measure whose support is the smallest ideal of S . Conversely under (ii) existence of a nontrivial (that is, not a unit mass) invariant measure implies the tightness of the sequence .n /. Now we present a law of large numbers for the mixed random walk. The following theorem of Furstenberg and Kifer is useful in this context. Theorem 4.22 (Furstenberg–Kifer). Let fZn W n D 0; 1; 2; : : : g be a Markov process with values in a compact metric space M with transition probability x such that P fZnC1 2 A j Z0 ; : : : ; Zn g D Zn .A/; where x is a continuous map from M into P .M / and P .M / is the compact metric space of probability measures on the Borel sets of M with respect to weak convergence. Then for any continuous real function f on M , the following results hold: Z n 1X (i) lim sup f .Zk / sup f d W  2 P .M / and for A  M; n!1 n kD0

Z .A/ D x .A/.dx/ ; (ii) If the set inside the parenthesis in (i) is a singleton and its value is f , then n 1X f .Zk / D f almost surely: n!1 n

lim

kD0

Proof. The proof is given in several steps. Step 1. Consider the Markov operator P defined on C.M /, the real continuous functions on M with the usual sup norm, by Z Pf .x/ D

f .y/x .dy/:

(4.171)

Then P maps C.M / into C.M /. Let f D g  P g with g 2 C.M /. Then we claim n 1X f .Zk / D 0 almost surely: n!1 n

lim

kD0

To prove (4.172), consider the sequence .Un / given by UnC1 D

n X kD0

1 ŒP g .Zk /  g .ZkC1 / : .k C 1/

(4.172)

346

4 Random Matrices

Then the following assertions hold: (i) E .UnC1 j Z0 ; Z1 ; : : : ; Zn / 1 ŒP g .Zn /  E fg .ZnC1 / j Z0 ; : : : ; Zn g D Un ; D Un C nC1 since P g .Zn / D E fg .ZnC1 / j Z0 ; : : : ; Zn g; (ii) For m < n, E fŒP g .Zm /  g .ZmC1 / ŒP g .Zn /  g .ZnC1 /g D E fE Œ.P g .Zm /  g .ZmC1 // .P g .Zn /  g .ZnC1 // j Zm ; ZmC1 ; Zn g D 0: It follows that

1 X  2  1  4kgk2  < 1: E UnC1 k2 kD1

By the Martingale convergence theorem (see [38]), the sequence .Un / converges almost surely. By Kronecker’s lemma (see [38, page 123]), n 1X ŒP g .Zk /  g .Zk / D 0 n!1 n

lim

kD0

almost surely. This establishes (4.172). Step 2. Let f 2 C.M /, f  0. Then given " > 0, we can write f D P g  g C h;

(4.173)

where g and h are in C.M / and Z khk  sup

f d W  2 P .M / and P  D  C "; Z

where P .A/ D

x .A/.dx/:

To prove (4.173), let V  C.M / be the subspace given by V D fP g  g W g 2 C.M /g and let dist.f; V / D ı. If ı D 0, then (4.173) is obvious. Let ı > 0. Then by the Hahn–Banach theorem (of functional analysis), there is a bounded linear functional x  on C.M / such that kx  k D 1;

x  .f / D ı and x  .h/ D 0

4.4 Invariant Measures for Random Walks in Nonnegative Matrices: Laws of Large Numbers 347

for h 2 V . By the Riesz-representation theorem, there is abounded signed measure ˇ such that Z x  .h/ D

hdˇ;

h 2 C.M /:

Since x  .h/ D x  .P h/ for h 2 C.M /, Z

Z hdˇ D

Z P h dˇ D

hd.P ˇ/;

Z

where P ˇ.A/ D

x .A/ˇ.dx/:

Decomposing ˇ into its positive and negative parts, we have ˇ D ˇC  ˇ ;

P ˇ D P ˇC  P ˇ :

Notice that if A is a positive set so that ˇ.A/ D ˇ C .A/ and Z C P ˇ .A/ D x .A/ˇ C .dx/ Z Z  x .A/ˇ C .dx/  x .A/ˇ  .dx/ D P ˇ.A/ D ˇ.A/ D ˇ C .A/: This means P ˇ C  Rˇ C . Similarly, P ˇ   ˇ  . It follows that P ˇ CD ˇC and P ˇ  D ˇ  . Since f dˇ D ı, f  0, and kˇk D 1, it is clear that ˇ C  > 0. Let ˇC D : kˇ C k Z

Then, P  D  and

Z f d 

f dˇ D ı:

Assertion (4.173) now follows immediately. Step 3. We observe that (i) in Theorem 4.22 follows immediately by proving (i) first for C C f , where C D kf k, so that C C f  0. By considering (i) first for f then for f , the Assertion (ii) also follows. t u Note that the Furstenberg–Kifer theorem exploited the compactness of the state space M . However, even in the absence of compactness, we can use the preceding proof to obtain a law of large numbers for the mixed random walk with some modifications. In what follows, we consider the mixed random walk of Theorem 4.20 (with the assumption that the zero matrix is not in S and under (i) and (ii) of this theorem) or the walk of Theorem 4.21 (under condition (ii) given in this theorem). Note that we still assume .n / is tight and define P as in (4.148) and T as in (4.141). It fol-

348

4 Random Matrices

lows that the unique invariant probability measure  (for P ) satisfies the following properties: (i) Under conditions of Theorem 4.20, .J / D 1; furthermore, every element in S is an idempotent, since S is a completely simple subsemigroup, where its group factor (the set eS e, e D e 2 2 S ) is a singleton (see the proof of Theorem 4.9). (ii) Under conditions of Theorem 4.21, fs 2 S W the last column of s is strictly positiveg D 1 if S consists of all rank 1 matrices in S . If there is no matrix in S with rank 1, then S consists of only matrices of the form   B0 ; B ¤ 0; B is d  d; (4.174) 0 1 (as shown in (4.121)), so that this case is the same as in Theorem 4.20 and as such, the results in (i) are valid in this case, identifying all .d C 1/  .d C 1/ matrices of the form given in (4.174) with simply the matrices B there. Note that when (i) holds, then given any compact subset K  S such that .K/ > 0, there is a function h in C0 .S /, the space of real continuous functions on S vanishing at infinity, such that 0  h  1 and h D 1 on K  K and since K  K  K (see Property (i) above), Z hd > 0I (4.175) also for any x 2 K, since K  x  K  K, Z h.yx/.dy/ > 0:

(4.176)

When (ii) holds and S consists of only rank 1 matrices (under conditions of Theorem 4.21), then x; y 2 S ) y  x D y: This means that given any compact subset K  S with .K/ > 0, we can again find h 2 C0 .S / satisfying (4.175) and (4.176). Let us now observe that the operator T , defined as in (4.141), is a bounded linear operator from C0 .S / into C0 .S /. The reason follows. Given " > 0, let A be a compact subset of S such that A  J c and .A/ > 1 ". If f 2 C0 .S /, B is compact and jf .s/j < " for s … B; then for y 2 A, jf .xy/j < " if x … BA1 , and

4.4 Invariant Measures for Random Walks in Nonnegative Matrices: Laws of Large Numbers 349

jf .xy/j < " if x … A1 B. Since A does not contain a matrix which has a zero row or a zero column and since S contains matrices with only nonnegative entries, the sets BA1 and A1 B are both compact. Then it follows easily that T W C0 .S / ! C0 .S /: Now following the proof of Theorem 4.22, we observe that n 1X h .Mk / D 0 almost surely; n!1 n

lim

(4.177)

kD0

where h D Tf  f , f R2 C0 .S /, and .Mk / is the mixed random walk. Let f 2 C0 .S / and f d D 0, where  is the unique invariant probability measure for .Mn /. Again considering the subspace V as in the proof of Theorem 4.22, we see that either the measure ˇ C or ˇ  there must be the zero measure (because of the uniqueness of the invariant measure for .Mn /) and consequently R the normalized measure ˇ=ˇ.S / obtained there must be either  or . Since f d D 0, it follows that f 2 closure.V /, and consequently from (4.177), n 1X f .Mk / D 0 almost surely: n!1 n

lim

(4.178)

kD0

R Now let f 2 C0 .S / and f d > 0. Again using the same argument as before and noting that  is the unique invariant probability measure for .Mn /, it follows that either the measure ˇ C or ˇ R (obtained in the proof of Theorem 4.22) again must be the zero measure. Since f d > 0, ˇ  must be the zero measure, ˇ D ˇ C , and  D ˇ=ˇ.S /. Since the functional x  there has norm 1 and the measure ˇ is induced by x  , ˇ.S / D 1 and consequently  D ˇ, so we have almost surely lim sup

n!1

Z n 1X f .Mk /  f d: n kD0

Note that

!

n 1X f .Mk / E lim sup n!1 n kD0

 lim sup E n!1

!

n 1X f .Mk / n kD0

n “ 1X f .y/P k .x; dy/.dx/: D lim sup n!1 n kD0

(4.179)

350

4 Random Matrices

Under the conditions we have assumed, it follows from Theorems 4.9 and 4.16 that n converges to a probability measure ˇ0 weakly. Following the proof of Theorem 4.18 (which works here even though S is not assumed to be compact, since we can restrict y and z in (4.142) with no loss of generality to a compact set using the tightness of .n /), it follows that for each x in S and f in C0 .S /, as in the proof of Theorem 4.18, lim T n f .x/ D T f .x/I n!1

Z

that is,

Z f .y/P n .x; dy/ D

lim

n!1

f .y/ˇ0 ıx ˇ0 .dy/:

(4.180)

This means “ lim f .y/P n .x; dy/.dx/ n!1 “ D f .y/ˇ0 ıx ˇ0 .dy/.dx/ • D f .yxz/ˇ0 .dy/.dx/ˇ0 .d z/ Z Z D f .y/ˇ0  ˇ0 .dy/ D f .y/ˇ0 .dy/; since ˇ0  ˇ0 D ˇ0 ˇ0 D ˇ0 . It follows that ! Z n 1X E lim sup f .Mk /  f dˇ0 : n!1 n

(4.181)

kD0

Since ˇ0 D  ˇ0 D ˇ0 , it follows from Proposition 2.5 that for each x 2 S , ˇ0 ıx ˇ0 D ˇ0 . Since Z P .x; A/ˇ0 .dx/ D ˛ˇ0 .A/ C .1  ˛/ ˇ0 .A/ D ˛ˇ0 .A/ C .1  ˛/ˇ0 .A/ D ˇ0 .A/; it follows that  D ˇ0 , where the invariant measure is unique. Now we observe that the inequality in (4.179) must actually be an equality almost surely. The reason is: If there is strict inequality in (4.179) on a set of positive probability, then Z

! n 1X f d > E lim sup f .Mk / ; n!1 n kD0

4.4 Invariant Measures for Random Walks in Nonnegative Matrices: Laws of Large Numbers 351

which contradicts (4.181), since  D ˇ. Thus, for f 2 C0 .S / and

R

Z n 1X lim sup f .Mk / D f d, almost surely: n!1 n

f d > 0, (4.182)

kD0

Now let X be a nonnegative random matrix, independent of the Xi s and the Yi s such that the distribution of X is . Then the process f .Mk X /, 1  k < 1 and f 2 C0 .S /, is a stationary process, so by the classical law of large numbers, P .B/ D 1, where (

) n 1X B D ! W lim f .Mk X / exists : n!1 n kD0

Z

Then 1D

P .B j X D x/.dx/:

This means given any subset A  S with .A/ > 0 and given any f 2 C0 .S /, there exists x 2 A such that there exists almost surely n 1X f .Mk x/ : n!1 n

lim

(4.183)

kD0

Let K be a compact subset of S such that .K/ > 0 and K  fs 2 S W the last column of s is strictly positiveg. Note that this is possible because of Observations (i) and (ii) just preceding (4.174). Choose h in C0 .S / for this K as in (4.175). By (4.183), there exists x 2 K such that there exists almost surely n 1X h .Mk x/: n!1 n

lim

(4.184)

kD0

Define h1 .y/ D Rh.yx/ for y 2 S . Since the last column of x is strictly R positive, h1 2 C0 .S / and h1 d > 0 by (4.176). Now let g 2 C0 .S / such that gd ¤ 0. Choose a real number p such that Z .p  h1  g/ d D 0: Using results from (4.178) and (4.184), it follows that there exists almost surely n 1X g .Mk /: n!1 n

lim

kD0

352

4 Random Matrices

It follows from (4.178) and (4.182) that if g 2 C0 .S / and

R

gd > 0, then

Z n 1X lim g .Mk / D gd almost surely n!1 n

(4.185)

kD0

R Replacing g by g when gd < 0, it follows that (4.185) holds for any g 2 C0 .S /. Finally, let g be a bounded continuous nonnegative function on S . Then given " > 0, there is some g0 in C0 .S / such that Z 0  g0  g and

.g  g0 / d < ":

Thus, n 1X g .Mk / n!1 n kD0 Z n 1X  lim inf g0 .Mk / D g0 d n!1 n kD0 Z > gd  ";

lim inf

almost surely. When g is an arbitrary bounded continuous function (such that jgj  M , a constant), then applying the preceding result on the functions M  g and M C g, we have Z n 1X lim g .Mk / D gd, almost surely: n!1 n

(4.186)

kD0

Thus, we have proven Theorem 4.23. Theorem 4.23. Consider the mixed random walk .Mn / in Theorem 4.20 (under the assumptions there) or the mixed random walk in Theorem 4.21 (under the assumption (ii) there). Assume that S does not contain the zero matrix. Then for any bounded continuous function f on S , Z n 1X f .Mk / D f d; n!1 n lim

kD0

almost surely, where  is the unique invariant probability for .Mk /.

4.4 Invariant Measures for Random Walks in Nonnegative Matrices: Laws of Large Numbers 353

We conclude Sect. 4.4 with a number of examples illustrating conditions used in Theorem 4.23 in the context of affine maps from Rd into Rd (or equivalently, .d C 1/  .d C 1/ matrices of the form (4.169)). Example 4.5. Consider maps f , g, and h given at the end of Sect. 4.2. Any probability measure  such that S D ff; g; hg meets all the conditions of Theorem 4.23. Example 4.6. Consider the family F of affine maps from Rd into Rd such that each f 2 F has the form x 2 Rd ;

f .x/ D A.f /x C B.f /; where f can be identified with the matrix   A.f / B.f / 0 1

(which has the form (4.169)) and there are positive numbers a with 0 < a < 1 and b such that for each f 2 F , kA.f /k < a and kB.f /k < b: If x is a d -column vector, then      f .x/ A.f / B.f / x D 1 0 1 1 and

      f ı g.x/ A.f / B.f / A.g/ B.g/ x D : 1 0 1 0 1 1

It is easily verified that if S  F , then S cl

1 [

!! Sn

nD1

is compact. Example 4.7. Consider an i.i.d. sequence of random .d C 1/.d C 1/ real matrices .Xn / such that   An Bn Xn D 0 1 is of the type described in (4.169). Suppose that almost surely, each kBn k < M < 1, and there are numbers ˛0 < ˛1 <    < ˛N such that   P ˛j 1 < kAn k  ˛j D ˇj ;

1  j  N;

354

4 Random Matrices

where the ˛i s and ˇi s satisfy conditions N X

ˇj D 1 and

j D1

N X

ˇj log ˛j . log r, say/ < 0:

(4.187)

j D1

Note that unlike Examples 4.5 and 4.6, the Xn s here do not need to be almost surely contraction maps, since some of the ˛i s can be much larger than 1 as long as (4.187) holds. Let  be the distribution of each Xn . We show that the convolution sequence .n / is tight, even though S is not necessarily contained in a compact matrix semigroup. Let us write ˚  B.j; n/ D ! W ˛j 1 < kAn .!/k  ˛j : (4.188) By the strong law of large numbers, then we have almost surely, ynj

n 1 X IB.j;m/ n mD1

converges to ˇj . Choose s > 0 such that log r1 log r C s

N X

j log ˛j j < 0:

(4.189)

j D1

Let 0 < t < s. There exists a positive integer-valued random variable n0 .!/ such that P .n0 < 1/ D 1 and n > n0 .!/ implies ˇ ˇ ˇ ˇ n.j; !/ ˇ (4.190)  ˇj ˇˇ < t; 1  j  N; ˇ n where n.j; !/ is the cardinality of the set ˚  m  n W ˛j 1 < kAm .!/k  ˛j : Here,

ˇ ˇ  ˚ n0 .!/ D inf m W ˇynj .!/  ˇj ˇ < s 8n  m :

Let L be a positive integer such that P .n0 < L/ > 1t. Let n > L and n0 .!/ < L. Then kA1 .!/A2 .!/ : : : An .!/k 

N Y 

˛j

j D1



N Y j D1

 n.ˇj Cs / ˛j D r1n :

n.j;!/

4.5 Notes and Commments

355

Since the first row of X1 X2 : : : Xn is .A1 A2 : : : An ; B1 C A1 B2 C A1 A2 : : : An1 Bn /; it is clear there exists C > 0 such that fn0 .!/ < Lg  fkX1 .!/ : : : Xn .!/k  C 8n > Lg :

Section 4.4 Exercises Exercise 4.27. Establish (4.140) and (4.142). Exercise 4.28. Prove the inequality just preceding (4.142). Exercise 4.29. Establish (4.142). Exercise 4.30. Discuss the difficulties in the development of the results without the assumption S1 D S2 at the beginning of this section. Exercise 4.31. Give versions of Theorem 4.18 in full details for the left random walk as well as for the right random walk. Exercise 4.32. Give the details of the proof of Theorem 4.21. Exercise 4.33. Explain the difficulties in the proof of Theorem 4.23 if S is allowed to contain the zero matrix.

4.5 Notes and Commments Section 4.2 Theorems 4.1 and 4.2 and Corollary 4.1 were originally results of an effort to understand how much of the classical theory due to Chung and Fuchs (see [19]) remains valid for random matrices. These results are taken from [156]. The reader may note that nonnegativity of the entries in the matrices has been crucial in proving these results. The recurrent sets R.Z/ and R.W / are of course the same in Rd ; but the picture is less clear for products of random matrices where we do not have commutativity of multiplication, and as such, natural questions arise regarding the structures of the sets R.Z/ and R.W / and how they are related. Theorems 4.3–4.6 answer some of these questions, and these are taken from [157]. The result that R.Z/ is nonempty in (4.46) is from [78].

356

4 Random Matrices

The concept of recurrence for matrices is used in the last part of this section to define attractors in a manner consistent with works of M. Barnsley and colleagues in [6]. Attractors are also introduced by various authors as supports of certain invariant probability measures. Theorem 4.7, taken from [157], shows how the two definitions are connected. Finally in this section, we show how the structure of the recurrent set R.W / can be exploited to give a complete description of the well-known attractor, the Sierpinski Gasket, and one of its unbounded versions. This was also given in [153]. Quite a few results in this section are less clear for matrices which are not necessarily nonnegative, even though many of the results in Sect. 4.3 are given for the general case of real matrices. A number of related results for real matrices can be found in [108]. Related results also appear in [101, 106, 107]. It is worthwhile to obtain analogs of all results in this section for random real matrices. See [63] and other papers mentioned there for more details on the Exercises 4.10–4.15.

Section 4.3 Many of the results in this section are obtained in the presence of one or both of the following conditions: (1) the set of matrices with at least one zero row or one zero column has zero measure, and (2) the set of strictly positive matrices in S is nonempty. As far as we know, these conditions were first explored by Kesten and Spitzer in [124] to avoid some pathologies in their study of convergence in distribution of products of i.i.d. nonnegative matrices. While some results in this section are for nonnegative matrices, a number of results such as Theorems 4.11– 4.17 and Proposition 4.2 are given for real matrices. It will be of course interesting to find whenever possible what the results are in the general situation. It is relevant to point out that papers by Bougerol [16] and Mukherjea [159] give results on tightness for the general case of real matrices. Theorems 4.11 and 4.12 are contained in [159] and Theorem 4.13 in [149]. A short readable account of some of the problems and results in this context is given in [104]. Theorem 4.17 follows from the work in [78], Theorem 4.8 appeared in [155], and Proposition 4.1 and Theorem 4.9 in [149]. Both Theorems 4.10 and 4.14 are due to Kesten and Spitzer [124]. Theorem 4.14 was only stated there. Proofs of Theorems 4.10 and 4.14 are taken from [149]. Analogs  d  d of these two results for affine maps from RC into RC are in Theorems 4.15 and 4.16, which appeared in [157]. Propositions 4.2–4.4 appeared in [16], [156], and [155], respectively. It is worthwhile pointing out the following interesting result (contained in Lemma 2 of [124]). Let .Xi / be i.i.d. d  d nonnegative matrices with distribution , Yn D Xn Xn1 : : : X1 , .J / D 0 and S \ J ¤ ;, where J , S and J are as in Theorem 4.10. Then the sequence Qn D kYYnn k has a limit distribution which is concentrated on the strictly positive matrices of rank 1. A “semigroup” proof of this result can be given along the following lines.

4.5 Notes and Commments

357

AB Note that if we define for matrices A and B, A  B D kABk , then .J c ; / is n a locally compact Hausdorff topological semigroup and  , the nth convolution power of  with respect to this new multiplication, is the distribution of Qn . Let T D fx j kxk D 1g. Then fn ; n  2g  P .T / and it is easy to see that the n P averages n1 k converges weakly to , k  k D  for k  1, J  S  J  kD1

J and .J / D 1 (since (4.55) holds). Also,   D . Restricting  to J c , it follows that the support of the 0 , the restriction of  to J c , is a completely simple semigroup X  G  Y by Theorem 2.8, where G is a compact group (with respect to ) of strictly positive matrices. (Note that for any x 2 S0 , the set x  S1  x, where S1 is the closed semigroup generated by S0 , can be taken as G.) Now we claim that G is a singleton set so that S0 consists of idempotent (with respect to ) matrices. To see this, we use the Frobenius theory. By Theorem 2.3 on page 546 n of [118], if e is the identity of G, then for some constant c, ec n converge pointwise to a rank 1 idempotent strictly positive matrix f . This means that since ke n k D 1, n n e D ec n kecn k ! kff k , so that e has rank 1. Since for x 2 G, x D x  e, x has rank 2

1. Thus, for x 2 G, x 2 D rx for some constant r so that x  x D kxx 2 k D x, since kxk D 1. Hence, G is a singleton, since every element G is idempotent. Next, considering fn ; n  2g  P .T /, we claim that for any weak limit point  of .n /,  .J c / D 1. The reason is the following: J D Jr [ Jc (see the definition on page 287). If  .Jr / > 0, then since   D  D   with n 1X k  ; n!1 n

 D .w/ lim

kD1

it follows that  .Jr /   .Jr / > 0 which contradicts that .J /1. Thus,  .Jr / D 0. Similarly,  .Jc / D 0 and therefore  .J c / D 1. This means that the weak limit points of n in P .T / can be regarded as weak limit points of n in P .J c /. It follows by Theorem 2.7 that n converges weakly to 0 , 0 .J / D 1, and S0 is a completely simple semigroup of rank 1 matrices with norm 1 and such matrices are easily seen to be of the form r `T , where r is a strictly positive column vector and `T is a strictly positive probability row vector. Note that since G is a singleton, we can regard S0 D X Y and YX D e D e  e. If .Xi / is a stationary, ergodic sequence of d  d matrices and Yn D Xn Xn1 : : : X1 , then it follows from Kingman’s subadditive ergodic theorem (see [127]) that when E logC kX1 k < 1, lim n1 log kYn k D E 2 Œ1; 1/, almost surely. The n!1 limit E is usually called the maximal Lyapunov exponent. Eric Key (see [126]) has used the Kesten and Spitzer result proved above to show that when   .Xi / are i.i.d. with distribution , .J / D 0, S \ J ¤ ; and E logC kX1 k < 1, then lim n1 log .Yn / D E almost surely, where .Yn / is the spectral radius of Yn . This result was earlier conjectured by Joel Cohen in [44]. In the context of computing the maximal Lyapunov exponent, let us mention [101, 125], and [185].

358

4 Random Matrices

Finally, in the context of weak convergence in 2  2 stochastic matrices, it is relevant to point out the following result: Let  be a probability measure on 2  2 stochastic matrices. Let .a; b/ denote the matrix whose first column elements are a and b. Then n converges weakly if and only if  is not the unit mass at .0; 1/. If S./, the support of , contains some .a; b/, where either 0 < a < 1 or 0 < b < 1, then the support of the weak limit of n contains only matrices with identical rows. For a proof of this, see [147]. In connection with continuous singularity of the weak limit in 2 by 2 and n by n stochastic matrices, see [161, 175]. The paper of Hennion (1997) covers extensive related results on products of random nonnegative matrices.

Section 4.4 As far as we know, mixed walks in compact semigroups were first studied by H¨ogn¨as (see Chap. 3); left and right random walks are special cases of these walks. The emphasis in this section is results showing the existence of a unique invariant probability measure for such walks. Theorems 4.18 through 4.21 are results in this direction; they are given in [109]. These results and Theorem 4.22, which is due to Furstenberg and Kifer [71], are used to obtain Theorem 4.23, which appeared in [109]. Example 4.7 is taken from [153]. This example describes an average contractivity condition (see (4.187)) for affine maps. It is shown here that this condition is really the tightness condition in disguise at least in the i.i.d. situation. A very similar condition was first introduced by Elton (see [6, 67]). For random affine maps, see also [158].

Appendix A

Products of I.I.D. Random Stochastic Matrices: Their Skeletons and Convergence in Distribution

Let 1 and 2 be two probability measures on the Borel subsets of d  d stochastic matrices with the usual Euclidean topology. Let S1 and S2 be, respectively, the compact multiplicative semigroups of d  d stochastic matrices generated by the supports S .1 / and S .2 /. For x in S1 [ S2 , we denote by xN the idempotent (identity) element in the group of limit points of fx n j n 1g. Throughout our discussion here, we make the following “skeleton” assumption. Assumption*. Let D D f1; 2; : : : ; d g. For each x 2 S .1 /, there is some y 2 S .2 / such that x and y have the same skeleton (that is, for all i; j in D, xij > 0 iff yij > 0). Similarly, for each y in S .2 /, there is some x in S .1 / such that x and y have the same skeleton. t u Our main purpose here is to show that under the above assumption, n1 converges weakly as n ! 1 iff n2 does so. Note that the skeleton assumption () easily 1 1 S S S .1 /m and S .2 /m , but may not extend to their closures, that extends to mD1

mD1

is, to S1 and S2 . For i D 1; 2, let us define the sets Ki by Ki D fx 2 Si j x has the minimal rank among all matrices in Si g : Then it is known (see Chap. 1) that Ki is the kernel of Si , i D 1; 2. We claim that under assumption (), the matrices in K1 [K2 have the same rank; that is, if x 2 K1 and y 2 K2 , then rank x D rank y. ˚ To prove this claim, let e be an idempotent element of K1 with basis T; C1 ; C2 ; : : : Cp such that among all idempotent elements in K1 , e has the minimum number of zero columns. See Theorem 1.11 for the definition of basis. 1 S First, suppose that e 2 S .1 /m . Then, of course, there is some z 2 1 S

mD1

S .2 /

m

such that e and z have the same skeleton. Thus, for 1  i  p,

mD1

1  j  p, each Ci T block, each Ci Cj (i ¤ j ) block, and the T T block of z is an all zero block, and each Ci Ci block of z is a strictly positive stochastic matrix.

359

360

A Products of I.I.D. Random Stochastic Matrices

The same is true for all powers of z, and therefore, for the element zN, an idempotent element in S2 . Note that since  n zN jCi Ci D lim z jCi Ci ;

1  i  p;

zN jCi Ci has rank one. Also, it is easily verified that if z jT Ci is strictly positive, for some i , 1  i  p, then the same is true for zN. Thus, e and zN have the same basis, and also the same skeleton. 1  ˚ S Next we assume that e 2 K1  S .1 /m , and e has basis T; C1 ; C2 ; : : : ; Cp . mD1

Then, with no loss of generality, we can assume that there are elements en in 1 S S .1 /m such that en ! e as n ! 1, for each i , 1  i  p, en jCi Ci is

mD1

strictly positive for each n, for all j in D D f1; 2; : : : ; d g and t 2 T , .en /jt < 2d1 2 , and en jT Ck is strictly positive whenever e jT Ck is so, for each n. Let Gn be the group of limit points of fenm j m 1g. Then eNn is the identity of Gn . Choose yn 2 Gn such that yn .en eNn / D .en eNn / yn D eNn : But since yn eNn D yn D eNn yn and en eNn D eNn en , we then have: yn en D en yn D eNn :

(A.1)

 ˚ Let the basis of eNn be T 0 ; C10 ; C20 ; : : : ; Cq0 . Note that eNn 2 S1 , but e 2 K1 , and thus q p. Now we claim that e and eNn have the same basis and the same skeleton. Let j 2 Ci0 , 1  i  q, and t 2 T 0 . Then we have 0 D .en eNn /jt D .eNn en /jt X D .eNn /jk .en /kt I k2Ci0

this implies that en jCi0 T 0 is an all zero block. Corresponding to yn in (A.1) by Theorem 1.14, there is a permutation  of f1; 2; : : : ; qg such that the Ci0  Cj0 of yn is a block with all entries strictly positive or zero accordingly as .i / D j or .i / ¤ j . Notice also that yn en D eNn , and thus, for u 2 Ck0 , we have: 1D

X j 2Ck0

.eNn /uj D

X 0 `2C.k/

.yn /u`

X

.en /`j ;

j 2Ck0

0 and this implies that en jC.k/ Ck0 is a stochastic matrix, whenever 1  k  q. We will now show that the permutation  corresponding to yn must be the identity permutation. If not, then there is some j , 1  j  q, such that j ¤ .j /, and consequently, ..j // ¤ .j /. Write: .j / D i and .k/ D j . Then i ¤ j , j ¤ k. It follows that en jCi0 Ci0 as well as en jCj0 Cj0 is an all zero block. Thus,

A Products of I.I.D. Random Stochastic Matrices

361

Ci0 [ Cj0  T , because of the way en was chosen, and Ci0  Cj0  T  T . But this contradicts the fact that en jCi0 Cj0 is a stochastic matrix, since for t1 2 T , t2 2 T ,

.en /t1 t2 < 2d1 2 , because of the way en was chosen. Thus,  must be the identity permutation, and thus, yn must be the element eNn (see Chap. 1), and this means that for k 1, eNn enk D enk eNn D eNn : It follows that for any y in Gn , eNn y D y eNn D eNn so that Gn is a singleton and lim enk D eNn . Also, since en eNn D eNn en D eNn , en must be of the form k!1

T0 0

C10

C20

T C10 0 Stochastic 0 C20 0 0 Stochastic Recall that en jT T is strictly sub-stochastic and thus, no Ci0 can be completely contained in T , and no Ci can intersect two different Cj0 (because of the way en was chosen). Thus, each Cj0 is completely contained in Ci [ T , for some i (i D i.j /, i depending on j ), and i.j / ¤ i.k/ for j ¤ k. It follows that p D q, T 0  T , and each Ci  some Cj0 . Now e, by choice, has the minimum number of zero columns among all idempotent elements of K1 ; this means that T 0 D T , and consequently, each Ci D some Cj0 . In other words, e and eNn have the same basis; in fact, they must also have the same skeleton since en eNn D eNn en D eNn . 1 S Now since en 2 S .1 /m , by our skeleton assumption, there must exist zn 2 1 S

mD1 m

S .2 / such that zn and en have the same skeleton for each n. Thus, zn , for

mD1

each n, must be of the form T C1

C2

T Stochastic (and all C1 0 0 entries positive) Stochastic (and all C2 0 0 entries positive) It is clear that each row sum in the zn jT T block is less˚ than 1 since the  same is true for en . It follows that zNn must have the same basis T; C1 ; : : : ; Cp and also the same skeleton as e. Thus, the minimal rank of the matrices in S2 is at most p. Reversing the roles of S1 and S2 , we have proven the following theorem.

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A Products of I.I.D. Random Stochastic Matrices

Theorem A.1. Under Assumption (), the minimal rank of the matrices in K1 is the same as that of the matrices in K2 . Furthermore, there are idempotent elements e in K1 and f in K2 such that e and f have the same basis and the same skeleton. t u The kernels K1 and K2 are completely simple subsemigroups of S1 and S2 , respectively, and as such, they have their usual Rees–Suskewitsch decompositions of the form X1  G1  Y1 and X2  G2  Y2 . Let us first describe how we choose the sets X1 ; X2 , G1 ; G2 , and Y1 ; Y2 . Theorem A.1 gives us idempotent elements e 2 K1 and f in K2 such that e and f have the same skeleton and the same basis. We take the set E .K1 e/ as X1 , E .K2 f / as X2 , E .eK1 / as Y1 , E .fK2 / as Y2 , the group eK1 e as G1 , and the group fK2 f as G2 . Here, E.A/ denotes the set of all idempotent elements in A. It is wellknown that K1 is isomorphic to X1  G1  Y1 and K2 to X2  G2  Y2 (see Chap. 1). We claim that under Assumption (), the group factors G1 and G2 are isomorphic. To prove this claim, let us suppose that the basis of the idempotent elements e and f is given by ˚  T; C1 ; C2 ; : : : ; Cp : Let x 2 K1 . Then exe 2 G1 . There exists a sequence xn 2

1 S

S .1 /m such that

mD1

xn ! x. Then exn e ! exe as n ! 1. We choose n sufficiently large so that whenever .exe/ij > 0, then .exn e/ij is also positive. This means that exe and exn e (for n large), both elements in G1 , must correspond to the same permutation  on f1; 2; : : : ; pg in the following sense: Ci  Cj block of exe has all elements strictly positive iff j D .i /. Since G1 is compact (in fact, finite being a compact group of stochastic matrices), the correspondence g !  from G1 to the group of permutations on f1; 2; : : : ; pg is an isomorphism (though possibly not onto). This means that there are elements yn in 1 S S .2 /m such that for each n, exn e and fyn f have the same skeleton so that the mD1

element exe in K1 and the element fyf in K2 (when y is a limit point of the sequence yn ) must correspond to the same permutation  on f1; 2; : : : ; pg. Reversing the roles of G1 and G2 , it is clear that the groups G1 and G2 are isomorphic. Thus, we have proven the following theorem. Theorem A.2. In the presence of Assumption (), the group factors G1 and G2 in the Rees–Suskewitsch product representations of K1 and K2 are isomorphic. t u Consider again the Rees–Suskewitsch product representations X1  G1  Y1 of K1 . Let N1 be the smallest normal subgroup of G1 such that Y1 X1  N1 . Define G1 the map ˚1 W S1 ! N , the factor group of G1 by N1 as follows: 1 ˚1 .s/ D ese N1 ;

A Products of I.I.D. Random Stochastic Matrices

363

where e is the fixed idempotent in K1 such that G1 D eK1 e, X1  E .K1 e/, and Y1  E .eK1 /. As is shown in Chap. 2, Theorem 2.18, ˚1 is a continuG1 ous homomorphism onto, and if we define the probability measure Q 1 on N by 1  1  n Q 1 .B/ D 1 ˚1 .B/ , then the sequence 1 on S1 converges weakly if the se1 quence Q n1 on G converges weakly. N1 As we defined N1 and ˚1 above, we similarly define N2 and ˚2 . Now we claim G2 1 that the homomorphic group images G N1 and N2 are isomorphic under Assumption (). To prove this claim, we consider again the Rees–Suskewitsch decomposition X1  G1  Y1 and X2  G2  Y2 of K1 and K2 (that we considered earlier), where G1 D eK1 e and G2 D fK2 f , as shown in Theorem A.2, are isomorphic. Notice that Y1 X1  G1 and Y2 X2  G2 . If we can show that the sets Y1 X1 and Y2 X2 both correspond to the same set of permutations under the usual isomorphisms from the groups G1 and G2 to the group of permutations on f1; 2; : : : ; pg, then it follows imG2 1 mediately that the factor groups G N1 and N2 are also isomorphic, where N1 and N2 are the smallest normal subgroups containing Y1 X1 and Y2 X2 , respectively. To show 1 1 S S this, let y1 2 Y1 and x1 2 X1 . Let y1n 2 S .1 /m , x1n 2 S .1 /m , mD1

mD1

y1n ! y1 , x1n ! x1 , and such that whenever .y1 /ij > 0, .y1n /ij is also so, and whenever .x1 /ij > 0, .x1n /ij is also so. 1 S Choose x2n and y2n in S .2 /m such that for each n, x2n and x1n have the mD1

same skeleton, and y2n and y1n have the ˚ same skeleton. Let  gn be the identity in the group of limit points of the sequence .f y˚ 2n /m W m 1 , and  hn be the identity in the group of limit points of the sequence .x2n f /m W mh 1 . Then, i fgn D gn and hn f D hn . Choose M and L sufficiently large so that .f y2n /M i h whenever .gn /ij > 0, and .x2n f /L > 0, whenever .hn /ij > 0.

ij

is positive,

ij

Note that the elements e .ey1n /M .x1n e/L e and f .f y2n /M .y1n f /L f have the same skeleton and, therefore, represent the group elements in G1 and G2 that h i correspond to the same permutation on f1; 2; : : : ; pg. Also, e .ey1n /M .x1n e/L e ij h i M L is positive whenever the element e .ey1 / .x1 e/ e D .y1 x1 /ij is positive. ij i h > 0, whenever the element Similarly, the element f .f y2n /M .x2n f /L f ij

Œfgn hn f ij D .gn hn /ij > 0, and thus, both represent the same group element gn hn . Notice that gn2 D gn D fgn 2 fK2 so that gn 2 Y2 , and h2n D hn D hn f 2 K2 f so that hn 2 X2 . Thus, with every element y1 x1 in Y1 X1 , we have associated an element gn hn in Y2 X2 that corresponds to the same permutation on f1; 2; : : : ; pg. Reversing the roles of K1 and K2 , it is clear that the sets Y1 X1 and Y2 X2 both correspond to the same set of permutation under the usual isomorphism from the groups G1 and G2 to the group of permutations on f1; 2; : : : ; pg. Thus, we have proven:

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A Products of I.I.D. Random Stochastic Matrices

Theorem A.3. Under Assumption (), the homomorphic group images are isomorphic.

G1 N1

and

G2 N2

t u

Now we are in a position to draw the promised conclusion of this appendix. Theorem A.4. Under Assumption (), the sequence n1 converges weakly iff n2 does so. Proof. Let S1 , S2 , K1 , K2 , G1 , G2 , N1 and N2 be as before. Let e 2 K1 (and also the identity of G1 ) and f 2 K2 (and also the identity of G2 ) be idempotents with 1 the same skeleton (and the same basis). For B  G , define the probability measure N1 Q 1 by   Q 1 .B/ D 1 ˚11 .B/ ; where ˚1 W S !

G1 N1

is defined by ˚1 .x/ D exe N1 :

We define ˚2 and Q 2 similarly. Notice that S .Q 1 / D fexe N1 W x 2 S .1 /g and S .Q 2 / D ffyf N2 W y 2 S .2 /g. It is known (Theorem 2.18 in Chap. 2) that n1 (respectively, n2 ) converges weakly iff Q n1 (respectively, Q n2 ) does. Notice that for x 2 S .1 /, there exists y 2 S .2 / such that x and y (and therefore, exe and fyf ) have the same skeleton. It follows from the proof of Theorem A.3 that the cosets exe N1 and fyf N2 correspond to the same set of permutations under the isomorphisms from the groups G1 and G2 to the group of permutations on f1; 2; : : : ; pg. Similarly, for each y in S .2 /, there exists x 2 S .1 / such that a similar assertion is again true for the sets fyf N2 and exe N1 . As a result, it follows G1 2 that under the isomorphism from N to G , observed in the proof of Theorem A.3, N2 1 S .Q 2 / is the isomorphic image of S .Q 1 /. Thus, Q n1 converges weakly iff Q n2 also does so. The proof of Theorem A.4 is now complete. t u For more details and generalizations, see [111].

Appendix B

An Example Due to Chamayou and Letac

Here, we present an example of a sequence of i.i.d. d  d (random) stochastic matrices .Xn /, where each row of X1 has Dirichlet density, the rows of X1 are independent such that the sequence of products Xn Xn1 : : : X1 converges almost surely to a (random) stochastic matrix Z with identical rows where each row of Z has also Dirichlet density. The Dirichlet density, a generalization of Beta density, with (positive) parameters ˛1 ; ˛2 ; : : : ; ˛d , is given by: 

d P i D1

! ˛i

 .˛1 /  .˛2 / : : :  .˛d /

1 1 x1˛1 1 x2˛2 1 : : : xd˛d1 .1  x1  x2      xd 1 /˛d 1 ;

(B.1)

where for 1  i  d  1, 0  xi  1. This density is zero if any of the xi s is less than 0 or greater than 1 or if x1 C x2 C    C xd 1 > 1. The lemma below is then easily verified. Lemma B.1. Let Y1 ; Y2 ; : : : ; Yk , k 2, be independent Gamma random variables so that their joint density function is given by L .y1 ; y2 ; : : : ; yk / D

k Y i D1

Let Wi D Yi =

k P j D1

1 ˛ 1 y i e yi I.0;1/ .yi / :  .˛i / i

Yj , 1  i  k  1, Wk D

k P j D1

Yj . Then the random vec-

tor .W1 ; W2 ; : : : ; Wk1 / is independent of Wk , such that .W1 ; W2 ; : : : ; Wk1 / has Dirichlet density with parameters ˛1 ; ˛2 ; : : : ; ˛k given by 

k P j D1

! ˛j

 .˛1 /  .˛2 / : : :  .˛k /

k1 1 w1˛1 1 w2˛2 1 : : : w˛kD1 .1  w1  w2      wk1 /˛k 1

365

366

B An Example Due to Chamayou and Letac

for w1 > 0; w2 > 0; : : : ; wk1 > 0, w1 C w2 C    C wk1  1, and D 0, otherwise, k P and Wk has Gamma density with parameter ˛j . t u j D1

  Now let A D aij be a d  d matrix with all entries strictly positive.  Let Y  .Y1 ; Y2 ; : : : ; Yd / be a 1  d random probability vector and X  Xij be a d  d random stochastic matrix such that Y is independent of X . Suppose that for 1  i  d , Xi1 ; Xi 2 ; : : : ; Xi.d 1/ has Dirichlet density with parameters .ai1 ; ai 2 ; : : : ; aid /, and that Y1 ; Y2 ; : : : ; Yd 1 has Dirichlet density with parameters d P aij , i D 1; 2; : : : ; d . Then we claim the following: j D1

Write W D YX . Then W has Dirichlet density with parameters

d P i D1

aij , j D

1; 2; : : : ; d . We prove this here using an idea due to Steffen Lauritzen (see [34]). Let Z be a d  d random nonnegative matrix with independent entries such that the random variable Zij has Gamma density with parameter aij . d d d d P P P P Zij , Kj D Zij , T D Li D Kj . Define: Xij D Write: Li D Zij Li

Yi

j D1 K Wj D Tj

i D1

i D1



j D1





. Then we have: W D Y X . Also, it is clear that each d P Kj has Gamma density with parameter aij , 1  j  d . It also follows from i D1    Lemma B.1 that W1 ; W2 ; : : : ; Wd1 (that is to say, W  ) has Dirichlet density d   P aij , j D 1; 2; : : : ; d , whereas Y  , that is Y1 ; : : : ; Yd1 has with parameters ,

D

Li T

,

i D1

d P Dirichlet density with parameters aij , i D 1; 2; : : : ; d . The Lemma also says j D1 

  is independent of Li (and also, independent that for each i , Xi1 ; Xi2 ; : : : ; Xi;d 1   of all Lj j ¤ i ). Thus, Y and X are independent. Note also that the rows of the stochastic matrix X  are independent, and the i th row of X  has Dirichlet density with parameters ai1 ; aij ; : : : ; aid , 1  i  d . This proves the claim. The following theorem will be now relevant and it is given in [34]. See also Exercise 4.22. The proof is not difficult.

Proposition B.1. Let X1 ; X2 ; : : : be a sequence of d  d i.i.d. random stochastic matrices such that P .Xi is strictly positive/ > 0. Then the sequence Xn Xn1 : : : X1 converges almost surely to a random d  d stochastic matrix Y with identical rows such that if  is the distribution of X1 and  that of Y so that for any Borel set B, .B/ D P .X1 2 B/ and .B/ D P .Y 2 B/, then    D  and  is the only solution of this convolution equation. t u

B An Example Due to Chamayou and Letac

367

The following is the last and the main result of this appendix. Theorem B.1. Let X1 ; X2 ; : : : be a sequence of i.i.d. random d  d stochastic matrices such that the rows of X1 are independent, and the i th row of X1 has Dirichlet density with parameters ai1 ; ai 2 ; : : : ; aid . Suppose that the d X j D1

aij D

d X

aj i D ri ;

1  i  d:

j D1

Then, if Z D lim Xn Xn1 : : : X1 , almost surely, Z has identical rows and each n!1 t u row of Z has Dirichlet density with parameters r1 ; r2 ; : : : ; rd . Proof. By Proposition B.1, Z exists almost surely, Z has identical rows, and if  is the distribution of Z, then  is the unique solution of    D , where  is the distribution of X1 . The discussion preceding Proposition B.1 shows that W  and Y  d d P P there will have the same distribution iff aij D aji for each i , 1  i  d . The i D1

proof follows.

i D1

t u

Let us mention the paper of Van Assche [226], where similar ideas (with different proofs) appeared first for 2  2 stochastic matrices.

Appendix C

Asymptotic Behavior of kXnXn1 : : : X0 uk for I.I.D. Random Nonnegative Matrices

Here, we follow the method of Furstenberg and Kifer [71] and use their arguments to describe the asymptotic behavior of kXn Xn1 : : : X0 uk, where .Xn / is a sequence of i.i.d. d  d nonnegative random matrices and u is a d  1 nonnegative vector. It is natural to do this here for nonnegative matrices as much of the Chap. 4 deals with such matrices. In [71], matrices were invertible real matrices, while here they are nonnegative, but not necessarily invertible. We show that results very similar to those in [71] also hold here, under slightly different conditions. The main theorem here is Theorem C.2. Let X0 ; X1 ; X2 ; : : : be i.i.d. d  d random nonnegative (that is, with nonnegative entries) matrices with distribution . Let S be the support of  and let S be the closed (with usual matrix topology) multiplicative semigroup generated by S , so that ! 1 [ n (C.1) S D cl S : nD1

We consider throughout the following conditions:  .Jc / D 0;

(C.2)

where Jc is the set of all matrices in S with at least one column with only zero entries; Z Œlog ksk C j log k.s/j .ds/ < 1; (C.3) where

( k.s/ D min

d X

) sij W 1  j  d :

(C.4)

i D1

 d Let us define the compact set Y  RC , with usual topology, by  ˚ Y D y D .y1 ; y2 ; : : : ; yd /W yi 0; 1  i  d; y12 C y22 C    C yd2 D 1 : (C.5)

369

370

C Asymptotic Behavior of kXn Xn1 : : : X0 uk for I.I.D. Random Nonnegative Matrices

Let A be a d  d nonnegative matrix with no zero columns. For y 2 Y , let A  y D .Ay/=kAyk; where .Ay/i D

d X

r Aij yj ;

kyk D

X

yi2 :

j D1

This action is then welldefined and .AB/  y D A  .B  y/; whenever A and B are two nonnegative matrices with no zero columns. It is also easily verified that for ˇ1 , ˇ2 in P .S /, the set of all (Borel) probability measures on S , and  2 P .Y /, Œˇ1  .ˇ2  / .A / D Œ.ˇ1  ˇ2 /   .A /; for any Borel subset A  Y , where Z ˇ2  .A / D

ˇ2 fs 2 S W s  y 2 A g.dy/:

Notice that P .Y / is compact in the weak topology (by Banach–Alaoglu’s theorem, see [164]), since Y is compact. Therefore, since k  ˇn  ˇn k ! 0 as n ! 1; where ˇn 

n 1X k  n

!  ˇ;

ˇ 2 P .Y /;

kD1

any weak cluster point of .ˇn / [and .ˇn / has at least one weak  cluster point] is a -invariant probability measure in P .Y /. From now on, we assume throughout condition (C.2). Let u 2 Y and Wn  Xn Xn1 : : : X0 . Write:   .XnC1 ; Wn  u/ : Z0  .X0 ; u/ ; : : : ; ZnC1

  Then Zn is a Markov process on the set M  S  Y with a transition function given by    ˚  P ZnC1 2 D j Zn D .s; y/ D  t 2 S W .t; s  y/ 2 D :

(C.6)

C Asymptotic Behavior of kXn Xn1 : : : X0 uk for I.I.D. Random Nonnegative Matrices

371

  Let  be an invariant probability measure for Zn on M so that Z .D/ D

P ..s; y/; D/.d.s; y//:

(C.7)

Let us define 0 in P .Y / by 0 .B/ D f.s; y/W s  y 2 Bg:

(C.8)

It follows from (C.6) and (C.7) that for D D A  B  M , .D/ D .A /0 .B/

(C.9)

so that 0 uniquely determines . Also if f is any bounded continuous function on Y , then for the function g on M defined by g.s; y/ D f .s  y/; we have Z

Z f d0 D

f .s  y/.d.s; y// Z

D

g.s; y/.d.s; y// “

D

g.t; s  y/.dt/.d.s; y// “

D

f .ts  y/.dt/.d.s; y// Z

Z D f .t  .s  y//.dt/ .d.s; y//

Z Z D f .t  y/.dt/ 0 .dy/; which implies   0 D 0 :

(C.10)

Now assume that S is compact; then M is compact. It is easily verified that condition (C.2) implies that n .Jc / D 0 for all positive integers n. Let us define the function h on M by 

kyk h.s; y/   log ksyk

 D log ksykI

(C.11)

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C Asymptotic Behavior of kXn Xn1 : : : X0 uk for I.I.D. Random Nonnegative Matrices

then h is continuous and  ks1  yk ks2 .s1  y/k   ks1 yk : D  log k.s2 s1 /yk 

h .s2 ; s1  y/ D  log

By Theorem 4.22, with probability 1, lim sup

n !1

Z  n 1 X   gdW  satisfies (C.7) : g Zk  sup n

(C.12)

kD0

This means that for any u 2 Y , almost surely lim sup

n !1

1 log kWn uk  sup n 

Z

 log ksykd.s; y/ ;

(C.13)

where the supremum is taken over all  satisfying (C.7). Let us now drop the assumption of compactness on S and instead, assume condition (C.3). We also continue to assume condition (C.2). Consider the set M  DS Y , where S is the one point compactification of S . With the functions h, defined as in (C.11), we define hm on M  as follows: hm D h for y 2 Y and S 2 S with ksk  m and k.s/

1 ; m

where k.s/ is as defined in (C.4). Let hm be extended continuously to all of M  and still bounded by log m C log d . Let us define the sets Am , Bm , and Cm by ˚  Am D s 2 S W ksk > m ;   1 and Bm D Am [ Cm : Cm D s 2 S W k.s/ < m Note that for any s 2 S and y 2 Y , 1  k.s/  ksyk  ksk: d

(C.14)

The second inequality is obvious. For the first, notice that for any y 2 Y , y D .y1 ; y2 ; : : : ; yd /, there is some j , 1  j  d , such that 1 yj p d

C Asymptotic Behavior of kXn Xn1 : : : X0 uk for I.I.D. Random Nonnegative Matrices

and for this j , d 1X 2 1 ksyk

sij 2 d d 2

i D1

d X

373

!2 sij

:

i D1

The last inequality follows from the observation that for any n real numbers a1 ; a2 ; : : : ; an , !2 ! n n X X ai n ai2 ; i D1

i D1

which can be established immediately by induction on n. This establishes (C.14). It follows from (C.14) that j log ksykj  log d C j log k.s/j C j log kskj:

(C.15)

It follows that we have almost surely ˇ n ˇ n ˇ1 X   1 X   ˇˇ ˇ  g Zk  gm Zk ˇ ˇ ˇn ˇ n kD0

kD0

n 1X  Œlog d C jlog k .Xk /j C jlog kXk kj C log m IBm .Xk / ; n kD0

which converges as n ! 1, by the ergodic theorem to Z Œj log k.s/j C jlogkskj .ds/ C Œlog d C log m  .Bm / : Bm

This expression goes to zero as m ! 1 because of our assumption of condition (C.3). Condition (C.3) also implies ˇZ ˇ Z ˇ ˇ lim ˇˇ gd  gm dˇˇ D 0: m!1 Since the inequality (C.20) holds for the function gm defined on all of the compact space M  , it follows from the preceding considerations that (C.13) holds under conditions (C.2) and (C.3) (and in this case, for (C.13) to hold, the compactness of S is not needed). Let us now write b./  supfa.; /W   D ; 2 P .Y /g; where Z a.; / 

log ksykd.s; y/ “

D

log ksyk.ds/ .dy/;

374

C Asymptotic Behavior of kXn Xn1 : : : X0 uk for I.I.D. Random Nonnegative Matrices

if is the measure 0 in (C.8). Notice that it follows from condition (C.3) and inequality (C.15) that b./ is finite, so that for each positive integer n, there exists

n in P .Y / such that 1 b./ < a .; n / C : n   Since n 2 P .Y / and Y is compact, there is a subsequence ni such that

ni ! 0 weakly so that Z

Z h1 d ni !

h1 d 0 ;

Z

where h1 .y/ D

log ksyk.ds/;

which is a bounded continuous function. Thus, b./ D a.; 0 / : This means there exists 0 2 P .Y /,   0 D 0 such that for any u 2 Y , we have almost surely 1 lim sup log kWn uk  a .; 0 / : (C.16) n!1 n Notice that if for each with   D , 2 P .Y /, a.; / is the same, then besides (C.13) we have almost surely, 1 lim sup Œ log kWn uk n “   log ksyk.ds/ .dy/ n!1

so that

1 log kWn uk a.; /: (C.17) n This means that if is the unique -invariant probability measure on Y (or if a.; / is the same for all -invariant in P .Y /, then for any u 2 Y , we have almost surely, lim inf

n!1

lim

n!1

1 log kWn uk D a.; /: n

(C.18)

Now let U be a Y -valued random variable with distribution 2 P .Y /,  D , such that U is independent of the Xi s. Then if h.s; y/ D log.kyk=ksyk/; the process h .XnC1 ; Wn  U / ;

n 1;

C Asymptotic Behavior of kXn Xn1 : : : X0 uk for I.I.D. Random Nonnegative Matrices

375

is stationary. It follows from the ergodic theorem that 1 log kWn U k ! H , almost surely; n

(C.19)

where H is a random variable such that E.H / D a.; /. This means lim inf

n!1

1 log kWn k a.; / n

(C.20)

on a set of positive probability, since H E.H / on a set of positive probability. Notice that for given N , 1  N < n, kWn k  kXn : : : XN C1 k kWn k so that lim inf

n!1

1 1 log kWn k  lim inf log kXn : : : XN C1 k : n!1 n n

This means  lim inf



n!1

1 log kWn k < a.; / n



1 lim inf log kXn : : : XN C1 k < a.; / n!1 n



Then these two events are equal almost surely, since the .Xi / are i.i.d. so they have the same probability. By the Kolmogorov zero-one law, these two events (see (C.20)) have probability zero. It follows that lim inf

n!1

1 log kWn k a.; /; n

(C.21)

almost surely. Since the left side of (C.21) is independent of , it follows from (C.16) that 1 lim log kWn k D a.; / D b./, almost surely: (C.22) n!1 n We call b./ the rate of growth. Note that     max sy .i /  ; i

where y .i / D .0; : : : ; 0; 1; 0; : : : ; 0/; the ith entry of y .i / being 1, is a norm; therefore, (C.16) and (C.20) lead to (C.21). If in (C.19), the random variable U has its distribution chosen as one of the extreme points of the compact convex set of -invariant probability measures in

376

C Asymptotic Behavior of kXn Xn1 : : : X0 uk for I.I.D. Random Nonnegative Matrices

P .Y /, then the process h .XnC1 ; Wn  U / is ergodic (see [19, p. 118, problem 6.11]). Consequently in (C.19), the random variable H becomes a constant almost surely, and 1 lim log kWn U k D a.; /, almost surely: (C.23) n It follows from (C.23) that   1 1 D P lim log kWn U k D a.; / n!1 n   Z 1 D P lim log kWn U k D a.; / j U D u .d u/; n!1 n which implies 1 log kWn uk D a.; /, almost surely; n for almost all u, where is an extremal -invariant measure in P .Y /. Notice that lim

n!1

(C.24)

supfa.; / W   D ; 2 P .Y /g D supfa.; / W   D ; is one of the extreme points in the set of -invariant measures in P .Y /g: By the Krein–Milman Theorem of functional analysis, it follows from (C.18) that  d for all u ¤ 0 in RC , lim

1 log kWn uk D b./, almost surely; n

(C.25)

if for each -invariant extremal measure , a.; / D b./.  d Suppose there exists u ¤ 0 in RC such that (C.25) is not valid. Then there exists an extremal -invariant measure such that a.; / < b./: Then for this , (C.24) holds and therefore

.L / D 1; where L is defined by    1 C  L D u 2 R W lim sup log Wn u  a.; / almost surely [ f0g: n!1 n 

d

C Asymptotic Behavior of kXn Xn1 : : : X0 uk for I.I.D. Random Nonnegative Matrices

Here,

377

uC D .ju1 j ; ju2 j ;    ; jud j/ ;

if u D .u; u; : : : ; ud /. It is easily verified that L is a subspace. Since there exists a -invariant extremal measure ˇ such that a.; / < a.; ˇ/ and since (C.24) holds for ˇ replacing there, it follows that L is a proper sub space.    Because of (C.24), .L / D 1. Let L be the smallest subspace such that

L D 1. Note: L D \fV W V is a subspace with .V / D 1g  c so that L is the union (and therefore, a countable union) of open sets with zero

-measure. Now     1 D L D   L Z   D x 1 L .dx/; where x 1 L D fy 2 Y W x  y 2 L g. This means that for -almost all x,  

x 1 L D 1: Since x 1 L is a subspace, because of the minimality of L , L  x 1 L or xL  L , almost everywhere ./: Thus, we have proved Theorem C.1. Theorem C.1. Suppose X0 ; X1 ; X2 : : : are i.i.d. d  d matrices with nonnegative entries with distribution  such that conditions (C.2) and (C.3) hold. Then for any  d u 2 RC , u ¤ 0, either lim

n!1

1 log kXn Xn1 : : : X0 uk D b./, almost surely; n

or for some proper -invariant subspace L (that is, xL  L for  almost all x)  Rd , for every u ¤ 0 in L, lim sup

n!1

1 log kXn Xn1 : : : X0 uk  ˛, almost surely; n

where ˛ is a constant less than b./. Notice that if L is a -invariant subspace of Rd , then every matrix g in the support of  has the form   g1 g2 gD ; (C.26) 0 g3

378

C Asymptotic Behavior of kXn Xn1 : : : X0 uk for I.I.D. Random Nonnegative Matrices

by choosing a basis of Rd whose initial vectors form a basis of L, where g1 represents the restriction of g to L. Note that      g1 g2 h1 h2 g1 h1 g1 h2 C g2 h3 D : 0 g3 0 h3 0 g3 h3 This means  induces a probability measure L on the set of matrices ˚  g1 W g 2 S ; g as in (C.26) and a probability measure 0 ( Rd jL ) on the set of matrices ˚  g3 W g 2 S ; g as in (C.26) Let us now establish that   ˚ b./ D max b .L / ; b 0 ;

(C.27)

where b .L / and b .0 / are defined for L and 0 , respectively, in the same way b./ was defined earlier for the measure . Notice that the result in  (C.22) can be established under weaker conditions, namely, when E logC kX1 k < 1, using Kingman’s subadditive ergodic theorem (see [127]). Indeed letting Wmn D log kXn1 Xn2 : : : Xm k ;

m < n;

we see that for m < n < p, Wmp  Wmn C Wnp . The distribution of Wmn depends C only on n  m and E W01 < 1, so that it follows from Kingman’s subadditive ergodic theorem that 1 lim log kWn k n!1 n exists with probability 1. This limit is almost surely a constant by the Kolmogorov zero-one law, as we explained when we established (C.22). Thus for the limit in (C.22) to exist, condition (C.3) alone is sufficient. Using the representation (C.26), X1 can be written as   X11 X12 : X1 D 0 X13 Then

˚  max logC kX11 k ; logC kX13 k  logC kX1 k

so that such results as (C.22) also hold for L and 0 . To establish (C.27), it is clear that ˚   max b .L / ; b 0  b./:

C Asymptotic Behavior of kXn Xn1 : : : X0 uk for I.I.D. Random Nonnegative Matrices

379

Suppose that the inequality here is strict. Using an argument from Furstenberg and Kifer [71], we consider the product W2N D .X2N X2N 1 : : : XN C1 / WN    0 WN1 WN 2 WN1 WN0 2 : D 0 WN0 3 0 WN 3 Now given ı > 0, " > 0, there exists N0 such that for N > N0 with probability greater than 1  ı, the following inequalities hold: kWN1 k  e .1C"/N b.L / ; kW2N k > e .1"/2N b./ ;

(C.28)

kWN 2 k  e .1C"/N b./ ; 0

kWN 3 k  e .1C"/N b. / :  0   0      These  0 inequalities (not counting the second one) hold also for WN1 , WN 2 , and W , respectively. Hence with probability greater than 1  7ı, N3 0

kW2N k  e .1C"/2N b.L / C e .1C"/2N b. / 0

Ce .1C"/N Œb. /Cb./ C e .1C"/N Œb.L /Cb./ ; which contradicts (C.28). This proves (C.27). It follows from (C.27) that if L1 and L2 are two -invariant subspaces of Rd , then L1 C L2 is also a -invariant subspace of Rd and  ˚      b L1 CL2 D max b L1 ; b L2 :   Thus, there is a unique maximal -invariant proper subspace L1 such that b L1 < b./. Now as in (C.26), we can assume with no loss of generality   Xn1 Xn2 Xn D : 0 Xn3 By (C.27),

˚     max b L1 ; b Rd =L1 D b./I

then

  b Rd =L1 D b./:

Let v 2 Rd =L1 . Then by Theorem C.1, if lim

n!1

  1 log Xn3 Xn1;3 : : : X0;3 vC  n

(C.29)

380

C Asymptotic Behavior of kXn Xn1 : : : X0 uk for I.I.D. Random Nonnegative Matrices

is not b./ with probability 1, then there is a proper subspace L2 of Rd =L1 with a rate of growth less than   b Rd =L1 D b./: But this means the rate of growth of the proper subspace L1 C L2 , which is also -invariant, is less than b./. This contradicts the maximality of L1 . Thus for representation (C.29), for any v 2 Rd =L1 , we have almost surely lim

n!1

  1 log Xn3 Xn1;3 : : : X0 vC  D b./: n

(C.30)

Now let w … L1 . Then we can write wD

  w1 ; w2

where w1 2 L1 and w2 2 Rd =L1 . Writing Xn as in (C.29), we have   Wn1 Wn2 ; Wn D 0 Wn3   C C    Wn wC  D Wn1 w1 C Wn2 w2    Wn3 wC 2   

Wn3 wC 2 : Thus, it follows from (C.30) and (C.16) that lim

n!1

  1 log Wn wC  D b./, almost surely: n

(C.31)

By repeating this procedure, we have now proven Theorem C.2. Theorem C.2. Let .Xn /, n 0, be a sequence of i.i.d. d  d nonnegative matrices with distribution  satisfying conditions (C.2) and (C.3). There is a sequence of subspaces f0g  Lr      L2  L1  Rd and a sequence of constants b./ > b1 ./ >    > br ./ such that if v 2 Li n Li C1 , then with probability 1,   1 log Xn Xn1 : : : X0 vC  D bi ./: n!1 n lim

We remark that in the context of Theorem C.2 above the most well-known result of this type for nonsingular matrices was proven by Valery Oseledets in [178].

C Asymptotic Behavior of kXn Xn1 : : : X0 uk for I.I.D. Random Nonnegative Matrices

381

In 1979, Raghunathan [190] gave another proof of the Oseledets’ multiplicative ergodic theorem in [178], using Kingman’s subadditive ergodic theorem in [127]. Ruelle in (Ergodic theory of differentiable dynamical systems, Inst. des Hautes Etudes Scient., Publ. Math. 50 (1979) 275–306) extended Oseledets’ theorem to Hilbert spaces, and R. Ma˜ne´ in “Geometric Dynamics” (Lecture Notes in Math., vol. 1007, Springer, 1983, 522–577) extended it to Banach spaces. If .Xn / is a stationary Markov chain with state space S and f is a Borel measurable map from S to the d  d invertible matrices GL.d; R/, then applying the Oseledets’ theorem on the sequence .f .Xn //, we can find a sequence of Lyapunov constants b˛  b˛1  : : :  b1 and a sequence of decreasing subspaces Li , 1  i  d , of Rd , such that   1 Li D v 2 Rd W lim log kf .Xn / f .Xn1 /    f .X1 / vk  bi : n In [76], Goldsheid and Margulis proved the following theorem: Let .Xn / be i.i.d. random matrices in GL.d; R/ with distribution . Let S be the semigroup generated by S./, the support of . If the algebraic closure of S in GL.d; R/, then the Lyapunov exponents bi , 1  i  d , of the matrix products fXn ; Xn1    X1 W n 1g are all distinct. Also, in this context, the paper of Guivarc’h (“Sur les exposants de Liapunoff des marches aleatoires”, C. R. A. S., t. 292 (1981) 327–329), and also the paper [84] are relevant. Finally, let us mention here that Kaimanovich (in J. Soviet Math. 47 (1989) 2387–2398) proved a multiplicative ergodic theorem for semisimple Lie groups, and that Karlsson and Ledrappier (Ann. Prob. 34 (2006) 1693–1706) proved a multiplicative ergodic theorem for the group ISOM .X / of a proper metric space X .

Appendix D

Continuous Singularity of the Limit Distribution of Products of I.I.D. Stochastic Matrices

Let  be a probability measure on the Borel subsets of 2  2 stochastic matrices (with the usual topology) and have support S ./. Let S be the smallest closed multiplicative semigroup generated by S ./, and .a; b/ denote the stochastic matrix whose first column is .a; b/. Then the sequence n converges weakly iff  is not the unit mass at .0; 1/. See [147]. In what follows, we make the following assumption: ./ Each matrix in S ./ is invertible, and there is at least one matrix in S ./; which has a power with a strictly positive column. It is clear that under condition ./, n converges weakly, and its weak limit  has its support consisting of matrices with rank one, since condition ./ guarantees that there is at least one matrix S which has rank one, this matrix being the pointwise limit of the matrix in S ./ ; which has a power with a strictly positive column. Lemma D.1. The measure  is continuous (that is,  fxg D 0 for each x 2 S ./) iff  is not degenerate (that is, its support is not a singleton) iff there are at least two points in S ./ which, as points on the plane are not collinear with the point .1; 0/. Proof. Suppose  is not degenerate. If it is not continuous, then b  sup f .fP g/ W P 2 S ./g is positive. After noticing that the set fP 2 S ./ W  .fP g/ > b=2g is finite, it is clear that the supremum is attained, that is, for some P 2 S ./,  .fP g/ D b, where this point P has rank one. For each y 2 S ./, y being invertible, the set fP g y 1  fQ 2 S W Qy D P g is a singleton. Now,    D , and for each y 2 S ./,    .fP g/  fP g y 1 :

383

384 D Continuous Singularity of the Limit Distribution of Products of I.I.D. Stochastic Matrices

Thus,

R

   .fP g/   fP g y 1  .dy/ D 0,

and therefore, for -almost all y,    fP g y 1 D  .fP g/ D b > 0.   The map y !  fP g y 1 is upper semicontinuous, and therefore, for all y 2 S ./,    fP g y 1 D b > 0. Similarly, since   .n/ D  , we can also show that for any y 2 S ./ and any positive integer n, y n 2 S .n/ , and therefore,  .fP g y n / D b > 0. Now notice that if y … f.1; 0/ ; .0; 1/g, then fP g D fP g y 1 if and only if y n converges to P if and only if the points P , y and .1; 0/ are collinear. If for each point y 2 S ./, fP g D fP g y 1 , then S will be contained in the line joining the points P and .1; 0/, and consequently, S ./ D fP g, which implies that  is degenerate. Thus, there is at least one point y 2 S ./ such that fP g is different from fP gy 1 ; this means that the points P , y and .1; 0/ are not collinear. We also know that for each positive integer m > 1, the points y, y m and .1; 0/ are collinear; this means that for any m > 1, the points y m , P and .1; 0/ are also not collinear. Therefore, for any two positive integers m and n, fP g y m 6D fP g y n , but this will contradict the fact that  is a probability measure. The following theorem, though known to experts, does not seem to be available in standard literature. The proof here follows that given in [160]. Theorem D.1. The measure  has exactly one of the following three properties: (i)  is discrete; (ii)  is continuous singular with respect to m, the Lebesgue measure on Œ0; 1 ; (iii)  is absolutely continuous with respect to m. (Note that the set of rank one stochastic matrices f.x; x/ W 0  x  1g is here identified with the interval Œ0; 1 ). Proof. Let E D fx 2 S ./ W  fxg > 0g, and for A  S ./, 1 .A/ D  .A \ E/, 2 .A/ D  .A \ E c /. Then there exist constants ˛, ˇ such that 0  ˛  1, 0  ˇ  1, ˛ C ˇ D 1, and probability measures 11 and 12 such that  D ˛11 C ˇ12 ,

D Continuous Singularity of the Limit Distribution of Products of I.I.D. Stochastic Matrices 385 1 1 where 11 D .E / 1 and 12 D .E c / 2 . In case  .E/ D 0,  D 12 is continuous, and when  .E c / D 0,  D 11 is discrete. Since each y in S ./ is assumed invertible, the set

fxg y 1 D fz 2 S W zy D xg is a singleton for each x in S , and consequently, 12 being continuous, 12   is also continuous. Also, since  D   , we have: ˛11 C ˇ12 D ˛11   C ˇ12  . We can also write: 11   D 21 C ı22 , where 0    1, 0  ı  1,  C ı D 1, 21 is a discrete probability measure, and 22 is a continuous probability measure. Then we have: ˛ Œ11  21 D ˛ı22 C ˇ .12    12 / : If ˛ D 0, then  is continuous. If ˛ > 0, then  D 1, ı D 0, 11 D 21 D 11   and 12 D 12  . Similarly, it follows that 11 D   11 and 12 D   12 . Thus, if ˛ > 0, 11 D 11   D   11 , and this means that 11 D 11  n D n  11 ,

n 1,

so that 11 D 11   D   11 . First, notice that S .11 /  S ./, S .11 / is an ideal of S ./, and S ./ is simple since  D   . Thus, S .11 / D S ./. Also, by Proposition 2.4 in Chap. 2, it follows that for B  S ./ and x 2 S ./,   11 Bx 1 D

Z

   Bx 1 y 1 11 .dy/

  D  Bx 1 , and  .B/ D    .B/ Z Z     D  Bx 1  .dx/ D 11 Bx 1  .dx/ D 11   .B/ D 11 .B/ ,

386 D Continuous Singularity of the Limit Distribution of Products of I.I.D. Stochastic Matrices

so that  D 11 and consequently, if ˛ > 0 then  is discrete. Suppose now that  is continuous. Consider S ./ (topologically) as a subspace of Œ0; 1 . Let m be the Lebesgue measure on Œ0; 1 . By the Lebesgue decomposition theorem, we can write:  D ˛3 31 C ˇ3 32 , where 0  ˛3  1, 0  ˇ3  1, 31 ? m and 32 0 is chosen such that whenever m .A/ < ı, A  Œ0; 1 , then 32 .A/ < . Notice that   m C x 1 D m fy W y .x1  x2 / C x2 2 C g < ı,     and this means that 32 C x 1 <  or 32 Bx 1 <  for x 2 S ./. In other words, 32   .B/ D 0. We have seen above that 32 b, c > d , 1a < d and .a  b/ C .c  d / D 1, then  is the uniform distribution on the interval o n1c

b d iff  f.a; b/g D a  b. For these and other related  x  1cCd .x; x/ W 1aCb results, we refer the reader to [172]. The main aim of this appendix is to prove the following general theorem, which first appeared in Mukherjea and Restrepo [165].

Theorem D.2. Let  be a probability measure on 2  2 stochastic matrices with a finite or countably infinite support such that S ./ D f.xi; yi / W 1  i < 1, xi ¤ yi , 0 < xi < 1, 0 < yi < 1g , where  f.xi ; yi /g D pi ,

0 < pi < 1

and

1 X

pi D 1.

i D1

Let i D jxi  yi j for each i such that one of the following three conditions hold: (i) (ii) (iii)

1 P i D1 1 P

i < 1; i D 1, and for at least one i , i ¤ pi ; pi i < 1. pi

i D1 1 Q i D1

We also assume: (iv)

1 P i D1

pi log pi > 1.

Then  D .w/ lim n is continuous singular with respect to the Lebesgue mean!1

sure m on Œ0; 1 , where Œ0; 1 is identified with the set of all 2  2 stochastic matrices with rank 1.

388 D Continuous Singularity of the Limit Distribution of Products of I.I.D. Stochastic Matrices

The proofs from [64] in similar contexts are adapted here in some of the following steps. Step 1: Let .S; d / be a compact metric space and let K.S / be the class of all closed subsets of S . For A; B in K.S / define D.A; B/  inffs > 0 W A  Ns .B/ and B  Ns .A/g where Ns .B/  fy 2 S W for some x 2 B, d.x; y/ < sg. Then .K.S /; D/ is a complete metric space. Let ffi W S ! S gi 2 , with  a countably infinite set, be a family of functions such that d.fi .x/; fi .y//  i d.x; y/ for all x; y 2 S and some i > 0 and let  D supfi W i 2 g. Then, as is well known for finite , here also if  < 1 then there is a unique compact subset A  S such that AD

S

fi .A/.

(D.1)

i 2

The idea of the proof is the same as is wellknown in the case when  is finite: Define for each A 2 K.S / the function F W K.S / ! K.S / such that F .A/ D S fi .A/. It is then easy to verify that for A; B 2 K.S /,

i 2

D.F .A/; F .B//  D.A; B/

(D.2)

and therefore (D.1) follows from the classical contraction mapping theorem and the completeness of the space .K.S /; D/. Along the same lines, let fpi gi 2 be a discrete probability measure on  and let I be a -valued random variable distributed according to fpi gi 2 . Assume that E ŒI < 1. Then there is also a unique invariant probability measure  on the Borel subsets of S such that P .A/ D pi .fi1 .A// (D.3) i 2

for all Borel subsets A  S . Briefly, the idea of the proof is the following: Since S is compact, there exists ˇ > 0 such that for any x; y 2 S , d.x; y/ < ˇ. Define the set Fˇ by Fˇ  fh W S ! R W jh.x/  h.y/j  d.x; y/ and jh.x/j  ˇ for all x; y 2 S g and define in P.S / (the space of all the probability measures defined on the Borel subsets of S ) the metric ˇ  ˚ˇR R .1 ; 2 /  sup ˇ hd1  hd2 ˇ W h 2 Fˇ . Then .P.S /; / is a complete metric space. To establish (D.3), we define ˚ W P.S / ! P.S / such that for any Borel set A  S , ˚./.A/ D

P

pi .fi1 .A//.

i 2

(D.4)

D Continuous Singularity of the Limit Distribution of Products of I.I.D. Stochastic Matrices 389

Now, taking a fixed point z 2 S , it follows that for h 2 Fˇ , 1 ; 2 2 P.S /, ˇ ˇR R ˇ hd˚.1 /  hd˚.2 /ˇ ˇ ˇ ˇ P R R ˇ ˇ D ˇ pi Œh ı fi .x/  h ı fi .z/ d1 .x/  Œh ı fi .x/  h ı fi .z/ d2 .x/ ˇˇ i 2 ˇ ˇ ˇP R ˇ R 1 ˇ Œh ı fi .x/  h ı fi .z/  ˇ pi i hi d1  hi d2 ˇˇ , where hi .x/ D i i 2  E ŒI .1 ; 2 /, since hi 2 Fˇ . Now (D.3) follows by applying the contraction mapping theorem on ˚. Notice that, in particular, E ŒI < 1 if  defined earlier (just preceding (D.1)) is less than 1. Step 2: The model map. To proceed, we need a “model” map like the one presented in the book by Edgar [64]. Let S , , ff Qi ; i W i 2 g and  be defined as before. Consider the product space .!/ D i , where i   for each i . The elements in .!/ can i 2

be considered as a point .e1 ; e2 ; e3 ; :::/, ei 2 i or as an infinite word e1 e2 e3 ::: where the letters are taken from the alphabet . If D e1 e2 :::ek k where e1 , e2 , ..., ek are the first k letters of and k 2 .!/ , then we denote the finite word e1 e2 :::ek by simply k , and by Πk we denote the set f 2 .!/ W k D k g, a natural topology over .!/ is the one generated by the sets of the form Πk , we will denote this topology by T . Also, in .!/ we are considering the distance function k Q . ; /  ei , where k is the largest number such that k D k . i D1

Given a probability distribution fpi gi 2 for , it induces the product probability measure P.1/ on .!/ . In particular, it follows that for any Borel subset B  .!/ , we have that n o P P.1/ .B/ D pi P.1/ 2 .!/ ji 2 B (D.5) i 2

We will need (D.5) after we construct a model map h W .!/ ! S , fashioned after [64], page 211, in Proposition D.1. In what follows, let I be a -valued random variable distributed according to fpi gi 2 . Remark D.1. Notice that if   supi 2 i < 1, then . ; / is a metric over .1/ that generates the topology T and that agrees with P .1/ , meaning that all Borel sets of .!/ under are measurable. Under the less restrictive condition   1, we should include also the assumption 1  E Œlog I < 0, that is trivially attained in the cases  < 1 or E ŒI < 1. Under this condition, we have that for P .1/ k Q almost all D e1 e2    2 .!/ , ei # 0 as k ! 1 i.e., . ; / D 0. Redefining i D1

over those that do not satisfy . ; / D 0, we obtain a metric on .!/ such that

390 D Continuous Singularity of the Limit Distribution of Products of I.I.D. Stochastic Matrices

for P .1/ -almost all 2 .!/ , . ; / D

k Q

ei for all  2 .!/ that coincide with

i D1

in exactly the first k letters. Proposition D.1. Let S be a compact metric space and for each i 2 , a countably infinite set, let fi be a map from S to S such that d.fi .x/; fi .y//  i d.x; y/ for all x; y 2 S , where i > 0 and 1  E Œlog I < 0. Consider the distance function on .!/ and the product probability measure P.1/ induced by fpi gi 2 . Then, there exists a unique measurable function h W .!/ ! S such that h.i / D fi .h. // for each 2 .!/ and each i 2 . This h also satisfies the following conditions: 1. The measure  2 P.S / such that for each Borel set A  S , .A/ D P.1/ .h1 .A// is the unique invariant probability measure given in (D.3).

    S 2. h .!/ D fi h .!/ . i 2

3. For P.1/ -almost all ;  2 .!/ , d.h. /; h.//  ˇ . ; / where ˇ D supfd.x; y/ W x 2 S , y 2 S g.   4. If  D supfi W i 2 g < 1 then h is continuous and h .!/ coincides with the unique invariant set A given in (D.1). The proof follows the same lines as in [64]. For completeness, we include it. Proof. Fix an arbitrary a 2 S . We define recursively a sequence of continuous functions gk , 1  k < 1, gk W .!/ ! S such that g0 . / D a for each 2 .!/ and gkC1 .i / D fi .gk . // for each 2 .!/ , i 2  and k 0. Notice that g0 .!/ is continuous let us assume that for k 0, gk is also continuous  .!/  on  . Now, in  ; T . Let i 2 .!/ . Given  > 0, let l be such that if  2 Œ l then d.gk . /; gk .// <  . Now, for any i  2 Œ.i /kC1 , i

d.gkC1 .i /; gkC1 .i // D d.fi .gk . //; fi .gk .///  i d.gk . /; gk .// < ,   proving that gkC1 is continuous in .!/ ; T . Similarly, it is possible to show that if ˇ  supfd.x; y/ W x 2 S , y 2 S g, then for any D e1 e2 ::: 2 .!/ and k 1, k Q d.gkCm . /; gk . //  ˇ ei , but the condition E Œlog I < 0 implies (by the i D1

strong law of large numbers) that for P.1/ -almost all 2 .!/ , lim sup

k Q

( ei D lim sup exp

k!1 i D1

k!1

k P i D1

) log ei

D 0,

(D.6)

proving that the sequence fgk . /gk1 is Cauchy for P.1/ -almost all 2 .!/ . Therefore, there exists a measurable function h. / such that for P.1/ -almost all

D Continuous Singularity of the Limit Distribution of Products of I.I.D. Stochastic Matrices 391

2 .!/ , lim gk . / D h. /. Notice that if   supfi ji 2  g < 1, then k!1

d.gkCm . /; gk . //  ˇ k implying that the convergence is uniform and therefore that h. / is continuous. Now, since gkC1 .i / D fi .gk . // for each 2 .!/ , i 2  and k 1, it follows that for P.1/ -almost all 2 .!/ , h.i / D fi .h. //. Also, from (D.5) it follows that for any Borel subset A  S , we have: P.1/ .h1 .A// D D

P i 2

P i 2

D

P

i 2

n o ˇ pi P.1/ 2 .!/ ˇi 2 h1 .A/ n o pi P.1/ 2 .!/ jfi .h. // 2 A pi .P.1/ ı h1 /.f 1 .A//,

  establishing property (1) in the proposition. It is now clear that h .!/ D   S  .!/  , and in the case that  < 1, h .!/ coincides with the unique fi h  i 2

invariant set given in (D.1). Finally, to prove property (3), let k be the greatest integer such that k D k . If  D e1 e2 :::ek k and D e1 e2 :::ek k , then d.h./; h. // D d.fe1 ı fe2 ı    ı fek ı h.k /; fe1 ı fe2 ı    ı fek ı h. k // 

k Q

ei d.h.k /; h. k //

i D1

 ˇ .; /. Step 3: A bound for the upper packing dimension. Let  be a Borel probability measure on a metric space S . Then the essential supremum (with respect to ) of L .x/, x 2 S defined by L .x/  lim sup ı!0

 log  Bı .x/ ,

log ı

where Bı .x/ is the open ball with center x and radius ı, is called the upper packing dimension of the measure . When S D Rd and -ess sup L .x/ < d , then  is singular with respect to the Lebesgue measure m on Rd . To see this suppose that   0g. For r!0

-almost all x 2 S ,

0 log @

1 f .y/dy A

Br .x/

L .x/ D lim sup r!0

D lim sup

R

log

R

log r



Br .x/ f .y/dy

log r  log m.Br .x// log ˛r .x/ C D lim sup log r log r r!0 r!0



D d. But .S / D

R

f .x/dx D

R

f .x/dx D .Rd /, so that for -almost all x 2 Rd ,

Rd

S

L .x/ D d and therefore, -ess sup L .x/ D d , which is a contradiction. Now, let S , , ffi W i 2 g, fpi W i 2 g, fi W i 2 g be as in Proposition D.1. The following proposition holds. Proposition D.2. Let  beP the unique invariant probability measure satisfying (see (D.3)) such that .A/ D pi .fi1 .A// for all Borel subsets A  S , and let I i 2

be a -valued random variable distributed according to fpi W i 2 g. Suppose that 1. 1  EŒlog I < 0. 2. EŒjlog pI j < 1. Then,   ess sup L .x/ 

P EŒlog pI pi log pi D Pi 2 . EŒlog I i 2 pi log i

(D.7)

Proof. Consider the product space .!/ together with the product measure P.1/ induced by fpi Wi 2g. Using the strong law of large numbers we have that for P.1/ almost all 2 .!/ , D e1 e2 e3 :::, Pk lim

i D1 log pei

k

k!1

and

D EŒlog pI

(D.8)

D EŒlog I .

(D.9)

Pk lim

k!1

i D1 log ei

k

D Continuous Singularity of the Limit Distribution of Products of I.I.D. Stochastic Matrices 393

Also, we have from (D.6) that the condition EŒlog I < 0 implies that for P.1/ almost all 2 .!/ , D e1 e2 e3 :::, lim

Qk

i D1 ei

k!1

D 0.

(D.10)

Let F  .!/ with P.1/ .F / D 1 be such that (D.8), (D.9), (D.10) and the property (3) of Proposition D.1 hold, for all in F . Fix x D h. / where 2 F , D e1 e2 e3 :::. For each  > 0, define   Q Q  .  k  max k W kiD1 ei <  k1 e i D1 i ˇ If 0 2 Œ k \ F , then we have by the property (3) of Proposition D.1, that   Q  ei <  d h. 0 /; x  ˇ . 0 ; /  ˇ ki D1   and therefore, h Œ k \ F  B .x/, so that

    log  B .x/ log  h Πk \ F     D log P.1/ h1 h Πk \ F  

log P.1/ Πk \ F D

k P i D1

log pei .

Now, we have:

 log  B .x/ lim sup !0

log 



1 Pk i D1 log pei k lim P k 1 1 !0 i D1 log ei k 1

D

1 Pk i D1 log pei k lim Pk1 1 k!1 i D1 log ei k1

D

EŒlog pI . EŒlog I





k k  1

k k1





This means that for all x 2 h.F /, P pi log pi L .x/  Pi 2 , i 2 pi log i but .h.F // D P.1/ .h1 .h.F /// P.1/ .F / D 1, which implies that

394 D Continuous Singularity of the Limit Distribution of Products of I.I.D. Stochastic Matrices

P pi log pi - ess sup L .x/  Pi 2 . i 2 pi log i Final step: Continuous singularity of . First, notice that  D limn!1 .n/ satisfies the convolution equation  D    and is the self-similar probability measure satisfying D

1 P

pi  ı fi1 ,

(D.11)

i D1

where fi .x/ D yi C .xi  yi /x, x 2 Œ0; 1 . In the present context, fi .h. // D h. /  .xi ; yi /, where h. / represents a stochastic matrix where the first element in each row is h. /, and .xi ; yi / represents a stochastic matrix where the first elements in the two rows are, respectively, xi and yi and the multiplication is matrix multiplication. With this observation, it is easy to verify that the last equation in step 2 involving P.1/ ı h1 can be written as

  P.1/ ı h1 D P.1/ ı h1  ,

implying that the invariant measure , being  the unique solution of the convolution equation  D   , must be the measure P.1/ ı h1 . Note that as is known,  can be considered as a probability measure on the closed unit interval Œ0; 1 , and its support K will then satisfy (as was shown earlier) the equality 1 S KD fi .K/. (D.12) i D1

Suppose now that m is the Lebesgue measure on Œ0; 1 . Let us assume one of the three conditions (i), (ii), and (iii), and also, condition (iv). If one of the conditions (i) and (ii) holds, then Jensen’s inequality implies that P1

i D1 pi

1 P

with equality if and only if

log

 P1 i  log i D1 i pi  0,

i D 1 and i D pi for all i 1. Therefore,

i D1



1 P

pi log pi < 

i D1

1 P

pi log i .

i D1

Thus, in case (i), (ii), or (iii) holds, we have: P1 pi log pi Pi1D1 0. 3. There exists a probability measure Q on G such that M  Q   D . Q

400

E Rate of Decay of Concentration Functions on Discrete Groups

4. limn!1 jS n j D d < 1. 5. N is finite. Proof. (1) ) (2) follows from Lemma E.1. (2) ) (3). If there exists a function f 2 L2 .G/ such that lim kT n f k22 D lim

n!1

n!1

T n TM n f; f

 > 0;

then the measure in Lemma E.2 is not zero. Then the measure Q D = .G/ satisfies (iii). (3) ) (4). Let M  Q   D . Q Because G is discrete and is a probability measure, there is a g0 2 G such that .g Q 0 / D supg2G Q > 0. For every positive integer n, we have .g Q 0/ D

X

.h Q 1 g0 h2 /n .h1 /M n .h2 /:

h1 ;h2

Thus, .h Q 1 g0 h2 / D .g Q 0 / for all h1 2 S n and h2 2 S n . If we now fix an h2 2 S n we have X 1

.h Q 1 g0 h2 / D jS n j .g Q 0 /: h1 2S n

Therefore, d D lim jS n j  n!1

1 : .g Q 0/

(4) ) (5). Since jS j is monotonic there is an N1 such that jS n j D d for all n N1 . Thus, for all g 2 S we have n

gS n D S n g D S nC1 ; for all n N1 . From this, we see that for all n N1 we have S .nCk/ S k D S k S .nCk/ D S n and

S .nCk/ S k D S k S .nCk/ D S n :

Clearly, the sets Rn;1 D S n S n and Rn;2 D S n S n are nondecreasing, symmetric and jRn;j j  d2 . Hence, Rn;j D RN0 ;j DW Rj for j D 1; 2 and for all n N0 where N0 N1 is sufficiently large. Then R1 R2 D S n S n S n S n D S n S 2n S n D S n S n D S n S n D R1 D R2 Since Rn;j is symmetric, this then implies that R WD R1 D R2 is a subgroup of G.

E.1 Preliminaries

401

Furthermore, if g 2 S , then g1 Rg D g 1 Rn;1 g D RnC1;1 D R for n N0 . Since S is adapted this implies that R is a normal subgroup of G. Finally, we have N D R D S n S n by Lemma E.3. (5) ) (1). Because N is normal for any g 2 S we have S n  g n N : Therefore, n .g n N / D 1 for all n 1.

t u

The following lemma will be useful to achieve our desired results. D

Lemma E.3. Suppose X  C0 Y k  2 C kZ, that Y =Z 1, and that 2

k  .2Y =Z/ DC2 < k C 1; 2

with X; Y; Z; C0 ; D; k > 0. Then X 1C D  C 0 Y is independent of X , Y , Z, and k.

2 D

Z for some constant C 0 which

Proof. D

2

D

X  C0 Y k  2 C kZ  C0 Y k  2 C Z .2Y =Z/ DC2 2

D

D

D C0 k  2 Y C Z DC2 .2Y / DC2

 D2 2 2 D  C0 Y .2Y =Z/ DC2  1 C Z DC2 .2Y / DC2 : 2

2

Since 1 < 2 DC2  .2Y =Z/ DC2 we see that 1 1 .2Y =Z/2=DC2 D1C  1 C 2=DC2 : .2Y =Z/2=DC2  1 .2Y =Z/2=DC2  1 2 1 So, setting C1 D 1 C

1 22=DC2 1

we have 1

f2Y =Zg

2 DC2

1



C1 2

f2Y =Zg DC2

:

Therefore,  D2

2 D 2 X  C0 C1D=2 Y f2Y =Zg DC2 C Z DC2 .2Y / DC2 D C2 Y D=2 D=DC2

where C2 D C0 C1

2

DC2=D

.

D

Z DC2 ;

C 22=DC2 . Finally, X

where C 0 D C2

2 DC2

DC2 D

 C 0 ZY

2 D

; t u

402

E Rate of Decay of Concentration Functions on Discrete Groups

Throughout the remainder of this appendix, unless otherwise stated, we will be assuming that G is an infinite discrete group and that F is an adapted probability density function on G with associated probability measure  and support S . Let FM .x/ D F .x 1 /. We will also let k  kp!q denote the operator norm from Lp .G/ to Lq .G/. The proof of the following Lemma by Varopoulos et al. lies outside of the scope of this appendix. Lemma E.4 (Varopoulos et al. [228] VII.2.6). Let Tk , k D 1; : : : be a sequence of operators which contract all the spaces Lp .G/, 1  p  1. Let us assume that these operators satisfy the family of inequalities:    C1 kTj f k22  kTk Tj f k22 kf k4=n kTj f k2C4=n 2 1 ; for all f 2 L1 .G/ \ L2 .G/;

j; k 1 and for some n > 0. Then we have

kTk : : : T1 k2!1  .C C1 n=k/n=4

for all k 1:

Moreover, if the adjoint operators TMk also satisfy the above hypotheses, we have  kTk : : : T1 k1!1 

C 0 C1 n k

n=2 for all k 2;

for some constants C and C 0 Lemma E.5. If T and T0 are contracting on L2 .G/, kT k1!2 0 with Vgp.A/ .m/ cmD [Vgp.B/ .m/ cmD ]. Then there exists a symmetric probability density function F0 [F1 ] such that F .n/  FM .n/ CF0 [FM .n/  F .n/ CF1 ] for     some constant C and kF0.k/ k1 D O k D=2 [kF1.k/ k1 D O k D=2 ]. Proof. By the definition of volume growth, there exists a symmetric finite set K D fk1 ; : : : ; kN g  gp.A/ such that jK m j cmD . It is possible to find a set fai;j g  A [ A1 such that ki D ai;1    ai;ni for each i . Let A0 D fai;j j 1  n n i  N; 1  j  ni g. Let ˝ D A0 [ A1 since 0 : Notice that ˝  S S n n D S S is symmetric. Also notice that Vgp.A0 / .m/ cm since K  gp.A0 /. 1 Then F0 D j˝j 1˝ satisfies the hypothesis of Theorem E.1 for gp.A0 / and therefore kF0.k/ k1 D O.k D=2 /. Because ˝  S n S n and since ˝ is a finite set, we can find a constant C such that F .n/  FM .n/ CF0 . Similar arguments work for the B  S n S n case. t u Theorem E.6 ([194]). Let G be a discrete group and let F be an adapted probability density function on G with S D supp.F / such that for some positive integer n and some sets A  S n S n and B  S n S n we have Vgp.A/ .m/ cmD and Vgp.B/ .m/ cmD . Then kF .k/ k1 D O.k D=2 /. Proof. Using Lemma E.10, we can find a symmetric probability density function F0 such that F .n/  FM .n/ CF0 . Likewise, we can find a symmetric probability density function F1 such that FM .n/  F .n/ CF1 . Let T ,T0 , and T1 be the operators of right convolution by F .n/ , F0 and F1 , respectively. Also by Lemma E.10 we know that kF0.k/ k1 D O.k D=2 / and therefore kT0k k1!2 D k.T01=2 /2 kk1!2 D O.k D=4 /. 1=2 1=2 Since T0 is contracting on L2 .G/ (see Lemma E.7), we see that k.T0 /k k1!2 D D=4 O.k /. Also by Lemma E.7 we have   kf k22  kT01=2 f k22  C 0 kf k22  kTf k22 ; for all f 2 L1 .G/ \ L2 .G/ and for some constant C 0 . By Lemma E.6 we see that kT k1!2  1. Putting all of this together with Lemma E.5 we see that 2C4=D

kTf k2

for all f 2 L1 .G/ \ L2 .G/.

  4=D  C 00 kTf k22  kT Tf k22 kf k1 ;

E.3 Adapted Measures

409

This entire argument works for the adjoint operator TM using FM .n/  F .n/ F1 and T1 . Therefore, by Lemma E.4 we see that 

kF .nk/ k1 D O k D=2 : Finally, by monotonicity we get 

kF .k/ k1 D O k D=2 : t u Since the sets S n S n and S n S n are obviously so important to our investigation, we will now consider these sets and certain of their subsets more closely. Lemma E.11 ([194]). Given a fixed integer m > 0 the following are equivalent: 1. 2. 3. 4. 5.

There exists an integer n > 0 such that S m  S n S n There exists an integer k > 0 such that S m \ S k S k ¤ ; There exists an integer k > 0 such that S mCk \ S k ¤ ; There exists an integer k > 0 such that S m \ S k S k ¤ ; There exists an integer l > 0 such that S m  S l S l

Proof. The equivalences of (2), (3), and (4) are obvious, as are (1) ) (2) and (5) ) (4). To show (2) ) (1): Let n D k C m. Then S m \ S k S k ¤ ; , S m \ S nm S .nm/ ¤ ; ) 9 y such that y 2 S m and y 2 S nm S .nm/ ) e D yy 1 2 S nm S .nm/ S m D S nm S n ) S m  S n S n : Similarly, (4) ) (5).

t u

Proposition E.3 ([201]). Let G be a group and let A  G be some nonempty subset. When there exists a; b 2 G with aA[bA  A, then either a and b generates a free subsemigroup or there exists z in the subsemigroup generated by a and b such that zA  aA \ bA. Proof. Suppose a and b do not generate a free subsemigroup. Then there exist two formal positive power words W .x; y/ ¤ V .x; y/ such that W .a; b/ D V .a; b/. Without loss of generality we may assume that the two words do not have the same first entry on the left. If one of the words, say W .x; y/, is empty, then A D eA D V .a; b/A. Note that since V .x; y/ ¤ W .x; y/ we know that V .x; y/ is not also empty, so it must have a first entry, let us call it v. Now, using aA [ bA  A we get V .a; b/A  vA. Let z D a if v D b or vice vera. Then zA  A D V .a; b/A  vA. Therefore zA  aA \ bA:

410

E Rate of Decay of Concentration Functions on Discrete Groups

If both V .x; y/ and W .x; y/ are nonempty then let v and w be the first entries on the left of V .a; b/ and W .a; b/, respectively. Let z D W .a; b/ D V .a; b/. Now, since aA [ bA  A we have zA  vA and zA  wA. However, V .x; y/ and W .x; y/ were assumed to start with different entries, so v ¤ w yielding the result. t u By taking inverses, we get the following corollary: Corollary E.2. Let G be a group and let A  G be some nonempty subset. When there exists a; b 2 G with Aa [ Ab  A, then either a and b generates a free subsemigroup or there exists z in the subsemigroup generated by a and b such that Az  Aa \ Ab. Lemma E.12 ([194]). Let G be a discrete group and S be an adapted subset such that S m \S n D ; when m ¤ n. Then either contains a free S subsemigroup S semigp(S) on two generators or both S m S m  S k S k and S m S m  S k S k for all m. Proof. Let A D [1 S k . If a1    am A \ b1    bm A D ; for some ai ; bi 2 S kD1 then, by Proposition E.3, ˛ D a1    am and ˇ D b1    bm would generate a free subsemigroup. We may then assume that ˛A \ ˇA ¤ ; for all ˛; ˇ 2 S m . For fixed ˛; ˇ 2 S m , ˛A \ ˇA ¤ ; implies ˛s1    sk D ˇt1    tl for some si ; ti 2 S . Furthermore, since the powers of S are pairwise disjoint, we have k D l. Thus, ˇ 1 ˛ D t1    tk sk1    s11 , and therefore ˇ 1 ˛ 2 S k S k . Since ˛ and ˇ are S S arbitrary elements of S m , S m S m  S k S k . We get S m S m  S k s k in a similar fashion using Corollary E.2 instead of Proposition E.3. t u Thus, there are three cases we will need to consider: 1. S m  S n S n and S m  S n S n for some m; n > 0 (equivalently S n \ S k ¤ ; for some n;S k > 0 by Lemma E.11); S 2. S m S m  S k S k and S m S m  S k S k for all m > 0; 3. G contains a free subsemigroup on two generators and S m \ S n D ; for all m ¤ n. We will now consider each of the three cases individually to prove Theorem E.2: Proof of Theorem E.2: Case I: Sm  Sn Sn and Sm  Sn Sn for some >0  S  m;k n k By Theorem E.3 we know that N D gp S S [ S k S k and therek k fore gp .S m /  N the sets S k S k and we also  S S are increasing, S. Since mk mk mk mk have N D gp . Thus N  gp .S m / and thus S S [S S N D gp .S m /. Now using Theorem E.6 with A D B D S m we get the desired result.

E.3 Adapted Measures

411

S

Case II: Sm Sm  Sk Sk and Sm Sm  Again by Theorem E.3 we know that N D gp

[

S k S k [ S k S k



S

D gp

Sk Sk for all m > 0

[



[  S k S k D gp S k S k

Since VN .m/ cmD there exists a finitely generated subgroup G0  N with VG0 .m/ cmD . Since G0 is finitely generated, G0  gp .S n S n / and G0  n n n n gp .S n S n / for some n. Letting AD  S S and B D S S in Theorem E.6, we  D=2 .k/ again obtain kF k1 D O k . Case III: G contains a free subsemigroup on two generators and Sm \ Sn D ; for all m ¤ n Since in this case we are assuming that G contains a free subsemigroup, VG .m/

cmD for all D > 0. Therefore we need to show that kF .k/ k1 D O.k D=2 / for all D > 0.

 .2/ Consider the probability density function F CF . This is clearly adapted 2 since its support is S [ S 2 S . Also, by Theorem E.3, N D G since S    1 2    S [ S 2 S [ S 2 . Furthermore, it is clear that S [ S 2 \ S [ S 2 ¤ ;. Therefore, using Lemma E.11 we see that Case I of this proof applies yielding  !.n/     F C F .2/    for all D > 0 there exists a C1 such that   2  

1



C1 : nD=2

Q Now, since in this case we are assuming that supp.F .m/ / D S m and supp.F .m/ /D m Q S have empty intersection for m ¤ m, Q we have the equality

 !.n/     F C F .2/      2  

1

 n  X nŠ 1 .nC`/    D F    `Š.n  `/Š 2n `D0 1    1  nŠ  .nC`/  : D max F  1 `D0;:::;n `Š.n  `/Š 2n

Letting n D 2k and considering the term with ` D k we get .2k/Š 1 kF .3k/ k1  kŠkŠ 22k



1 .2k/Š max kF .2kC`/ k1 `D0;:::;2k `Š.2k  `/Š 22k  2k    .2/   C1 :  D  F CF  .2k/D=2 1



412

E Rate of Decay of Concentration Functions on Discrete Groups

But by a simple calculation involving Stirling’s formula we see that p1 . So, k C2 .2k/Š 1 C3

p Dp 2k kŠkŠ 2 k k

.2k/Š 1 kŠkŠ 22k



Thus, for each D > 0 there exists a C1 such that

C3 kF .3k/ k1 C1 p ;  D=2 .2k/ k

whence for each D > 0 there exists a C4 such that kF .3k/ k1 

C4 : k .D1/=2

Since this is true for all D > 0 we can replace D by D C 1 (notice that the constant may change in so doing since the constant may depend on the specific value of D) to get for each D > 0 there exists a C5 such that kF .3k/ k1 

C5 : k D=2

By monotonicity, we finally arrive at for each D > 0 there exists a C6 such that kF .k/ k1  This is the desired result.

C6 : k D=2 t u

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Index

A Abelian group, 32, 50, 81, 126, 127, 137, 169, 210, 222, 240 Abelian semigroup, 2, 26, 37, 88, 147, 151, 154–156, 168, 194, 222 discrete, 63, 156, 159 Absolutely continuous component, 129–131 Absolutely continuous measure, 131, 139, 176, 205 Act, 6, 53, 54 Adapted, 24, 128, 198, 204, 206, 207, 397–399, 401, 402, 408–412 Affine contraction, 214, 222, 250 Affine maps, 251, 254, 260, 264, 265, 276, 313, 344, 353, 356, 358 Almost sure convergence, 150 Antihomomorphism, 6, 44, 45, 59, 60 Antiisomorphism, 6 Attractor, 276–283, 356 Average contractivity condition, 280

B Basis of a matrix, 38, 40, 42, 45–49, 55, 58, 59 Bernoulli distribution, 248 Bicancellative, 72, 78, 79 Bicyclic semigroup, 21, 22, 60, 197 Bilateral, 173, 175 ideal, 2, 26, 180, 181, 220, 224 identity, 209 Bilateral random walk, 173, 183, 187, 208, 211, 212, 220, 224, 229, 230, 236, 238, 247 Bilateral walk, 172, 178, 180, 183–185, 188, 190, 193, 198, 199, 209, 212, 220, 221, 223, 226–228, 231, 232, 234, 242, 243 Birkhoff’s theorem, 48, 164, 165

Bistochastic matrix, 48, 49, 59, 61, 143, 164, 165, 222, 250 Bistochastic stochastic matrix, 48

C Cancellative semigroup, 5, 32, 95, 131, 273 Cell, 9 group, 11, 18, 19, 31, 32, 35 null, 11, 18, 19, 31, 32, 35 C-group, 128 Chain, 48, 175, 192, 195, 204, 208, 214, 215, 217–219, 239 Circulant matrix, 5, 50, 137 Code, 248 Coding theory, 61, 248 Coloring semigroup, 51 Communicate, 177, 180, 181, 185, 209, 215, 220, 240, 243 Communication relation, 171, 177, 215, 219, 220 Commutative semigroup. See Abelian semigroup Commute, 2 Compact group, 32, 33, 49, 63, 74–77, 82, 87, 88, 90, 94, 95, 106, 113, 115, 119, 127, 129–131, 133, 136, 139, 144, 147, 160, 198–219, 232, 238, 239, 241, 244, 249, 286, 287, 289–291, 293–295, 297, 298, 303, 317, 332, 341, 357, 362, 397 connected, 130, 138, 140 of nonnegative matrices, 42, 45, 46, 61 Compact group element, 33 Compact group factor, 33, 87, 88, 90, 94, 106, 113, 119, 241, 286, 297, 303, 332, 341 Compactness condition, 33, 51, 99, 102, 103, 105, 106, 115, 116, 250

423

424 Compact semigroup, 26–29, 32, 63, 100, 141, 152, 219–242, 246–250, 279, 341, 358 abelian, 26, 50, 63, 81, 88, 126, 137, 147, 151, 154, 222, 240 Completely simple minimal ideal. See Kernel Completely 0-simple semigroup, 18–22, 28, 30, 33, 61, 246 Completely simple (sub)semigroup, 145, 241 of nonnegative matrices, 316 Complete set of representatives, 3, 4 Component of a transition probability operator, 173, 176, 177, 179, 211, 215, 220, 224, 229, 230, 238, 240, 249, 250 continuous, 176, 179 equivalent, 173, 176 nontrivial, 176, 238 Composition of functions, 3, 61, 182 Composition of relations, 3, 5 Concentration functions, 126, 128, 397–412 (CL) condition, 251 (CR) condition, 251 Congruence, 6, 7, 9, 51 Conservative element, 177, 209 Constant mapping, 3, 13, 222 Continuous component, 129–131, 176, 204, 206–209, 238, 239, 245, 247, 250 Continuous measure, 176, 383, 385 Continuous singular measure, 205, 206, 384, 387 Contractive coset, 128 Convergence almost sure, 156, 166, 167, 192, 235–237, 253, 282, 322, 325, 331–333, 346, 366 vague, 66, 84, 99, 105 to 0, 85, 101, 107–109, 115, 116, 119, 122, 124, 134, 211, 245, 306 weak, 63, 64, 66, 67, 83–168, 192, 284–333, 345, 358 of convolution sequences, 122 of empirical distributions, 192 of products of affine maps, 356, 358 of products of nonnegative matrices, 356 of products of real matrices, 299, 302 Convolution, 63–68, 72, 83–168, 172, 176, 197, 206, 207, 211, 222, 247, 249, 284, 341, 366, 394, 399, 403, 404, 408 power, 68, 87, 135, 177, 204, 253, 289, 357, 397 Convolution sequence, 121, 171, 211, 247, 251, 354

Index tight, 83, 87–89, 96, 97, 101, 122, 123, 132, 133, 137, 150, 171, 286, 289–291, 293, 296, 305, 306, 309, 310, 312, 315, 354 weak convergence of, 63, 87, 101, 122, 150, 192, 289, 299 Cross section of a partition, 4, 8, 9, 16, 20 Cyclic classes, 184, 185, 192, 250

D Density, 129–131, 169, 205–207, 209, 210, 217, 248, 365–367, 397, 398, 402–404, 408, 411 Direct product, 7–11, 17, 22, 27, 126, 128, 194, 204, 251 Dirichlet density, 365–367 Discrete group, 63, 126, 138, 156, 159, 194, 247, 249, 397–412 Discrete measure, 63, 138, 180, 186, 192, 385, 388, 397–400, 402 Discrete semigroup, 122, 124, 135, 136, 138, 169, 180–197, 213, 220, 241, 247, 248 abelian, 63, 159 Distinguished subgroup, 134 Divisor of zero, 10–12, 19, 27–30, 37 Doubly stochastic matrix. See Bistochastic stochastic matrix

E Edge coloring, 51 Effective act, 6 Ellis’s theorem, 24, 26–28, 30, 42, 198, 216 Embedding, 52, 81 Empirical distributions convergence of, 192, 212 Equicontinuous operator, 229, 230 Equidistributed random walk, 212 Equivalent component, 176 Ergodic, 192, 231, 234, 236–240, 250, 357, 373, 375, 376, 378, 381 Essential class, 181, 183, 184, 191–194, 198, 199, 209, 210, 212, 228, 239, 240, 242, 243 Essential element, 177, 179–187, 189, 190, 197–199, 216, 220, 221, 239, 240, 242 Exponential growth, 249

Index F Factor, 2, 7, 11, 15, 20, 22, 30, 171, 172, 175, 185, 190, 192, 196, 213, 224, 241, 245 group, 123, 124, 189, 193–195, 232, 244–246, 248, 250, 312, 317, 348, 362 compact, 33, 87, 88, 90, 94, 106, 113, 119, 171, 241, 251, 286, 297, 303, 332, 341 finite, 124, 131, 189, 193, 195, 248 left, 224 middle, 192, 194, 232, 244, 246 right, 15, 224 Factor group, 131, 133, 362, 363 Factor semigroup, 6, 7, 20 Feller operator, 177, 215 Finite group factor, 124, 131, 189, 193, 195, 248 First hitting time, 176 First return time, 176, 218 Free subsemigroup, 409–412 Frobenius norm, 33 Full transformation semigroup, 3, 6, 8, 20 Furstenberg-Kifer theorern, 345–352, 358

G Gamma random variable, 365 General linear group, 33, 34, 41 Generating set, 398, 407 Group, 2, 63, 171, 256, 359, 381, 397–412 with 0, 10, 12, 18, 19, 29, 31 abelian, 32, 50, 81, 126, 127, 137, 169, 210, 222, 240 compact, 32, 33, 45, 49, 63, 74–77, 82, 87, 88, 90, 94, 95, 106, 113, 115, 119, 127, 129–131, 133, 136, 139, 144, 147, 160, 198–219, 232, 238, 239, 241, 244, 249, 286, 287, 289–291, 293–295, 297, 298, 303, 317, 332, 341, 357, 362, 397 connected, 130, 138, 140, 249 of nonnegative matrices, 46, 61 discrete, 63, 126, 138, 156, 159, 194, 247, 249, 397–412 factor, 131, 133, 362, 363 general linear, 33, 34, 41 locally compact, 45, 49, 127, 198–219, 238, 241, 244, 249, 397 neat, 127 noncompact, 109, 124–128 of nonnegative matrices, 42, 45, 46 permutation, 8, 44, 45, 58, 59, 362–364

425 recurrent, 194, 195, 204, 211 strange, 127 symmetric, 8, 9 topological, 24–33, 42, 45, 72, 75, 81, 94–96, 128, 151, 179, 198, 204, 213 transient, 194, 217, 218 Group cell, 11, 18, 19, 31, 32, 35 Group factor, 133, 194, 232, 244, 348, 362, 363 compact, 33, 87, 88, 90, 94, 106, 113, 119, 123, 171, 241, 245, 246, 250, 251, 286, 297, 303, 312, 317, 332, 341 finite, 124, 131, 189, 193, 195, 248 Growth, 210, 249, 253, 375, 380, 397, 398, 404, 407, 408 Guivarc’h-Keane conjecture, 249

H Haar measure, 76, 78, 79, 129–132, 138, 139, 169, 176, 192, 204–207, 210–212, 228, 232, 239, 245, 249 unimodular, 72, 81, 211, 397 Harmonic, 211 Harris recurrent, 204, 207, 208, 217 Hitting time, 176, 203 Homomorphism, 6–10, 14, 17, 34, 45, 59, 80, 81, 133, 206, 248, 363

I Ideal, 1, 75, 179–191, 254–264, 385 completely simple minimal (see Kernel) 0-minimal, 10, 12, 18 Idempotent, 2, 64–83, 182, 256, 359 primitive, 12–14, 17, 18, 28–30, 34, 35, 242, 256 Idempotent matrix, 17, 34, 36, 38, 42, 48, 49, 52, 56, 57, 290, 302, 304, 316, 329 Idempotent probability measure, 63–83, 87, 88, 96, 115, 137, 143, 241, 293, 329, 330 Identity, 2–6, 8–14, 18–21, 23, 25–27, 29, 31, 32, 34–36, 40–43, 45, 46, 50, 52–55, 58, 59, 75, 87–90, 93, 94, 108, 130–133, 136, 137, 142, 143, 159–161, 163, 177, 194, 196, 198, 200, 201, 204, 206–209, 213, 216, 223, 239, 242, 243, 248, 249, 259, 264, 268, 273, 286, 289, 290, 292, 293, 316, 317, 332, 357, 360, 361, 363, 364, 397, 405–407

426

Index

Inessential class, 181 Invariance semigroup, 215–218, 240 Invariant measure, 63, 192, 211, 224, 233, 235, 236, 240, 246, 253, 333–355, 376, 394. See also Probability measure l , 72, 75, 76, 79, 142, 144 left, 81 r*, 72, 77 right, 80, 81 Inverse, 2–4, 15, 19–24, 27, 28, 34, 42, 44, 46, 53, 54, 61, 194, 248, 273, 295, 407, 410 of a transition probability matrix, 48, 61 Isomorphism, 6, 8, 14, 28, 41, 42, 362–364 topological, 27, 79, 81, 87 Iterated function system, 214, 239, 280

Left random walk, 172, 173, 177, 180, 182–186, 190, 193, 198, 200, 201, 204, 206, 207, 211, 212, 220, 226, 228, 231, 234, 236, 241–246, 254, 355 Left simple, 2, 5, 7, 8, 13, 14, 32, 182, 236, 239 Left 0-simple, 10, 11 Left translation, 6, 23 Left zero, 2, 3 Left zero semigroup, 3, 9, 14, 133 l -invariant measure, 72, 75, 76, 79, 142, 144 Locally compact group, 45, 49, 127, 198–219, 238, 241, 244, 249, 397 Locally compact semigroup, 27, 29, 225, 238, 241, 242, 246, 250

J Jointly continuous multiplication, 24, 30, 33, 35, 168, 220 Juxtaposition, 1, 20

M Markov chain, 48, 49, 171, 174–177, 179, 180, 183, 184, 186, 189, 191, 192, 195, 196, 204, 208, 213–218, 240, 247, 381 canonical, 175 Markov property, 176, 208 Markov random walk, 195, 196, 213, 214, 216, 218, 219, 239, 240, 248 Matrix semigroup, 27, 37, 60, 61, 251, 280, 354 MaxPlus algebra, 251 Measure. See also Probability measure Haar, 76, 78, 79, 129–132, 138, 139, 169, 176, 192, 204–207, 210–212, 228, 232, 239, 245, 249 unimodular, 72, 81, 211, 397 left invariant, 81 l -invariant, 72, 75, 76, 79, 142, 144 Radon, 203, 211, 244 right invariant, 80, 81 r -invariant, 72, 77 symmetric, 210 Middle factor, 11, 192, 194, 232, 244, 246 Minimal ideal, 2, 12, 15, 16, 22, 28, 37, 54, 119, 124, 133, 181, 182, 184–191, 193, 234, 235, 238, 241, 242, 248, 254, 256, 258, 260–262, 286, 287, 290, 293, 303, 311, 330, 341 completely simple (see Kernel) Minimal rank, 13, 16, 17, 35, 53, 260, 266, 279, 296, 311, 315, 329, 359, 361 Mixed random walk, 172, 173, 175, 181, 183, 187, 188, 190, 193, 198, 208, 211, 212, 219, 220, 224, 228, 229,

K Kernel, 15, 26, 27, 32, 33, 50–54, 59, 61, 75, 87–89, 94, 97, 100, 116, 118, 135–137, 147, 175, 176, 194, 195, 204, 205, 207, 208, 213, 214, 221, 222, 224, 227, 228, 232, 238–240, 248, 250, 251, 258, 289, 334, 344, 359, 362 Kesten conjecture, 249 Koutsky process, 196, 214–216, 239, 240

L Law of a random variable, 172 Laws of large numbers, 270, 276, 319, 320, 345, 347, 351, 354, 390, 392 Leads to, 65, 145, 215, 332 infinitely often, 107, 108, 177, 217 Left cancellative, 7, 72 Left factor, 15 Left group, 7, 15, 20, 51, 52, 75–79, 83 representation, 77 Left ideal, 2, 3, 13, 15, 16, 26, 27, 31, 75, 118, 179, 180, 182, 191, 220, 221, 231, 234, 238, 239, 242, 244, 258, 260, 264, 267, 269, 311 0-minimal, 18 Left identity, 2, 4, 5, 8, 10, 14, 27 Left invariant measure, 81

Index 232–234, 236, 242, 247, 248, 253, 270, 333, 334, 338, 342, 344, 345, 347, 349, 352 Mod (Modulo), 6, 7, 11, 184, 192 Model map, 389 Monoid, 248 Monothetic subsemigroup, 27, 222 Multiplication, 1–3, 5–9, 11, 13, 14, 21–24, 27–30, 33, 35, 46, 50, 51, 59, 62, 85–87, 95, 135, 137, 168, 182, 197, 204, 205, 219, 221, 222, 225, 228, 230, 241, 280, 286, 303–305, 344, 355, 357, 394 jointly continuous multiplication, 24, 30, 33, 35, 168, 220 separately continuous multiplication, 23, 24

N Neat group, 127 Nonzero elements, 10, 18, 19, 48, 179, 246, 261, 262 Norm, asymptotic behavior of, 369, 374–378, 380–381 Null cell, 11, 18, 19, 31, 32, 35 Null recurrent, 178, 189, 190, 194, 246 Null semigroup, 10, 18 Null space, 17, 34–36, 41

O Open translations, 81

P Parallel random walks, 250 Partition of a transformation, 20 Period of random walk, 183 Permutation group, 8 Permutation matrix, 6, 7 Polynomial growth, 210 Positive point, 177, 189, 208, 212, 223–224, 227, 246 Positive Harris, 218 Positive recurrent, 178, 189–196, 208, 211, 212, 217, 225, 227, 244–246, 250, 253, 258, 265, 266, 273, 281 Primitive idempotent, 12–14, 17, 18, 20–22, 28–30, 256 Principal ideal, 2, 26 left, 2 right, 2

427 Probability measure absolutely continuous component of, 129, 131 adapted, 206, 207, 399 convergence of, 83–168 (see also Convergence) convolution of, 83–168 idempotent, 63–81, 87, 88, 96, 115, 137, 143, 224, 241, 293, 317, 329, 330 invariant, 61, 77, 83, 97, 212, 225, 232, 234, 235, 250, 265, 266, 277, 278, 283, 321, 337, 338, 342–345, 348, 356, 358, 370, 371, 374, 375, 388, 390, 392 for mixed random walks on nonnegative matrices, 333 for products of I.I.D. random matrices, 284–333 recurrent, 189, 193 null, 189 positive, 189 regular, 64, 65, 75, 173, 180 spread-out, 128 strictly aperiodic, 128 support of, 177, 179, 180 tightness of, 119 Product measure, 74, 172, 192, 205, 232, 392 Product –talgebra, 172 Projection matrix, 29, 34, 219 Proper divisors of zero, 10–12, 19, 27, 28 Pseudoinvertible, 22, 54

Q Quotient semigroup, 6, 7, 34 R Radon measure, 203, 211, 244 Random walk, 108, 150, 171–251, 253–284, 333–355, 358. See also Bilateral random walk; Left random walk; Mixed random walk; Right random walk equidistributed, 212, 238, 239 Harris, 204, 207, 208, 217 parallel, 237 period of, 183 recurrent, 178, 194, 195, 217, 218, 246, 250, 254–284 null, 178, 194, 246 positive, 189–196, 208, 211, 212, 217, 225, 227, 245, 246 positive Harris, 218

428 transient, 178, 187, 188, 197, 203, 217 unilateral, 184, 188 Range of a matrix, 34, 36, 41, 51 of a transformation, 3–4, 8–10, 16, 20 Rank. See also Minimal rank of a matrix, 16, 27, 29, 31, 33–53, 56, 57, 59 of a transformation, 3, 12, 19, 20, 22 Rate of decay of the concentration function, 397–412 Recurrent Harris, 204, 207, 208, 217 null, 178, 189, 190, 194, 246 positive, 178, 189–196, 208, 211, 212, 217, 225, 227, 244–246, 250, 253, 258, 265, 266, 273, 281 Recurrent sets (R, R(Z), R(W)) of random walks on matrices, 258, 355 Rees factor (quotient) semigroup, 7 Rees product, 9, 11, 12, 14, 15, 17, 18, 22, 27–29, 35, 49, 61, 197, 228 over a group with 0, 12 representation, 15 Rees–Suschkewitsch representation, 12–22, 26, 35, 184, 185, 194, 231, 238, 245, 246 Regular probability measures, 64, 65, 75, 180 Regular sandwich function, 11 Regular semigroup, 12, 18 Return point, 177, 178, 180, 181, 183 Return time, 176, 194, 196, 200, 202, 204, 208, 218 Right cancellative, 2, 7, 72, 77, 81 Right convolution operator, 399, 403, 404 Right factor, 15 Right group, 7–10, 16, 20, 27, 31, 75, 76, 256 Right ideal, 2, 3, 9–11, 13–16, 18, 22, 26, 32, 34, 53, 117, 118, 180, 182, 183, 191, 220, 221, 224, 228, 231, 232, 234, 236, 239, 240, 242, 250, 255, 256, 258–260, 311 0-minimal, 10 Right identity, 2, 6, 10, 14 Right-invariant measure, 80, 81 Right random walk, 173, 174, 180, 182, 183, 185–188, 190–192, 194, 195, 198, 200, 201, 203, 205, 207, 211, 220, 224, 226–228, 231, 232, 234–238, 242, 245, 246, 250, 253, 254, 355, 358 Right reversible, 81 Right simple, 2, 7, 8, 25, 26, 31, 72, 235 Right 0-simple, 10–12

Index Right translation, 6, 23, 81 Right zero, 2, 3, 35 Right zero semigroup, 7, 9–11, 14, 27, 29, 31, 42, 93, 133 r*-invariant measure, 72, 77 Road-coloring problem, 61

S Sandwich function, 9, 11, 12, 14, 17, 18, 22, 35, 185, 194, 195, 241, 248 regular, 11 trivial, 9 Semigroup, 1–251, 253–256, 259–261, 265, 266, 268, 270, 272, 273, 279, 280, 284, 287, 289, 293, 296, 297, 302, 303, 305, 311–313, 316, 322, 331, 333, 334, 337, 341, 342, 348, 354, 356–359, 362, 369, 381, 383, 397, 398, 404–411 abelian, 2, 26, 63, 147, 151, 154–156, 159, 168, 194, 222 discrete, 63, 156, 159 of affine maps, 214, 222, 251 bicyclic, 21, 22, 60, 197 cancellative, 32, 95, 131 commutative (see Abelian) compact, 26–29, 32, 63, 100, 141, 147, 152, 168, 179, 219–242, 246–250, 279, 305, 341, 358 abelian, 26, 63, 147, 151, 154, 155, 222 completely simple, 13–17, 26, 31, 42, 46, 47, 49, 52, 59, 60, 63, 82, 87, 88, 91, 106, 109, 115, 119, 123, 134, 136, 166, 171, 221, 241–246, 251, 273, 312, 357 of nonnegative matrices, 46, 47 completely 0-simple, 18, 19, 21, 30, 33, 61, 246 direct product of, 7, 251 discrete, 63, 122, 124, 135, 136, 138, 169, 180–197, 213, 220, 241, 247, 248 abelian, 63, 156, 159 factor, 6, 7, 20 invariance, 215–218, 240 left zero, 3, 9, 14 locally compact, 27, 29, 225, 238, 241, 242, 246, 250 of matrices, 16, 33–60, 168 of nonnegative matrices, 46, 47, 51, 105, 259, 261 null, 10, 18 positive recurrent, 189 pseudoinvertible, 54

Index quotient, 34 recurrent, 189, 194 positive, 189 Rees factor (quotient), 7, 20 regular, 11, 12 right reversible, 81 right zero, 7, 9–11, 14, 27, 29, 31, 42, 93, 133 semitopological, 23, 168 topological, 22–34, 48, 60, 63–169, 171, 177, 179, 219, 222, 241, 289, 322, 337, 357 of transformations, 1, 3, 4, 6–8, 12, 15, 16, 20, 21, 52, 197 full, 3, 6, 8, 20 transient, 218 with 0, 10, 12, 17, 27, 29, 225 0-simple, 11, 12, 18–21, 21, 28–30, 61, 246, 260 Semigroup homomorphism, 6 Semigroup isomorphism, 6, 14, 27 Semigroup monothetic, 3, 29 Semigroup of relations, 4, 5, 60 Separately continuous multiplication, 23, 24, 168 Shift operator, 175, 231 Sierpinski gasket, 278–280, 356 Simple, 2 Skeleton of a matrix, 253 Solvable transformation semigroup, 52 Spread-out, 128, 169, 204–208, 210, 211 State space, 53, 55, 171, 177, 189, 196, 198, 204, 215, 240, 241, 250, 284, 347, 381 Stationary distributions, 231, 238, 249 Stochastic matrix, 27, 48, 165, 332, 360, 361, 365, 366, 383, 386, 394 Stopping time, 175, 194, 195, 238 Strange group, 127 Strictly aperiodic, 128 Strongly continuous operator, 176 Strong Markov property, 176, 208 Subgroup, 2 distinguished, 134 Subharmonic, 211 Subsemigroup, 2–6, 8, 9, 11–13, 15, 19, 21–23, 25, 27, 30, 32, 33, 35, 37, 54, 61, 68, 74, 75, 79, 81, 87, 88, 90, 92, 94, 95, 97, 98, 113, 142, 143, 145, 149–151, 161, 165, 177, 179, 182, 185, 198, 199, 206, 215, 222, 228, 238, 241–244, 249, 256, 259, 260, 266, 268, 270, 293, 297, 311, 312, 316, 348, 362, 397, 398,

429 404, 408–411. See also Semigroup monothetic Substochastic matrix, 27, 48 Superharmonic, 211 Support of a measure, 177, 225 Suschkewitsch, 12–22, 26, 29, 30, 35, 60, 184, 185, 194, 231, 238, 241, 244–246, 248, 251 Symmetric group, 8, 9 Symmetric measure, 210

T Taboo probability, 195 Tail-idempotent, 160, 161 Tight, 83, 87–89, 96–99, 101, 105, 106, 110–113, 118–124, 132, 133, 136, 137, 141, 150, 159, 163, 164, 171, 191, 258, 266, 269, 275, 283, 285, 286, 288–291, 293, 296, 302, 303, 305–307, 309, 310, 312, 313, 315, 317–321, 325–329, 331–334, 338, 339, 341–344, 347, 354 Tightness criterion for products of affine maps, 250, 254, 313 for products of random matrices, 284–333 for products of random 2 x 2 matrices, 318 Topological group, 24–33, 42, 45, 72, 75, 81, 94–96, 128, 151, 179, 198, 204, 213 Topological isomorphism, 27, 79, 81, 87 Topological semigroups, 22–33, 48, 60, 63–169, 171, 177, 179, 219, 222, 241, 289, 322, 337, 357 Trace, 36, 40, 46, 52 Trace of transformation, 52 Transformation, 1, 3–8, 12, 16, 20, 21, 33, 34, 48, 52, 62, 197, 260 semigroup of, 3, 4, 20 Transformation matrix, 7, 48 Transient, 49, 178, 187–189, 194, 197, 203, 208, 211, 217, 218, 222, 223, 225, 249 Transition probability function, 173, 177, 180, 334, 338 Transition probability matrix, 61, 184, 240 inverse, 61 Transition probability operator, 173, 176, 177, 179, 211, 215, 220, 224, 229, 230, 238, 240, 249, 250 Translation, 6, 23, 81, 175, 219, 231 open, 81 Trivial sandwich function, 9 Two sided. See Bilateral Type of a point, 189

430 U Unconditionally recurrent, 186 Unilateral walk, 182, 184, 187, 188, 198, 201, 223, 238, 239, 242, 244 Unimodular, 72, 81, 211, 397, 405 Upper packing dimension, 391 V Vague convergence, 66, 84, 85, 99, 101, 105, 107, 108, 116, 122, 124, 211 to 0, 85, 101, 105, 107, 108, 116, 122, 134, 211 Volume growth rate, 398, 404, 407, 408 W Weak convergence of convolution sequences, 63, 64, 83–168

Index of empirical distributions, 192 of products of affine maps, 313 of products of nonnegative matrices, 287 of products of real matrices, 302 Weak*-convergence, 67, 101 Weakly continuous operator, 176

Z Zero, 2, 10 divisor of, 10, 28 proper, 10 Zero semigroup, 3, 7, 9–11, 14, 27, 29, 31, 42, 93, 133 0-minimal ideal, 10 0-minimal left (right) ideal, 10, 18 0-simple, 10–12, 18–22, 28–30, 33–35, 37, 61, 246, 251, 259, 260, 262, 264