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BILLINGSLEY Tee University of Chicago Chicago, Illinois Convergence of Probability Measures WILEY SERIES IN PROBABILITY AND STATISTICS PROBABILITY AND STATISTICS SECTION Established by WALTER A. SHEWHART and SAMUEL S. WILKS Editors: Vic Barnett, Noel A. C. Cressie, Nicholas I. Fisher, Iain M, Johnstone, J. B. Kadane, David G. Kendall, David W. Scott, Bernard W. Silverman, Adrian F. M. Smith, Jozef L. Teugels; Ralph A. Bradley, Emeritus, J. Stuart Hunter, Emeritus A complete list of the titles in this series appears at the end of this volume. Convergence of Probability Measures Second Edition PATRICK BILLINGSLEY TEe University of Chicago Chicago, Illinois A Wiley-Interscience Publication JOHN WILEY & SONS,INC. NewYork Chichester Weinheim Brisbane Singapore Toronto This text is printed on acid-free paper. 8 Copyright 0 1999 by John Wiley & Sons, Inc. All rights reserved. Published simultaneously in Canada. No patt of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4744. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012, (212) 850-601 1, fax (212) 850-6008, E-Mail: PERMREQ @, WILEY.COM. For ordering and customer service, call 1-800-CALL-WILEY. Library of Congress Cataloging in Publication Data: Billingsley, Patrick. Convergence of probability measures / Patrick Billingsley. - 2nd ed. p. cm. - (Wiley series in probability and statistics. Probability and statistics) “Wiley-Interscience publication.” Includes bibliographical references and indexes. ISBN 0-471-19745-9 (alk. paper) 1. Probability measures. 2. Metric spaces. 3. Convergence. I. Title. 11. Series: Wiley series in probability and statistics. Probability and statistics. QA273.6.BS5 1999 5 19.24~21 99-30372 CIP Printed in the United States of America 109876 PREFACE From the preface to the first edition. Asymptotic distribution theo- rems in probability and statistics have from the beginning depended on the classical theory of weak convergence of distribution functions in Euclidean spat-onvergence, that is, at continuity points of the limit function. The past several decades have seen the creation and extensive application of a more inclusive theory of weak convergence of probability measures on metric spaces. There are many asymptotic results that can be formulated within the classical theory but require for their proofs this more general theory, which thus does not merely study itself. This book is about weak-convergence methods in metric spaces, with applications sufficient to show their power and utility. The second edition. A person who read the first edition of this book when it appeared thirty years ago could move directly on to the periodical literature and to research in the subject. Although the book no longer takes the reader to the current boundary of what is known in this area of probability theory, I think it is still useful as a textbook one can study before tackling the industrial-strength treatises now available. For the second edition I have reworked most of the sections, clarifying some and shortening others (most notably the ones on dependent random variables) by using discoveries of the last thirty years, and I have added some new topics. I have written with students in mind; for example, instead of going directly to the space D[0, oo), I have moved, in what I hope are easy stages, from C[O, I] to D[O,11 to D[O,oo). In an earlier book of mine, I said that I had tried to follow the excellent example of Hardy and Wright, who wrote their Introduction to the Theory of Numbers with the avowed aim, as they say in the preface, of producing an interesting book, and I have again taken them as my model. Chicago, Patrick Billingsley January 1999 V For mathematical information and advice, I thank Richard Arratia, Peter Donnelly, Walter Philipp, Simon Tavar6, and Michael Wichura. For essential help on Tex and the figures, I thank Marty Billingsley and Mitzi Nakatsuka. PB CONTENTS Introduction 1 Chapter 1. Weak convergence in Metric Spaces 7 Section 1. Measures on Metric Spaces, 7 Measures and Integrals. Tightness. Some Examples. Problems. Section 2. Properties of Weak Convergence, 14 The Portmanteau Theorem. Other Criteria. ,The Mapping Theorem. Product Spaces. Problems. Section 3. Convergence in Distribution, 24 Random Elements. Convergence in Distribution. Convergence in Probability. Local us. Integral Laws. Integration to the Limit. Relative measure.* Three Lemmas.* Problems. Section 4. Long Cycles and Large Divisors,* 38 Long Cycles. The Space A. The Poisson-Dirichlet Distribu- tion. Size-Biased Sampling. Large Prime Divisors. Technical Arguments. Problems. Section 5. Prohorov’s Theorem, 57 Relative Compactness. Tightness. The Proof. Problems. Section 6. A Miscellany,* 65 The Ball a-Field. Slcorohod’s Representation Theorem. The Prohorov Metric. A Coupling Theorem. Problems. Chapter 2. The Space C 80 Section 7. Weak Convergence and Tightness in C, 80 * Starred topics can be omitted on a first reading vii viii CONTENTS Tightness and Compactness in C. Random Functions. Coordi- nate Variables. Problems. Section 8. Wiener Measure and Donsker’s Theorem, 86 Wiener Measure. Construction of Wiener Measure. Donsker ’s Theorem. An Application. The Brownian Bridge. Problems. Section 9. Functions of Brownian Motion Paths, 94 Maximum and Minimum. The Arc Sane Law. The Brownian Bridge. Problems. Section 10. Maximal Inequalities,lO5 Maxima of Partial Sums. A More General Inequality. A Fur- ther Inequality. Problems. Section 11. Trigonometric Series: 113 Lacunary Series. Incommensurabl e Arguments. Problem. Chapter 3. The Space D 121 Section 12. The Geometry of D, 121 The Definition. The Skorohod Topology. Separability and Com- pleteness of D. Compactness in D. A Second Characterization of Compactness. Finite-Dimensional Sets. Random Functions in D. The Poisson Lamat.* Problems. Section 13. Weak Convergence and Tightness in D, 138 Finite-Dimensional Distributions. Tightness. A Criterion for Convergence. A Criterion for Existence.“ Problem. Section 14. Applications, 146 Donsker’s Theorem Again. An Extension. Dominated Mea- sures. Empirical Distribution Functions. Random Change of Time. Renewal Theory. Problems. Section 15. Uniform Topologies,” 156 The Uniform Metric on D[0,1]. A Theorem of Dudley’s. Em- pirical Processes Indexed by Convex Sets. Section 16. The Space D[0, oo), 166 Definitions. Properties of the Metric. SeparabilitQ and Com- pleteness. Compactness. Finite-Dimensional Sets. Weak Con- vergence. Tightness. Aldous ’s Tightness Criterion. CONTENTS ix Chapter 4. Dependent Variables 180 Section 17. More on Prime Divisors: 180 Introduction. A General Limit Theorem. The Brownian Mo- tion Limit. The Poisson-Process Limit. Section 18. Martingales, 193 Triangular Arrays. Ergodic Martingale Differences. Section 19. Ergodic Processes, 196 The Basic Theorem. Uniform Mixing. Functions of Mixing Processes. Diophantine Approximation. Chapter 5. Other Modes of Convergence 207 Section 20. Convergence in Probability, 207 A Convergence-in-Probability Version of Donsker ’s Theorem. Section 21. Approximation by Independent Normal Sequences, 211 Section 22. Strassen’s Theorem, 220 The Theorem. Preliminaries on Brownian Motion. Proof of Strassen’s Theorem. Applications. Appendix M 236 Metric Spaces. Analysis. Convexity. Probability. Some Notes on the Problems 264 Bibliographical Notes 267 Bibliography 270 Index 275 Convergence of Probability Measures Patrick Billingsley Copyright 01999 by John Wiley & Sons, Inc INTRODUCTION The De Moivre-Laplace limit theorem says that, if Fn(z)= P['" m--np is the distribution function of the normalized number of successes in n Bernoulli trials, and if is the standard normal distibution function, then (3) Fn(z) + F(z) for all z (n-+ 00, the probability p of success fixed). We say of arbitrary distribution functions F, and F on the line that F, converges weakly to F, which we indicate by writing F, + F, if (3) holds at every continuity point z of F. Thus the De Moivre-Laplace theorem says that (1) converges weakly to (2); since (2) is everywhere continuous, the proviso about continuity points is vacuous in this case. If Fn and F are defined by (4) Fn(z) = I[n-l,m) (I for the indicator function) and (5) F(z)= 1[o,CO)(z)7 then again Fn + F, and this time the proviso does come into play: (3) fails at z = 0. 1 2 INTRODUCTION For a better understanding of this notion of weak convergence, which underlies a large class of limit theorems in probability, consider the probability measures P. and P generated by arbitrary distribution functions F, and F. These probability measures, defined on the class of Borel subsets of the line, are uniquely determined by the requirements P~(-oo,z]= F~(s),P(-co,z] = F(z). Since F is continuous at 2 if and only if the set {z} consisting of z alone has P-measure 0, Fn =+ F means that the implication holds for each 2. Let dA denote the boundary of a subset A of the line; dA consists of those points that are limits of sequences of points in A and are also limits of sequences of points outside A. Since the boundary of (oo,~] consists of the single point 2, (6) is the same thing as (7) &(A) ---$ P(A) if P(dA) = 0, where we have written A for (--oo,z]. The fact of the matter is that Fn =+ F holds if and only if the implication (7) is true for every Borel set A-a result proved in Chapter 1. Let us distinguish by the term P-continuity set those Borel sets A for which P(8A) = 0, and let us say that P.
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