Probability Theory: An Analytic View

551 Pages·2010·4.69 MB·English

by  Daniel W. Stroock

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Probability Theory
An Analytic View, Second Edition
This second edition of Daniel W. Stroock’s text is suitable for first-year graduate
students with a good grasp of introductory undergraduate probability. It provides
a reasonably thorough introduction to modern probability theory with an empha-
sis on the mutually beneficial relationship between probability theory and analy-
sis. It includes more than 750 exercises and offers new material on Levy processes,
large deviations theory, Gaussian measures on a Banach space, and the relationship
between a Wiener measure and partial differential equations.
Thefirstpartofthebookdealswithindependentrandomvariables,CentralLimit
phenomena,thegeneraltheoryofweakconvergenceandseveralofitsapplications,as
wellaselementsofboththeGaussianandMarkoviantheoriesofmeasuresonfunction
space. The introduction of conditional expectation values is postponed until the
secondpartofthebook,whereitisappliedtothestudyofmartingales.Thispartalso
explorestheconnectionbetweenmartingalesandvariousaspectsofclassicalanalysis
and the connections between a Wiener measure and classical potential theory.
Dr. Daniel W. Stroock is the Simons Professor of Mathematics Emeritus at the
Massachusetts Institute of Technology. He has published many articles and is the
author of six books, most recently Partial Differential Equations for Probabilists
(2008).

Probability Theory
An Analytic View
Second Edition
Daniel W. Stroock
Massachusetts Institute of Technology
cambridgeuniversitypress
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Cambridge University Press
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Information on this title:www.cambridge.org/9780521132503
(cid:2)c DanielW.Stroock 1994,2011
This publication is in copyright.Subject to statutory exception
and to the provisions ofrelevant collective licensing agreements,
no reproduction ofany part may take place without the written
permission ofCambridge University Press.
First edition published 1994
First paperback edition 2000
Second edition published 2011
Printed in the United States ofAmerica
A catalog record for this publication is available from the British Library.
Library of Congress Cataloging in Publication data
Stroock,DanielW.
Probability theory :an analytic view/ DanielW.Stroock.– 2nd ed.
p. cm.
Includes bibliographicalreferences and index.
ISBN 978-0-521-76158-1 (hardback) – ISBN 978-0-521-13250-3 (pbk.)
1.Probabilities. I.Title.
QA 273.S763 2010
519.2–dc22 2010027652
ISBN 978-0-521-76158-1 Hardback
ISBN 978-0-521-13250-3 Paperback
Cambridge University Press has no responsibility for the persistence or accuracy ofURLs for
externalor third-party Internet Web sites referred to in this publication and does not guarantee
that any content on such Web sites is,orwillremain,accurate or appropriate.
This book is dedicated to my teachers:
M. Kac, H.P. McKean,Jr., and S.R.S. Varadhan

Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
Table of Dependence . . . . . . . . . . . . . . . . . . . . . . xxi
Chapter 1 Sums of Independent Random Variables . . . . . . . . 1
1.1 Independence . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1. Independent σ-Algebras . . . . . . . . . . . . . . . . . . . 1
1.1.2. Independent Functions . . . . . . . . . . . . . . . . . . . . 4
1.1.3. The Rademacher Functions . . . . . . . . . . . . . . . . . . 5
Exercises for §1.1 . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 The Weak Law of Large Numbers . . . . . . . . . . . . . . . 14
1.2.1. Orthogonal Random Variables . . . . . . . . . . . . . . . 14
1.2.2. Independent Random Variables . . . . . . . . . . . . . . . 15
1.2.3. Approximate Identities . . . . . . . . . . . . . . . . . . . 16
Exercises for §1.2 . . . . . . . . . . . . . . . . . . . . . . . . 20
1.3 Cram´er’s Theory of Large Deviations . . . . . . . . . . . . . . 22
Exercises for §1.3 . . . . . . . . . . . . . . . . . . . . . . . . 31
1.4 The Strong Law of Large Numbers . . . . . . . . . . . . . . . 35
Exercises for §1.4 . . . . . . . . . . . . . . . . . . . . . . . . 42
1.5 Law of the Iterated Logarithm . . . . . . . . . . . . . . . . 49
Exercises for §1.5 . . . . . . . . . . . . . . . . . . . . . . . . 56
Chapter 2 The Central Limit Theorem . . . . . . . . . . . . . 59
2.1 The Basic Central Limit Theorem . . . . . . . . . . . . . . . 60
2.1.1. Lindeberg’s Theorem . . . . . . . . . . . . . . . . . . . 60
2.1.2. The Central Limit Theorem . . . . . . . . . . . . . . . . 62
Exercises for §2.1 . . . . . . . . . . . . . . . . . . . . . . . . 65
2.2 The Berry–Esseen Theorem via Stein’s Method . . . . . . . . . 71
2.2.1. L1-Berry–Esseen . . . . . . . . . . . . . . . . . . . . . 72
2.2.2. The Classical Berry–Esseen Theorem . . . . . . . . . . . . 75
Exercises for §2.2 . . . . . . . . . . . . . . . . . . . . . . . . 81
2.3 Some Extensions of The Central Limit Theorem . . . . . . . . . 82
2.3.1. The Fourier Transform . . . . . . . . . . . . . . . . . . . 82
2.3.2. Multidimensional Central Limit Theorem . . . . . . . . . . . 84
2.3.3. Higher Moments . . . . . . . . . . . . . . . . . . . . . 87
Exercises for §2.3 . . . . . . . . . . . . . . . . . . . . . . . . 90
vii
viii Contents
2.4 An Application to Hermite Multipliers . . . . . . . . . . . . . 96
2.4.1. Hermite Multipliers . . . . . . . . . . . . . . . . . . . . 96
2.4.2. Beckner’s Theorem . . . . . . . . . . . . . . . . . . . . 101
2.4.3. Applications of Beckner’s Theorem . . . . . . . . . . . . . 105
Exercises for §2.4 . . . . . . . . . . . . . . . . . . . . . . . . 110
Chapter 3 Infinitely Divisible Laws . . . . . . . . . . . . . . . 115
3.1 Convergence of Measures on RN . . . . . . . . . . . . . . . . 115
3.1.1. Sequential Compactness in M (RN) . . . . . . . . . . . . . 116
1
3.1.2. L´evy’s Continuity Theorem . . . . . . . . . . . . . . . . . 117
Exercises for §3.1 . . . . . . . . . . . . . . . . . . . . . . . . 119
3.2 The L´evy–Khinchine Formula . . . . . . . . . . . . . . . . . 122
3.2.1. I(RN) Is the Closure of P(RN) . . . . . . . . . . . . . . . 123
3.2.2. The Formula . . . . . . . . . . . . . . . . . . . . . . . 126
Exercises for §3.2 . . . . . . . . . . . . . . . . . . . . . . . . 137
3.3 Stable Laws . . . . . . . . . . . . . . . . . . . . . . . . . 139
3.3.1. General Results . . . . . . . . . . . . . . . . . . . . . . 139
3.3.2. α-Stable Laws . . . . . . . . . . . . . . . . . . . . . . 141
Exercises for §3.3 . . . . . . . . . . . . . . . . . . . . . . . . 147
Chapter 4 L´evy Processes . . . . . . . . . . . . . . . . . . . 151
4.1 Stochastic Processes, Some Generalities . . . . . . . . . . . . . 152
4.1.1. The Space D(RN) . . . . . . . . . . . . . . . . . . . . . 153
4.1.2. Jump Functions . . . . . . . . . . . . . . . . . . . . . . 156
Exercises for §4.1 . . . . . . . . . . . . . . . . . . . . . . . . 159
4.2 Discontinuous L´evy Processes . . . . . . . . . . . . . . . . . 160
4.2.1. The Simple Poisson Process . . . . . . . . . . . . . . . . . 161
4.2.2. Compound Poisson Processes . . . . . . . . . . . . . . . . 163
4.2.3. Poisson Jump Processes . . . . . . . . . . . . . . . . . . 168
4.2.4. L´evy Processes with Bounded Variation . . . . . . . . . . . 170
4.2.5. General, Non-Gaussian L´evy Processes . . . . . . . . . . . . 171
Exercises for §4.2 . . . . . . . . . . . . . . . . . . . . . . . . 174
4.3 Brownian Motion, the Gaussian L´evy Process . . . . . . . . . . 177
4.3.1. Deconstructing Brownian Motion . . . . . . . . . . . . . . 178
4.3.2. L´evy’s Construction of Brownian Motion . . . . . . . . . . . 180
4.3.3. L´evy’s Construction in Context . . . . . . . . . . . . . . . 182
4.3.4. Brownian Paths Are Non-Differentiable . . . . . . . . . . . 183
4.3.5. General L´evy Processes . . . . . . . . . . . . . . . . . . 185
Exercises for §4.3 . . . . . . . . . . . . . . . . . . . . . . . . 187