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Durrett概率论理论与实例.pdf

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    Half-title
    Series-title
    Title
    Copyright
    Contents
    Preface
    1 Measure Theory
    1.1 Probability Spaces
    Measures on Rd
    Exercises
    1.2 Distributions
    Exercises
    1.3 Random Variables
    Exercises
    1.4 Integration
    Exercises
    1.5 Properties of the Integral
    Exercises
    1.6 Expected Value
    1.6.1 Inequalities
    1.6.2 Integration to the Limit
    1.6.3 Computing Expected Values
    Exercises
    1.7 Product Measures, Fubini's Theorem
    Exercises
    2 Laws of Large Numbers
    2.1 Independence
    2.1.1 Sufficient Conditions for Independence
    2.1.2 Independence, Distribution, and Expectation
    2.1.3 Sums of Independent Random Variables
    2.1.4 Constructing Independent Random Variables
    Exercises
    2.2 Weak Laws of Large Numbers
    2.2.1 L2 Weak Laws
    2.2.2 Triangular Arrays
    2.2.3 Truncation
    Exercises
    2.3 Borel-Cantelli Lemmas
    Exercises
    2.4 Strong Law of Large Numbers
    Exercises
    2.5 Convergence of Random Series
    2.5.1 Rates of Convergence
    2.5.2 Infinite Mean
    Exercises
    2.6 Large Deviations
    3 Central Limit Theorems
    3.1 The De Moivre-Laplace Theorem
    Exercises
    3.2 Weak Convergence
    3.2.1 Examples
    3.2.2 Theory
    Exercises
    3.3 Characteristic Functions
    3.3.1 Definition, Inversion Formula
    3.3.2 Weak Convergence
    3.3.3 Moments and Derivatives
    3.3.4 Polya's Criterion
    3.3.5 The Moment Problem
    3.4 Central Limit Theorems
    3.4.1 i.i.d. Sequences
    Exercises
    3.4.2 Triangular Arrays
    Exercises
    3.4.3 Prime Divisors (Erdos-Kac)
    3.4.4 Rates of Convergence (Berry-Esseen)
    3.5 Local Limit Theorems
    3.6 Poisson Convergence
    3.6.1 The Basic Limit Theorem
    3.6.2 Two Examples with Dependence
    3.6.3 Poisson Processes
    3.7 Stable Laws
    Exercises
    3.8 Infinitely Divisible Distributions
    3.9 Limit Theorems in Rd
    4 Random Walks
    4.1 Stopping Times
    4.2 Recurrence
    4.3 Visits to 0, Arcsine Laws
    4.4 Renewal Theory
    5 Martingales
    5.1 Conditional Expectation
    5.1.1 Examples
    5.1.2 Properties
    5.1.3 Regular Conditional Probabilities
    5.2 Martingales, Almost Sure Convergence
    Exercises
    5.3 Examples
    5.3.1 Bounded Increments
    5.3.2 Polya's Urn Scheme
    5.3.3 Radon-Nikodym Derivatives
    5.3.4 Branching Processes
    5.4 Doob's Inequality, Convergence in Lp
    5.4.1 Square Integrable Martingales
    5.5 Uniform Integrability, Convergence in L1
    5.6 Backwards Martingales
    Exercises
    5.7 Optional Stopping Theorems
    Exercises
    6 Markov Chains
    6.1 Definitions
    6.2 Examples
    Exercises
    6.3 Extensions of the Markov Property
    Exercises
    6.4 Recurrence and Transience
    6.5 Stationary Measures
    6.6 Asymptotic Behavior
    Exercises
    6.7 Periodicity, Tail sigma -field
    6.8 General State Space
    6.8.1 Recurrence and Transience
    6.8.2 Stationary Measures
    6.8.3 Convergence Theorem
    6.8.4 GI/G/1 Queue
    7 Ergodic Theorems
    7.1 Definitions and Examples
    Exercises
    7.2 Birkhoff's Ergodic Theorem
    7.3 Recurrence
    7.4 A Subadditive Ergodic Theorem
    7.5 Applications
    8 Brownian Motion
    8.1 Definition and Construction
    8.2 Markov Property, Blumenthal's 0-1 Law
    8.3 Stopping Times, Strong Markov Property
    8.4 Path Properties
    8.4.1 Zeros of Brownian Motion
    8.4.2 Hitting Times
    8.4.3 Levy's Modulus of Continuity
    8.5 Martingales
    8.5.1 Multidimensional Brownian Motion
    8.6 Donsker's Theorem
    8.7 Empirical Distributions, Brownian Bridge
    8.8 Laws of the Iterated Logarithm
    Appendix A: Measure Theory Details
    A.1 Caratheodory's Extension Theorem
    A.2 Which Sets Are Measurable?
    A nonmeasurable subset of [0,1)
    Banach-Tarski theorem
    Solovay’s theorem
    A.3 Kolmogorov's Extension Theorem
    A.4 Radon-Nikodym Theorem
    A.5 Differentiating under the Integral
    References
    Index
    This page intentionally left blank
    Probability Theory and Examples Fourth Edition This book is an introduction to probability theory covering laws of large numbers, central limit theorems, random walks, martingales, Markov chains, ergodic theorems, and Brownian motion. It is a comprehensive treatment concentrating on the results that are the most useful for appli- cations. Its philosophy is that the best way to learn probability is to see it in action, so there are 200 examples and 450 problems. Rick Durrett received his Ph.D. in operations research from Stanford University in 1976. After nine years at UCLA and twenty-five at Cornell University, he moved to Duke University in 2010, where he is a professor of mathematics. He is the author of 8 books and more than 170 journal articles on a wide variety of topics, and he has supervised more than 40 Ph.D. students. He is a member of the National Academy of Science and the American Academy of Arts and Sciences and a Fellow of the Institute of Mathematical Statistics.
    CAMBRIDGE SERIES IN STATISTICAL AND PROBABILISTIC MATHEMATICS Editorial Board: Z. Ghahramani, Department of Engineering, University of Cambridge R. Gill, Department of Mathematics, Utrecht University F. Kelly, Statistics Laboratory, University of Cambridge B. D. Ripley, Department of Statistics, University of Oxford S. Ross, Department of Industrial & Systems Engineering, University of Southern California M. Stein, Department of Statistics, University of Chicago This series of high-quality upper-division textbooks and expository monographs covers all aspects of stochastic applicable mathematics. The topics range from pure and applied statistics to probability theory, operations research, optimization, and mathematical pro- gramming. The books contain clear presentations of new developments in the field and also of the state of the art in classical methods. While emphasizing rigorous treatment of theoretical methods, the books also contain applications and discussions of new techniques made possible by advances in computational practice. Already Published 1. Bootstrap Methods and Their Application, by A. C. Davison and D. V. Hinkley 2. Markov Chains, by J. Norris 3. Asymptotic Statistics, by A. W. van der Vaart 4. Wavelet Methods for Time Series Analysis, by Donald B. Percival and Andrew T. Walden 5. Bayesian Methods, by Thomas Leonard and John S. J. Hsu 6. Empirical Processes in M-Estimation, by Sara van de Geer 7. Numerical Methods of Statistics, by John F. Monahan 8. A User’s Guide to Measure Theoretic Probability, by David Pollard 9. The Estimation and Tracking of Frequency, by B. G. Quinn and E. J. Hannan 10. Data Analysis and Graphics Using R, by John Maindonald and John Braun 11. Statistical Models, by A. C. Davison 12. Semiparametric Regression, by D. Ruppert, M. P. Wand, and R. J. Carroll 13. Exercise in Probability, by Loic Chaumont and Marc Yor 14. Statistical Analysis of Stochastic Processes in Time, by J. K. Lindsey 15. Measure Theory and Filtering, by Lakhdar Aggoun and Robert Elliott 16. Essentials of Statistical Inference, by G. A. Young and R. L. Smith 17. Elements of Distribution Theory, by Thomas A. Severini 18. Statistical Mechanics of Disordered Systems, by Anton Bovier 19. The Coordinate-Free Approach to Linear Models, by Michael J. Wichura 20. Random Graph Dynamics, by Rick Durrett 21. Networks, by Peter Whittle 22. Saddlepoint Approximations with Applications, by Ronald W. Butler 23. Applied Asymptotics, by A. R. Brazzale, A. C. Davison, and N. Reid 24. Random Networks for Communication, by Massimo Franceschetti and Ronald Meester 25. Design of Comparative Experiments, by R. A. Bailey 26. Symmetry Studies, by Marlos A. G. Viana 27. Model Selection and Model Averaging, by Gerda Claeskens and Nils Lid Hjort 28. Bayesian Nonparametrics, by Nils Lid Hjort, Peter M¨uller, and Stephen G. Walker 29. From Finite Sample to Asymptotic Methods in Statistics, by Pranab K. Sen, Julio M. Singer, and Antonio C. Pedroso de Lima 30. Brownian Motion, by Peter M¨orters and Yuval Peres
    Probability Theory and Examples Fourth Edition RICK DURRETT Department of Mathematics, Duke University
    cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, S˜ao Paulo, Delhi, Dubai, Tokyo, Mexico City Cambridge University Press 32 Avenue of the Americas, New York, NY 10013-2473, USA www.cambridge.org Information on this title: www.cambridge.org/9780521765398 C Rick Durrett 1991, 1995, 2004, 2010 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First edition published 1991 by Wadsworth Publishing Second edition published 1995 by Duxbury Press Third edition published 2004 by Duxbury Press Fourth edition published 2010 by Cambridge University Press Printed in the United States of America A catalog record for this publication is available from the British Library. Library of Congress Cataloging in Publication data Durrett, Richard, 1951– Probability : theory and examples / Rick Durrett. – 4th ed. p. cm. Includes bibliographical references and index. ISBN 978-0-521-76539-8 (hardback) 1. Probabilities. QA273.D865 519.2–dc22 2010013387 I. Title. 2010 ISBN 978-0-521-76539-8 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet Web sites referred to in this publication and does not guarantee that any content on such Web sites is, or will remain, accurate or appropriate.
    Contents Preface 1 Measure Theory 1.1 Probability Spaces 1.2 Distributions 1.3 Random Variables 1.4 Integration 1.5 Properties of the Integral 1.6 Expected Value 1.6.1 Inequalities 1.6.2 Integration to the Limit 1.6.3 Computing Expected Values 1.7 Product Measures, Fubini’s Theorem 2 Laws of Large Numbers 2.1 Independence 2.1.1 Sufficient Conditions for Independence 2.1.2 Independence, Distribution, and Expectation 2.1.3 Sums of Independent Random Variables 2.1.4 Constructing Independent Random Variables 2.2 Weak Laws of Large Numbers 2.2.1 L2 Weak Laws 2.2.2 Triangular Arrays 2.2.3 Truncation 2.3 Borel-Cantelli Lemmas 2.4 Strong Law of Large Numbers 2.5 Convergence of Random Series* 2.5.1 Rates of Convergence 2.5.2 Infinite Mean 2.6 Large Deviations* 3 Central Limit Theorems 3.1 The De Moivre-Laplace Theorem 3.2 Weak Convergence 3.2.1 Examples 3.2.2 Theory v page ix 1 1 9 14 17 23 27 27 29 30 36 41 41 43 45 47 50 53 53 56 59 64 73 78 82 84 86 94 94 97 97 100
    vi Contents 3.3 Characteristic Functions 3.3.1 Definition, Inversion Formula 3.3.2 Weak Convergence 3.3.3 Moments and Derivatives 3.3.4 Polya’s Criterion* 3.3.5 The Moment Problem* 3.4 Central Limit Theorems 3.4.1 i.i.d. Sequences 3.4.2 Triangular Arrays 3.4.3 Prime Divisors (Erd¨os-Kac)* 3.4.4 Rates of Convergence (Berry-Esseen)* 3.5 Local Limit Theorems* 3.6 Poisson Convergence 3.6.1 The Basic Limit Theorem 3.6.2 Two Examples with Dependence 3.6.3 Poisson Processes 3.7 Stable Laws* 3.8 Infinitely Divisible Distributions* 3.9 Limit Theorems in Rd 4 Random Walks 4.1 Stopping Times 4.2 Recurrence 4.3 Visits to 0, Arcsine Laws* 4.4 Renewal Theory* 5 Martingales 5.1 Conditional Expectation 5.1.1 Examples 5.1.2 Properties 5.1.3 Regular Conditional Probabilities* 5.2 Martingales, Almost Sure Convergence 5.3 Examples 5.3.1 Bounded Increments 5.3.2 Polya’s Urn Scheme 5.3.3 Radon-Nikodym Derivatives 5.3.4 Branching Processes 5.4 Doob’s Inequality, Convergence in Lp 5.4.1 Square Integrable Martingales* 5.5 Uniform Integrability, Convergence in L1 5.6 Backwards Martingales 5.7 Optional Stopping Theorems 6 Markov Chains 6.1 Definitions 6.2 Examples 6.3 Extensions of the Markov Property 6.4 Recurrence and Transience 6.5 Stationary Measures 6.6 Asymptotic Behavior 106 106 112 114 118 120 124 124 129 133 137 141 146 146 151 154 158 169 172 179 179 189 201 208 221 221 223 226 230 232 239 239 241 242 245 249 254 258 264 269 274 274 277 282 288 296 307
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