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    Probability Theory
    Preface to the Second Edition
    Preface to the First Edition
    Contents
    Chapter 1: Basic Measure Theory
    1.1 Classes of Sets
    1.2 Set Functions
    1.3 The Measure Extension Theorem
    1.4 Measurable Maps
    1.5 Random Variables
    Chapter 2: Independence
    2.1 Independence of Events
    2.2 Independent Random Variables
    2.3 Kolmogorov's 0-1 Law
    2.4 Example: Percolation
    Uniqueness of the Infinite Open Cluster*
    Chapter 3: Generating Functions
    3.1 Definition and Examples
    3.2 Poisson Approximation
    3.3 Branching Processes
    Chapter 4: The Integral
    4.1 Construction and Simple Properties
    4.2 Monotone Convergence and Fatou's Lemma
    4.3 Lebesgue Integral Versus Riemann Integral
    Chapter 5: Moments and Laws of Large Numbers
    5.1 Moments
    5.2 Weak Law of Large Numbers
    5.3 Strong Law of Large Numbers
    Entropy and Source Coding Theorem*
    5.4 Speed of Convergence in the Strong LLN
    5.5 The Poisson Process
    Chapter 6: Convergence Theorems
    6.1 Almost Sure and Measure Convergence
    6.2 Uniform Integrability
    6.3 Exchanging Integral and Differentiation
    Chapter 7: Lp-Spaces and the Radon-Nikodym Theorem
    7.1 Definitions
    7.2 Inequalities and the Fischer-Riesz Theorem
    7.3 Hilbert Spaces
    7.4 Lebesgue's Decomposition Theorem
    7.5 Supplement: Signed Measures
    7.6 Supplement: Dual Spaces
    Chapter 8: Conditional Expectations
    8.1 Elementary Conditional Probabilities
    8.2 Conditional Expectations
    8.3 Regular Conditional Distribution
    Chapter 9: Martingales
    9.1 Processes, Filtrations, Stopping Times
    9.2 Martingales
    9.3 Discrete Stochastic Integral
    9.4 Discrete Martingale Representation Theorem and the CRR Model
    Chapter 10: Optional Sampling Theorems
    10.1 Doob Decomposition and Square Variation
    10.2 Optional Sampling and Optional Stopping
    10.3 Uniform Integrability and Optional Sampling
    Chapter 11: Martingale Convergence Theorems and Their Applications
    11.1 Doob's Inequality
    11.2 Martingale Convergence Theorems
    11.3 Example: Branching Process
    Chapter 12: Backwards Martingales and Exchangeability
    12.1 Exchangeable Families of Random Variables
    Heuristic for the Structure of Exchangeable Families
    12.2 Backwards Martingales
    12.3 De Finetti's Theorem
    Chapter 13: Convergence of Measures
    13.1 A Topology Primer
    13.2 Weak and Vague Convergence
    13.3 Prohorov's Theorem
    13.4 Application: A Fresh Look at de Finetti's Theorem
    Chapter 14: Probability Measures on Product Spaces
    14.1 Product Spaces
    14.2 Finite Products and Transition Kernels
    14.3 Kolmogorov's Extension Theorem
    14.4 Markov Semigroups
    Chapter 15: Characteristic Functions and the Central Limit Theorem
    15.1 Separating Classes of Functions
    15.2 Characteristic Functions: Examples
    15.3 Lévy's Continuity Theorem
    15.4 Characteristic Functions and Moments
    15.5 The Central Limit Theorem
    15.6 Multidimensional Central Limit Theorem
    Chapter 16: Infinitely Divisible Distributions
    16.1 Lévy-Khinchin Formula
    16.2 Stable Distributions
    Symmetric Stable Distributions
    General Stable Distributions
    Convergence to Stable Distributions
    Chapter 17: Markov Chains
    17.1 Definitions and Construction
    17.2 Discrete Markov Chains: Examples
    17.3 Discrete Markov Processes in Continuous Time
    17.4 Discrete Markov Chains: Recurrence and Transience
    17.5 Application: Recurrence and Transience of Random Walks
    17.6 Invariant Distributions
    17.7 Stochastic Ordering and Coupling
    Chapter 18: Convergence of Markov Chains
    18.1 Periodicity of Markov Chains
    18.2 Coupling and Convergence Theorem
    18.3 Markov Chain Monte Carlo Method
    Metropolis Algorithm
    Gibbs Sampler
    Perfect Sampling
    18.4 Speed of Convergence
    Chapter 19: Markov Chains and Electrical Networks
    19.1 Harmonic Functions
    19.2 Reversible Markov Chains
    19.3 Finite Electrical Networks
    19.4 Recurrence and Transience
    19.5 Network Reduction
    Reduced Network
    Step-by-Step Reduction of the Network
    Application to Example 19.32
    Alternative Solution
    19.6 Random Walk in a Random Environment
    Chapter 20: Ergodic Theory
    20.1 Definitions
    20.2 Ergodic Theorems
    20.3 Examples
    20.4 Application: Recurrence of Random Walks
    20.5 Mixing
    20.6 Entropy
    Chapter 21: Brownian Motion
    21.1 Continuous Versions
    21.2 Construction and Path Properties
    21.3 Strong Markov Property
    21.4 Supplement: Feller Processes
    21.5 Construction via L2-Approximation
    Lévy Construction of Brownian Motion
    Brownian Motion and White Noise
    21.6 The Space C([0,infty))
    21.7 Convergence of Probability Measures on C([0,infty))
    21.8 Donsker's Theorem
    21.9 Pathwise Convergence of Branching Processes*
    21.10 Square Variation and Local Martingales
    Chapter 22: Law of the Iterated Logarithm
    22.1 Iterated Logarithm for the Brownian Motion
    22.2 Skorohod's Embedding Theorem
    Supplement: Proof of Remark 22.6
    22.3 Hartman-Wintner Theorem
    Chapter 23: Large Deviations
    23.1 Cramér's Theorem
    23.2 Large Deviations Principle
    23.3 Sanov's Theorem
    23.4 Varadhan's Lemma and Free Energy
    Chapter 24: The Poisson Point Process
    24.1 Random Measures
    24.2 Properties of the Poisson Point Process
    24.3 The Poisson-Dirichlet Distribution*
    The Chinese Restaurant Process
    Chapter 25: The Itô Integral
    25.1 Itô Integral with Respect to Brownian Motion
    25.2 Itô Integral with Respect to Diffusions
    25.3 The Itô Formula
    25.4 Dirichlet Problem and Brownian Motion
    25.5 Recurrence and Transience of Brownian Motion
    Chapter 26: Stochastic Differential Equations
    26.1 Strong Solutions
    26.2 Weak Solutions and the Martingale Problem
    26.3 Weak Uniqueness via Duality
    Notation Index
    References
    Name Index
    Subject Index
    Universitext Achim Klenke Probability Theory A Comprehensive Course Second Edition
    Universitext
    Universitext Series Editors: Sheldon Axler San Francisco State University, San Francisco, CA, USA Vincenzo Capasso Università degli Studi di Milano, Milan, Italy Carles Casacuberta Universitat de Barcelona, Barcelona, Spain Angus MacIntyre Queen Mary, University of London, London, UK Kenneth Ribet University of California, Berkeley, Berkeley, CA, USA Claude Sabbah CNRS, École Polytechnique, Palaiseau, France Endre Süli University of Oxford, Oxford, UK Wojbor A. Woyczynski Case Western Reserve University, Cleveland, OH, USA Universitext is a series of textbooks that presents material from a wide variety of mathematical disciplines at master’s level and beyond. The books, often well class-tested by their author, may have an informal, personal, even experimental approach to their subject matter. Some of the most successful and established books in the series have evolved through several editions, always following the evolution of teaching curricula, into very polished texts. Thus as research topics trickle down into graduate-level teaching, first textbooks written for new, cutting-edge courses may make their way into Universitext. For further volumes: www.springer.com/series/223
    Achim Klenke Probability Theory A Comprehensive Course Second Edition
    Achim Klenke Institut für Mathematik Johannes Gutenberg-Universität Mainz Mainz, Germany Translation from the German language edition: ‘Wahrscheinlichkeitstheorie’ by Achim Klenke Copyright © 2013 Springer Berlin Heidelberg Springer Berlin Heidelberg is a part of Springer Science+Business Media All Rights Reserved ISSN 0172-5939 Universitext ISBN 978-1-4471-5360-3 DOI 10.1007/978-1-4471-5361-0 Springer London Heidelberg New York Dordrecht ISSN 2191-6675 (electronic) ISBN 978-1-4471-5361-0 (eBook) Library of Congress Control Number: 2013945095 Mathematics Subject Classification: 60-01, 28-01 © Springer-Verlag London 2008, 2014 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of pub- lication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
    Preface to the Second Edition In the second edition of this book many errors have been corrected. Furthermore, the text has been extended carefully in many places. In particular, there are more exercises and a lot more illustrations. I would like to take the opportunity to thank all of those who helped improv- ing the first edition of this book, in particular: Michael Diether, Maren Eckhoff, Christopher Grant, Matthias Hammer, Heiko Hoffmann, Martin Hutzenthaler, Mar- tin Kolb, Manuel Mergens, Thal Nowik, Felix Schneider, Wolfgang Schwarz and Stephan Tolksdorf. A constantly updated list of errors can be found at www.aklenke.de. Mainz March 2013 Achim Klenke v
    Preface to the First Edition This book is based on two four-hour courses on advanced probability theory that I have held in recent years at the universities of Cologne and Mainz. It is implicitly as- sumed that the reader has a certain familiarity with the basic concepts of probability theory, although the formal framework will be fully developed in this book. The aim of this book is to present the central objects and concepts of proba- bility theory: random variables, independence, laws of large numbers and central limit theorems, martingales, exchangeability and infinite divisibility, Markov chains and Markov processes, as well as their connection with discrete potential theory, coupling, ergodic theory, Brownian motion and the Itô integral (including stochas- tic differential equations), the Poisson point process, percolation and the theory of large deviations. Measure theory and integration are necessary prerequisites for a systematic prob- ability theory. We develop it only to the point to which it is needed for our purposes: construction of measures and integrals, the Radon–Nikodym theorem and regular conditional distributions, convergence theorems for functions (Lebesgue) and mea- sures (Prohorov) and construction of measures in product spaces. The chapters on measure theory do not come as a block at the beginning (although they are written such that this would be possible; that is, independent of the probabilistic chapters) but are rather interlaced with probabilistic chapters that are designed to display the power of the abstract concepts in the more intuitive world of probability theory. For example, we study percolation theory at the point where we barely have measures, random variables and independence; not even the integral is needed. As the only exception, the systematic construction of independent random variables is deferred to Chapter 14. Although it is rather a matter of taste, I hope that this setup helps to motivate the reader throughout the measure-theoretical chapters. Those readers with a solid measure-theoretical education can skip in particular the first and fourth chapters and might wish only to look up this or that. In the first eight chapters, we lay the foundations that will be needed in all subse- quent chapters. After that, there are seven more or less independent parts, consisting of Chaps. 9–12, 13, 14, 15–16, 17–19, 20 and 23. The chapter on Brownian motion vii
    viii Preface to the First Edition (21) makes reference to Chaps. 9–15. Again, after that, the three blocks consisting of Chaps. 22, 24 and 25–26 can be read independently. I should like to thank all those who read the manuscript and the German original version of this book and gave numerous hints for improvements: Roland Alkemper, René Billing, Dirk Brüggemann, Anne Eisenbürger, Patrick Jahn, Arnulf Jentzen, Ortwin Lorenz, L. Mayer, Mario Oeler, Marcus Schölpen, my colleagues Ehrhard Behrends, Wolfgang Bühler, Nina Gantert, Rudolf Grübel, Wolfgang König, Pe- ter Mörters and Ralph Neininger, and in particular my colleague from Munich Hans-Otto Georgii. Dr John Preater did a great job language editing the English manuscript and also pointing out numerous mathematical flaws. I am especially indebted to my wife Katrin for proofreading the English manuscript and for her patience and support. I would be grateful for further suggestions, errors etc. to be sent by e-mail to math@aklenke.de Mainz October 2007 Achim Klenke
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