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Advanced Probability Advanced Probability Alexander Sokol Anders Rønn-Nielsen Department of Mathematical Sciences University of Copenhagen Department of Mathematical Sciences University of Copenhagen Universitetsparken 5 DK-2100 Copenhagen Copyright 2013 Alexander Sokol & Anders Rønn-Nielsen ISBN 978-87-7078-999-8 Contents Preface v 1 Sequences of random variables 1 1.1 Measure-theoretic preliminaries . 1 1.2 Convergence of sequences of random variables . 3 1.3 Independence and Kolmogorov's zero-one law . 15 1.4 Convergence of sums of independent variables . 21 1.5 The strong law of large numbers . 24 1.6 Exercises . 30 2 Ergodicity and stationarity 35 2.1 Measure preservation, invariance and ergodicity . 35 2.2 Criteria for measure preservation and ergodicity . 40 2.3 Stationary processes and the law of large numbers . 44 2.4 Exercises . 55 3 Weak convergence 59 3.1 Weak convergence and convergence of measures . 60 3.2 Weak convergence and distribution functions . 67 3.3 Weak convergence and convergence in probability . 69 3.4 Weak convergence and characteristic functions . 72 3.5 Central limit theorems . 85 3.6 Asymptotic normality . 92 3.7 Higher dimensions . 95 3.8 Exercises . 98 4 Signed measures and conditioning 103 4.1 Decomposition of signed measures . 103 4.2 Conditional Expectations given a σ-algebra . 115 iv CONTENTS 4.3 Conditional expectations given a random variable . 124 4.4 Exercises . 128 5 Martingales 133 5.1 Introduction to martingale theory . 134 5.2 Martingales and stopping times . 137 5.3 The martingale convergence theorem . 145 5.4 Martingales and uniform integrability . 151 5.5 The martingale central limit theorem . 164 5.6 Exercises . 175 6 The Brownian motion 191 6.1 Definition and existence . 192 6.2 Continuity of the Brownian motion . 197 6.3 Variation and quadratic variation . 206 6.4 The law of the iterated logarithm . 215 6.5 Exercises . 223 7 Further reading 227 A Supplementary material 229 A.1 Limes superior and limes inferior . 229 A.2 Measure theory and real analysis . 233 A.3 Existence of sequences of random variables . 239 A.4 Exercises . 240 B Hints for exercises 241 B.1 Hints for Chapter 1 . 241 B.2 Hints for Chapter 2 . 244 B.3 Hints for Chapter 3 . 246 B.4 Hints for Chapter 4 . 248 B.5 Hints for Chapter 5 . 251 B.6 Hints for Chapter 6 . 256 B.7 Hints for Appendix A . 258 Bibliography 263 Preface The purpose of this monograph is to present a detailed introduction to selected fundamentals of modern probability theory. The focus is in particular on discrete-time and continuous-time processes, including the law of large numbers, Lindeberg's central limit theorem, martingales, the martingale convergence theorem and the martingale central limit theorem, as well as basic results on Brownian motion. The reader is assumed to have a reasonable grasp of basic analysis and measure theory, as can be obtained through Hansen (2009), Carothers (2000) or Ash (1972), for example. We have endeavoured throughout to present the material in a logical fashion, with detailed proofs allowing the reader to perceive not only the big picture of the theory, but also to understand the finer elements of the methods of proof used. Exercises are given at the end of each chapter, with hints for the exercises given in the appendix. The exercises form an important part of the monograph. We strongly recommend that any reader wishing to acquire a sound understanding of the theory spends considerable time solving the exercises. While we share the responsibility for the ultimate content of the monograph and in partic- ular any mistakes therein, much of the material is based on books and lecture notes from other individuals, in particular \Videreg˚aendesandsynlighedsregning" by Martin Jacobsen, the lecture notes on weak convergence by Søren Tolver Jensen, \Sandsynlighedsregning p˚a M˚alteoretiskGrundlag" by Ernst Hansen as well as supplementary notes by Ernst Hansen, in particular a note on the martingale central limit theorem. We are also indebted to Ketil Biering Tvermosegaard, who diligently translated the lecture notes by Martin Jacobsen and thus eased the migration of their contents to their present form in this monograph. We would like to express our gratitude to our own teachers, particularly Ernst Hansen, Martin Jacobsen and Søren Tolver Jensen, who taught us measure theory and probability theory. vi CONTENTS Also, many warm thanks go to Henrik Nygaard Jensen, who meticulously read large parts of the manuscript and gave many useful comments. Alexander Sokol Anders Rønn-Nielsen København, August 2012 Since the previous edition of the book, a number of misprints and errors have been cor- rected, and various other minor amendments have been made. We are grateful to the many students who have contributed to the monograph by identifying mistakes and suggesting improvements. Alexander Sokol Anders Rønn-Nielsen København, June 2013 Chapter 1 Sequences of random variables In this chapter, we will consider sequences of random variables and the basic results on such sequences, in particular the strong law of large numbers, which formalizes the intuitive notion that averages of independent and identically distributed events tend to the common mean. We begin in Section 1.1 by reviewing the measure-theoretic preliminaries for our later results. In Section 1.2, we discuss modes of convergence for sequences of random variables. The results given in this section are fundamental to much of the remainder of this monograph, as well as modern probability in general. In Section 1.3, we discuss the concept of independence for families of σ-algebras, and as an application, we prove the Kolmogorov zero-one law, which shows that for sequences of independent variables, events which, colloquially speaking, depend only on the tail of the sequence either have probability zero or one. In Section 1.4, we apply the results of the previous sections to prove criteria for the convergence of sums of independent variables. Finally, in Section 1.5, we prove the strong law of large numbers, arguably the most important result of this chapter. 1.1 Measure-theoretic preliminaries As noted in the introduction, we assume given a level of familiarity with basic real analysis and measure theory. Some of the main results assumed to be well-known in the following are reviewed in Appendix A. In this section, we give an independent review of some basic 2 Sequences of random variables results, and review particular notation related to probability theory. We begin by recalling some basic definitions. Let Ω be some set. A σ-algebra F on Ω is a set of subsets of Ω with the following three properties: Ω 2 F, if F 2 F then F c 2 F as 1 well, and if (Fn)n≥1 is a sequence of sets with Fn 2 F for n ≥ 1, then [n=1Fn 2 F as well. We refer to the second condition as F being stable under complements, and we refer to the third condition as F being stable under countable unions. From these stability properties, 1 it also follows that if (Fn)n≥1 is a sequence of sets in F, \n=1Fn 2 F as well. We refer to the pair (Ω; F) as a measurable space, and we refer to the elements of F as events. A probability measure P on (Ω; F) is a mapping P : F! [0; 1] such that P (;) = 0, P (Ω) = 1 P1 and whenever (Fn) is a sequence of disjoint sets in F, it holds that n=1 P (Fn) is convergent 1 P1 and P ([n=1Fn) = n=1 P (Fn). We refer to the latter property as the σ-additivity of the probability measure P . We refer to the triple (Ω; F;P ) as a probability space. Next, assume given a measure space (Ω; F), and let H be a set of subsets of Ω. We may then form the set A of all σ-algebras on Ω containing H, this is a subset of the power set of the power set of Ω. We may then define σ(H) = \F2AF, the intersection of all σ-algebras in A, that is, the intersection of all σ-algebras containing H. This is a σ-algebra as well, and it is the smallest σ-algebra on Ω containing H in the sense that for any σ-algebra G containing H, we have G 2 A and therefore σ(H) = \F2AF ⊆ G. We refer to σ(H) as the σ-algebra generated by H, and we say that H is a generating family for σ(H). Using this construction, we may define a particular σ-algebra on the Euclidean spaces: The d d Borel σ-algebra Bd on R for d ≥ 1 is the smallest σ-algebra containing all open sets in R . We denote the Borel σ-algebra on R by B. Next, let (Fn)n≥1 be a sequence of sets in F. If Fn ⊆ Fn+1 for all n ≥ 1, we say that (Fn)n≥1 is increasing. If Fn ⊇ Fn+1 for all n ≥ 1, we say that (Fn)n≥1 is decreasing. Assume that D is a set of subsets of Ω such that the following holds: Ω 2 D, if F; G 2 D with F ⊆ G 1 then G n F 2 D and if (Fn)n≥1 is an increasing sequence of sets in D then [n=1Fn 2 D. If this is the case, we say that D is a Dynkin class. Furthermore, if H is a set of subsets of Ω such that whenever F; G 2 H then F \ G 2 H, then we say that H is stable under finite intersections.
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