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Preface 1 Sets and Events 1.1 Introduction 1.2 Basic Set Theory 1.2.1 Indicator functions 1.3 Limits of Sets 1.4 Monotone Sequences 1.5 Set Operations and Closure 1.5.1 Examples 1.6 The ¦Ò-field Generated by a Given Class C 1.7 Borel Sets on the Real Line 1.8 Comparing Borel Sets 1.9 Exercises 2 Probability Spaces 2.1 Basic Definitions and Properties 2.2 More on Closure 2.2.1 Dynkin's theorem 2.2.2 Proof of Dynkin's theorem 2.3 Two Constructions 2.4 Constructions of Probability Spaces 2.4.1 General Construction of a Probability Model 2.4.2 Proof of the Second Extension Theorem 2.5 Measure Constructions 2.5.1 Lebesgue Measure on (0,1] 2.5.2 Construction of a Probability Measure on R with Given Distribution Function F(x) 2.6 Exercises 3 Random Variables,Elements,and Measurable Maps 3.1 Inverse Maps 3.2 Measurable Maps,Random Elements, Induced Probability Measures 3.2.1 Composition 3.2.2 Random Elements of Metric Spaces 3.2.3 Measurability and Continuity 3.2.4 Measurability and Limits 3.3 ¦Ò-Fields Generated by Maps 3.4 Exercises 4 Independence 4.1 Basic Definitions 4.2 Independent Random Variables 4.3 Two Examples of Independence 4.3.1 Records,Ranks,Renyi Theorem 4.3.2 Dyadic Expansions of Uniform Random Numbers 4.4 More on Independence:Groupings 4.5 Independence,Zero-One Laws,Borel-Cantelli Lemma 4.5.1 Borel-Cantelli Lemma 4.5.2 Borel Zero-One Law 4.5.3 Kolmogorov Zero-One Law 4.6 Exercises 5 Integration and Expectation 5.1 Preparation for Integration 5.1.1 Simple Functions 5.1.2 Measurability and Simple Functions 5.2 Expectation and Integration 5.2.1 Expectation of Simple Functions 5.2.2 Extension of the Definition 5.2.3 Basic Properties of Expectation 5.3 Limits and Integrals 5.4 Indefinite Integrals 5.5 The Transformation Theorem and Densities 5.5.1 Expectation is Always an Integral on R 5.5.2 Densities 5.6 The Riemann vs Lebesgue Integral 5.7 Product Spaces 5.8 Probability Measures on Product Spaces 5.9 Fubini's theorem 5.10 Exercises 6 Convergence Concepts 6.1 Almost Sure Convergence 6.2 Convergence in Probability 6.2.1 Statistical Terminology 6.3 Connections Between a.s.and i.p.Convergence 6.4 Quantile Estimation 6.5 Lp Convergence 6.5.1 Uniform Integrability 6.5.2 Interlude:A Review of Inequalities 6.6 More on Lp Convergence 6.7 Exercises 7 Laws of Large Numbers and Sums of Independent Random Variables 7.1 Truncation and Equivalence 7.2 A General Weak Law of Large Numbers 7.3 Almost Sure Convergence of Sums of Independent Random Variables 7.4 Strong Laws of Large Numbers 7.4.1 Two Examples 7.5 The Strong Law of Large Numbers for IID Sequences 7.5.1 Two Applications of the SLLN 7.6 The Kolmogorov Three Series Theorem 7.6.1 Necessity of the Kolmogorov Three Series Theorem 7.7 Exercises 8 Convergence in Distribution 8.1 Basic Definitions 8.2 Scheffe's lemma 8.2.1 Scheffe's lemma and Order Statistics 8.3 The Baby Skorohod Theorem 8.3.1 The Delta Method 8.4 Weak Convergence Equivalences; Portmanteau Theorem 8.5 More Relations Among Modes of Convergence 8.6 New Convergences from Old 8.6.1 Example:The Central Limit Theorem for m-Dependent Random Variables 8.7 The Convergence to Types Theorem 8.7.1 Application of Convergence to Types: Limit Distributions for Extremes 8.8 Exercises 9 Characteristic Functions and the Central Limit Theorem 9.1 Review of Moment Generating Functions and the Central Limit Theorem 9.2 Characteristic Functions:Definition and First Properties 9.3 Expansions 9.3.1 Expansion of eix 9.4 Moments and Derivatives 9.5 Two Big Theorems: Uniqueness and Continuity 9.6 The Selection Theorem,Tightness,and Prohorov's theorem 9.6.1 The Selection Theorem 9.6.2 Tightness,Relative Compactness, and Prohorov's theorem 9.6.3 Proof of the Continuity Theorem 9.7 The Classical CLT for iid Random Variables 9.8 The Lindeberg-Feller CLT 9.9 Exercises 10 Martingales 10.1 Prelude to Conditional Expectation: The Radon-Nikodym Theorem 10.2 Definition of Conditional Expectation 10.3 Properties of Conditional Expectation 10.4 Martingales 10.5 Examples of Martingales 10.6 Connections between Martingales and Submartingales 10.6.1 Doob's Decomposition 10.7 Stopping Times 10.8 Positive Super Martingales 10.8.1 Operations on Supermartingales 10.8.2 Upcrossings 10.8.3 Boundedness Properties 10.8.4 Convergence of Positive Super Martingales 10.8.5 Closure 10.8.6 Stopping Supermartingales 10.9 Examples 10.9.1 Gambler's Ruin 10.9.2 Branching Processes 10.9.3 Some Differentiation Theory 10.10 Martingale and Submartingale Convergence 10.10.1 Krickeberg Decomposition 10.10.2 Doob's (Sub) martingale Convergence Theorem 10.11 Regularity and Closure 10.12 Regularity and Stopping 10.13 Stopping Theorems 10.14 Wald's Identity and Random Walks 10.14.1 The Basic Martingales 10.14.2 Regular Stopping Times 10.14.3 Examples of Integrable Stopping Times 10.14.4 The Simple Random Walk 10.15 Reversed Martingales 10.16 Fundamental Theorems of Mathematical Finance 10.16.1 A Simple Market Model 10.16.2 Admissible Strategies and Arbitrage 10.16.3 Arbitrage and Martingales 10.16.4 Complete Markets 10.16.5 Option Pricing 10.17 Exercises References Index