Probability: Theory and Examples

Description

This lively introduction to measure-theoretic probability theory covers laws of large numbers, central limit theorems, random walks, martingales, Markov chains, ergodic theorems, and Brownian motion. Concentrating on results that are the most useful for applications, this comprehensive treatment is a rigorous graduate text and reference. Operating under the philosophy that the best way to learn probability is to see it in action, the book contains extended examples that apply the theory to concrete applications. This fifth edition contains a new chapter on multidimensional Brownian motion and its relationship to partial differential equations (PDEs), an advanced topic that is finding new applications. Setting the foundation for this expansion, Chapter 7 now features a proof of Itô's formula. Key exercises that previously were simply proofs left to the reader have been directly inserted into the text as lemmas. The new edition re-instates discussion about the central limit theorem for martingales and stationary sequences. 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 Renewal Theory* 2.7 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 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 3.7 Poisson Processes 3.7.1 Compound Poisson Processes 3.7.2 Thinning 3.7.3 Conditioning 3.8 Stable Laws* 3.9 Infinitely Divisible Distributions* 3.10 Limit Theorems in Rd 4 Martingales 4.1 Conditional Expectation 4.1.1 Examples 4.1.2 Properties 4.1.3 Regular Conditional Probabilities* 4.2 Martingales, Almost Sure Convergence 4.3 Examples 4.3.1 Bounded Increments 4.3.2 Polya’s Urn Scheme 4.3.3 Radon-Nikodym Derivatives 4.3.4 Branching Processes 4.4 Doob’s Inequality, Convergence in Lp, p > 1 4.5 Square Integrable Martingales* 4.6 Uniform Integrability, Convergence in L1 4.7 Backwards Martingales 4.8 Optional Stopping Theorems 4.8.1 Applications to Random Walks 4.9 Combinatorics of Simple Random Walk* 5 Markov Chains 5.1 Examples 5.2 Construction, Markov Properties 5.3 Recurrence and Transience 5.4 Recurrence of Random Walks Stararred Section 5.5 Stationary Measures 5.6 Asymptotic Behavior 5.7 Periodicity, Tail σ-Field* 5.8 General State Space* 5.8.1 Recurrence and Transience 5.8.2 Stationary Measures 5.8.3 Convergence Theorem 5.8.4 GI/G/1 Queue 6 Ergodic Theorems 6.1 Definitions and Examples 6.2 Birkhoff’s Ergodic Theorem 6.3 Recurrence 6.4 A Subadditive Ergodic Theorem 6.5 Applications 7 Brownian Motion 7.1 Definition and Construction 7.2 Markov Property, Blumenthal’s 0-1 Law 7.3 Stopping Times, Strong Markov Property 7.4 Path Properties 7.4.1 Zeros of Brownian Motion 7.4.2 Hitting Times 7.5 Martingales 7.6 Ito’s Formula* 8 Applications to Random Walk 8.1 Donsker’s Theorem 8.2 CLTs for Martingales 8.3 CLTs for Stationary Sequences 8.3.1 Mixing Properties 8.4 Empirical Distributions, Brownian Bridge 8.5 Laws of the Iterated Logarithm 9 Multidimensional Brownian Motion 9.1 Martingales 9.2 Heat Equation 9.3 Inhomogeneous Heat Equation 9.4 Feynman-Kac Formula 9.5 Dirichlet Problem 9.5.1 Exit Distributions 9.6 Green’s Functions and Potential Kernels 9.7 Poisson’s Equation 9.7.1 Occupation Times 9.8 Schrodinger Equation Appendix A Measure Theory Details A.1 Carath ´ eodory’s Extension Theorem A.2 Which Sets Are Measurable? A.3 Kolmogorov’s Extension Theorem A.4 Radon-Nikodym Theorem A.5 Differentiating under the Integral References Index