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 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 Exercises 1.7 Product Measures, Fubini’s Theorem Exercises 2 Laws of Large Numbers 2.1 Independence Exercises 2.2 Weak Laws of Large Numbers Exercises 2.3 Borel-Cantelli Lemmas Exercises 2.4 Strong Law of Large Numbers Exercises 2.5 Convergence of Random Series* Exercises 2.6 Renewal Theory* Exercises 2.7 Large Deviations* Exercises 3 Central Limit Theorems 3.1 The De Moivre-Laplace Theorem Exercises 3.2 Weak Convergence Exercises 3.3 Characteristic Functions Exercises Exercises Exercises 3.4 Central Limit Theorems Exercises Exercises 3.5 Local Limit Theorems* 3.6 Poisson Convergence 3.7 Poisson Processes 3.8 Stable Laws* Exercises 3.9 Infinitely Divisible Distributions* Exercises 3.10 Limit Theorems in Rd Exercises 4 Martingales 4.1 Conditional Expectation Exercises 4.2 Martingales, Almost Sure Convergence Exercises 4.3 Examples Exercises 4.4 Doob’s Inequality, Convergence in Exercises 4.5 Square Integrable Martingales* 4.6 Uniform Integrability, Convergence in Exercises 4.7 Backwards Martingales Exercises 4.8 Optional Stopping Theorems Exercises 4.9 Combinatorics of Simple Random Walk* Exercises 5 Markov Chains 5.1 Examples Exercises 5.2 Construction, Markov Properties Exercises 5.3 Recurrence and Transience Exercises 5.4 Recurrence of Random Walks Stararred Section Exercises 5.5 Stationary Measures Exercises 5.6 Asymptotic Behavior Exercises 5.7 Periodicity, Tail Field* 5.8 General State Space* Exercises 6 Ergodic Theorems 6.1 Definitions and Examples Exercises 6.2 Birkhoff’s Ergodic Theorem Exercises 6.3 Recurrence Exercises 6.4 A Subadditive Ergodic Theorem 6.5 Applications Exercises 7 Brownian Motion 7.1 Definition and Construction Exercises 7.2 Markov Property, Blumenthal’s 0-1 Law Exercises 7.3 Stopping Times, Strong Markov Property Exercises 7.4 Path Properties Exercises 7.5 Martingales Exercises 7.6 Itˆo’s Formula* Exercises 8 Applications to RandomWalk 8.1 Donsker’s Theorem Exercises 8.2 CLTs for Martingales 8.3 CLTs for Stationary Sequences 8.4 Empirical Distributions, Brownian Bridge Exercises 8.5 Laws of the Iterated Logarithm Exercises 9 Multidimensional Brownian Motion 9.1 Martingales Exercises 9.2 Heat Equation 9.3 Inhomogeneous Heat Equation 9.4 Feynman-Kac Formula 9.5 Dirichlet Problem Exercises 9.6 Green’s Functions and Potential Kernels 9.7 Poisson’s Equation 9.8 Schr¨odinger 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