Mathematics of Probability (Graduate Studies in Mathematics) (Graduate Studies in Mathematics, 149)

This book covers the basics of modern probability theory. It begins with probability theory on finite and countable sample spaces and then passes from there to a concise course on measure theory, which is followed by some initial applications to probability theory, including independence and conditional expectations. The second half of the book deals with Gaussian random variables, with Markov chains, with a few continuous parameter processes, including Brownian motion, and, finally, with martingales, both discrete and continuous parameter ones. The book is a self-contained introduction to probability theory and the measure theory required to study it. Readership: Graduate students and researchers interested in probability. Preface Chapter 1. Some Background and Preliminaries §1.1. The Language of Probability Theory 1.1.1. Sample Spaces and Events 1.1.2. Probability Measures Exercises for §1.1 §1.2. Finite and Countable Sample Spaces 1.2.1. Probability Theory on a Countable Space 1.2.2. Uniform Probabilities and Coin Tossing 1.2.3. Tournaments 1.2.4. Symmetric Random Walk 1.2.5. De Moivre's Central Limit Theorem 1.2.6. Independent Events 1.2.7. The Arc Sine Law 1.2.8. Conditional Probability Exercises for §1.2 §1.3. Some Non-Uniform Probability Measures 1.3.1. Random Variables and Their Distributions 1.3.2. Biased Coins 1.3.3. Recurrence and Transience of Random Walks Exercises for §1.3 §1.4. Expectation Values 1.4.1. Some Elementary Examples 1.4.2. Independence and Moment Generating Functions 1.4.3. Basic Convergence Results Exercises for §1.4 Comments on Chapter 1 Chapter 2. Probability Theory on Uncountable Sample Spaces §2.1. A Little Measure Theory 2.1.1. Sigma Algebras, Measurable Functions, and Measures 2.1.2. ∏- and ∧-Systems Exercises for §2.1 §2.2. A Construction of P_p on {0, 1}^Z^+ 2.2.1. The Metric Space {0, 1}^Z^+ 2.2.2. The Construction Exercises for §2.2 §2.3. Other Probability Measures 2.3.1. The Uniform Probability Measure on [0, 1] 2.3.2. Lebesgue Measure on R 2.3.3. Distribution Functions and Probability Measures Exercises for §2.3 §2.4. Lebesgue Integration 2.4.1. Integration of Functions 2.4.2. Some Properties of the Lebesgue Integral 2.4.3. Basic Convergence Theorems 2.4.4. Inequalities 2.4.5. Fubini's Theorem Exercises for §2.4 §2.5. Lebesgue Measure on R^N 2.5.1. Polar Coordinates 2.5.2. Gaussian Computations and Stirling's Formula Exercises for §2.5 Comments on Chapter 2 Chapter 3. Some Applications to Probability Theory §3.1. Independence and Conditioning 3.1.1. Independent σ-Algebras 3.1.2. Independent Random Variables 3.1.3. Conditioning 3.1.4. Some Properties of Conditional Expectations Exercises for §3 .1 §3.2. Distributions that Admit a Density 3.2.1. Densities 3.2.2. Densities and Conditioning Exercises for §3.2 §3.3. Summing Independent Random Variables 3.3.1. Convolution of Distributions 3.3.2. Some Important Examples 3.3.3. Kolmogorov's Inequality and the Strong Law Exercises for §3.3 Comments on Chapter 3 Chapter 4. The Central Limit Theorem and Gaussian Distributions §4.1. The Central Limit Theorem 4.1.1. Lindeberg's Theorem Exercises for §4.1 §4.2. Families of Normal Random Variables 4.2.1. Multidimensional Gaussian Distributions 4.2.2. Standard Normal Random Variables 4.2.3. More General Normal Random Variables 4.2.4. A Concentration Property of Gaussian Distributions 4.2.5. Linear Transformations of Normal Random Variables 4.2.6. Gaussian Families Exercises for §4.2 Comments on Chapter 4 Chapter 5. Discrete Parameter Stochastic Processes §5.1. Random Walks Revisited 5.1.1. Immediate Rewards 5.1.2. Computations via Conditioning Exercises for §5.1 §5.2. Processes with the Markov Property 5.2.1. Sequences of Dependent Random Variables 5.2.2. Markov Chains 5.2.3. Long-Time Behavior 5.2.4. An Extension Exercises for §5.2 §5.3. Markov Chains on a Countable State Space 5.3.1. The Markov Property 5.3.2. Return Times and the Renewal Equation 5.3.3. A Little Ergodic Theory Exercises for §5.3 Comments on Chapter 5 Chapter 6. Some Continuous-Time Processes §6.1. Transition Probability Functions and Markov Processes 6.1.1. Transition Probability Functions Exercises for §6.1 §6.2. Markov Chains Run with a Poisson Clock 6.2.1. The Simple Poisson Process 6.2.2. A Generalization 6.2.3. Stationary Measures Exercises for §6.2 §6.3. Brownian Motion 6.3.1. Some Preliminaries 6.3.2. Levy's Construction 6.3.3. Some Elementary Properties of Brownian Motion 6.3.4. Path Properties 6.3.5. The Ornstein-Uhlenbeck Process Exercises for §6.3 Comments on Chapter 6 Chapter 7. Martingales §7.1. Discrete Parameter Martingales 7.1.1. Doob's Inequality Exercises for §7.1 §7.2. The Martingale Convergence Theorem 7.2.1. The Convergence Theorem 7.2.2. Application to the Radon-Nikodym Theorem Exercises for §7.2 §7.3. Stopping Times 7.3.1. Stopping Time Theorems 7.3.2. Reversed Martingales 7.3.3. Exchangeable Sequences Exercises for §7.3 §7.4. Continuous Parameter Martingales 7.4.1. Progressively Measurable Functions 7.4.2. Martingales and Submartingales 7.4.3. Stopping Times Again 7.4.4. Continuous Martingales and Brownian Motion 7.4.5. Brownian Motion and Differential Equations Exercises for §7.4 Comments on Chapter 7 Notation Bibliography Index