Probability Theory: A Comprehensive Course

Description

Probabilistic concepts play an increasingly important role in mathematics, physics, biology, financial engineering and computer science. They help us to understand magnetism, amorphous media, genetic diversity and the perils of random developments on the financial markets, and they guide us in constructing more efficient algorithms.This text is a comprehensive course in modern probability theory and its measure-theoretical foundations. Aimed primarily at graduate students and researchers, the book covers a wide variety of topics, many of which are not usually found in introductory textbooks, such as: * limit theorems for sums of random variables; * martingales; * percolation; * Markov chains and electrical networks; * construction of stochastic processes; * Poisson point processes and infinite divisibility; * large deviation principles and statistical physics; * Brownian motion; and * stochastic integral and stochastic differential equations.The theory is developed rigorously and in a self-contained way, with the chapters on measure theory interlaced with the probabilistic chapters in order to display the power of the abstract concepts in the world of probability theory. In addition, plenty of figures, computer simulations, biographic details of key mathematicians, and a wealth of examples support and enliven the presentation. Front Matter....Pages I-XII Basic Measure Theory....Pages 1-48 Independence....Pages 49-75 Generating Functions....Pages 77-84 The Integral....Pages 85-99 Moments and Laws of Large Numbers....Pages 101-127 Convergence Theorems....Pages 129-142 L p -Spaces and the Radon-Nikodym Theorem....Pages 143-167 Conditional Expectations....Pages 169-187 Martingales....Pages 189-203 Optional Sampling Theorems....Pages 205-215 Martingale Convergence Theorems and Their Applications....Pages 217-229 Backwards Martingales and Exchangeability....Pages 231-243 Convergence of Measures....Pages 245-270 Probability Measures on Product Spaces....Pages 271-292 Characteristic Functions and the Central Limit Theorem....Pages 293-326 Infinitely Divisible Distributions....Pages 327-343 Markov Chains....Pages 345-378 Convergence of Markov Chains....Pages 379-402 Markov Chains and Electrical Networks....Pages 403-430 Ergodic Theory....Pages 431-445 Brownian Motion....Pages 447-494 Law of the Iterated Logarithm....Pages 495-504 Large Deviations....Pages 505-524 The Poisson Point Process....Pages 525-542 The Itô Integral....Pages 543-566 Stochastic Differential Equations....Pages 567-590 Back Matter....Pages 591-621