A Modern Approach to Probability Theory

Students and teachers of mathematics and related fields will find in this second edition, as previously, a comprehensive and modern approach to probability theory, providing the background and techniques to go from the beginning graduate level to the point of specialization in research areas of current interest. The book is designed for a two- or three-semester course, assuming only courses in undergraduate real analysis or rigorous advanced calculus, and some elementary linear algebra.Revisions and additions to the second edition:* A variety of applications?Bayesian statistics, financial mathematics, information theory, tomography, and signal processing?appear as threads in conjunction with the relevant mathematics. The goal is to both enhance the understanding of the mathematics and motivate students whose main interests are outside of pure areas.* The relevant measure theory is integrated with the standard topics of probability theory. The latter part of the book examines stochastic processes in both discrete and continuous time: martingales, renewal sequences, Markov processes, exchangeable sequences, stationary sequences, point processes, diffusions, and stochastic calculus. The treatment of stochastic calculus has been expanded considerably.* Numerous examples illustrate the richness and variety of the subject, from sophisticated results in gambling theory to concrete calculations involving random sets.* Over 1,000 exercises are designed to give a deep intuitive feel for the far-reaching implications of the theory.* A solutions manual is available, containing information for about 30 per cent of the exercises, ranging from a simple answer in some cases to a full-detailed calculation with accompanying proofs in others.

Table of Contents:
List of Tables * Preface * Part I: Probability Spaces, Random Variables, and Expectations * Probability Spaces * Random Variables * Distribution Functions * Expectations: Theory * Expectations: Applications * Calculating Probabilities and Measures * Measure Theory: Existence and Uniqueness * Integration Theory * Part 2: Independence and Sums * Stochastic Independence * Sums of Independent Random Variables * Random Walk * Theorems of A.S. Convergence * Characteristic Functions * Part 3: Convergence in Distribution * Convergence in Distribution on the Real Line * Distributional Limit Theorems for Partial Sums * Infinitely Divisible and Stable Distributions as Limits * Convergence in Distribution on Polish Spaces * The Invariance Principle and Brownian Motion * Part 4: Conditioning * Spaces of Random Variables * Conditional Probabilities * Construction of Random Sequences * Conditional Expectations * Part 5: Random Sequences * Martingales * Renewal Sequences * Time-homogeneous Markov Sequences * Exchangeable Sequences * Stationary Sequences * Part 6: Stochastic Processes * Point Processes * Diffusions and Stochastic Calculus * Applications of Stochastic Calculus * Part 7: Appendices * Appendix A. Notation and Usage of Terms * Appendix B. Metric Spaces * Appendix C. Topological Spaces * Appendix D. Riemann?Stieltjes Integration * Appendix E. Taylor Approximations, C-Valued Logarithms * Appendix F. Bibliography * Appendix G. Comments and Credits * Index