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. A variety of applicationsBayesian statistics, financial mathematics, information theory, tomography, and signal processingappear as threads to both enhance the understanding of the relevant mathematics and motivate students whose main interests are outside of pure areas.

A variety of applicationsBayesian statistics, financial mathematics, information theory, tomography, and signal processingnow appear as threads in conjunction with the relevant mathematics. Bridges the gap between undergraduate and advanced graduate courses. The treatment of stochastic calculus has been expanded considerably. Numerous examples and over 1000 exercises, with a solutions manual for about 30%. Includes supplementary material: sn.pub/extras

Inhalt
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. RiemannStieltjes Integration * Appendix E. Taylor Approximations, C-Valued Logarithms * Appendix F. Bibliography * Appendix G. Comments and Credits * Index