Synopses & Reviews
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% of the exercises, ranging from a simple answer in some cases to a full-detailed calculation with accompanying proofs in others.
"Covers the essentials in a clear and readable fashion... A must for professionals and an attractive text for a graduate course." --American Mathematical Monthly (review of the first edition) "This ambitious book is intended as 'a textbook in probability for graduate students in mathematics and related areas such as economics, statistics, physics and operations research'...The coverage is careful and thorough...Quite a lot of fairly recent material is incorporated, and this is certainly one of the book's strengths. The selection of material is sensible, and the quality of exposition is good...In sum: the book contains a lot of good mathematics, nicely done, and should prove useful to students and teachers, and to specialists in probability theory." --Mathematical Reviews (review of the first edition) "The book takes the reader from a relatively low level.... To the point where he or she can specialize in research topics of current interest. ...an outstanding basis for teaching a graduate course in probability theory. The exhaustive compilation of results and detailed index also make it a very useful reference text for the more advanced probabilist... a good buy for anyone looking for a very accessible and complete mathematical account of modern probability theory." --Journal of the American Statistical Association (review of the first edition)
Overview This book is intended as a textbook in probability for graduate students in math ematics and related areas such as statistics, economics, physics, and operations research. Probability theory is a 'difficult' but productive marriage of mathemat ical abstraction and everyday intuition, and we have attempted to exhibit this fact. Thus we may appear at times to be obsessively careful in our presentation of the material, but our experience has shown that many students find them selves quite handicapped because they have never properly come to grips with the subtleties of the definitions and mathematical structures that form the foun dation of the field. Also, students may find many of the examples and problems to be computationally challenging, but it is our belief that one of the fascinat ing aspects of prob ability theory is its ability to say something concrete about the world around us, and we have done our best to coax the student into doing explicit calculations, often in the context of apparently elementary models. The practical applications of probability theory to various scientific fields are far-reaching, and a specialized treatment would be required to do justice to the interrelations between prob ability and any one of these areas. However, to give the reader a taste of the possibilities, we have included some examples, particularly from the field of statistics, such as order statistics, Dirichlet distri butions, and minimum variance unbiased estimation."
Students and teachers of mathematics and related fields will find this book 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 applications--Bayesian statistics, financial mathematics, information theory, tomography, and signal processing--appear as threads to both enhance the understanding of the relevant mathematics and motivate students whose main interests are outside of pure areas.
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