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Description
This 2nd edition textbook offers a rigorous introduction to measure theoretic probability with particular attention to topics of interest to mathematical statisticians—a textbook for courses in probability for students in mathematical statistics. It is recommended to anyone interested in the probability underlying modern statistics, providing a solid grounding in the probabilistic tools and techniques necessary to do theoretical research in statistics. For the teaching of probability theory to post graduate statistics students, this is one of the most attractive books available.
Of particular interest is a presentation of the major central limit theorems via Stein's method either prior to or alternative to a characteristic function presentation. Additionally, there is considerable emphasis placed on the quantile function as well as the distribution function. The bootstrap and trimming are both presented. Martingale coverage includes coverage of censored data martingales. The text includes measure theoretic preliminaries, from which the authors own course typically includes selected coverage.
This is a heavily reworked and considerably shortened version of the first edition of this textbook. "Extra" and background material has been either removed or moved to the appendices and important rearrangement of chapters has taken place to facilitate this book's intended use as a textbook.
Table of Contents
Preface.- Use of This Text.- Definition of Symbols.- Chapter 1. Measures.- Chapter 2. Measurable Functions and Convergence.- Chapter 3. Integration.- Chapter 4 Derivatives via Signed Measures.- Chapter 5. Measures and Processes on Products.- Chapter 6. Distribution and Quantile Functions.- Chapter 7. Independence and Conditional Distributions.- Chapter 8. WLLN, SLLN, LIL, and Series.- Chapter 9. Characteristic Functions and Determining Classes.- Chapter 10. CLTs via Characteristic Functions.- Chapter 11. Infinitely Divisible and Stable Distributions.- Chapter 12. Brownian Motion and Empirical Processes.- Chapter 13. Martingales.- Chapter 14. Convergence in Law on Metric Spaces.- Chapter 15. Asymptotics Via Empirical Processes.- Appendix A. Special Distributions.- Appendix B. General Topology and Hilbert Space.- Appendix C. More WLLN and CLT.- References.- Index.
