Probability for statisticians

Description

This 2nd edition textbookoffers 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. Front Matter ....Pages i-xxii Measures (Galen R. Shorack)....Pages 1-21 Measurable Functions and Convergence (Galen R. Shorack)....Pages 23-38 Integration (Galen R. Shorack)....Pages 39-66 Derivatives via Signed Measures (Galen R. Shorack)....Pages 67-87 Measures and Processes on Products (Galen R. Shorack)....Pages 89-105 Distribution and Quantile Functions (Galen R. Shorack)....Pages 107-126 Independence and Conditional Distributions (Galen R. Shorack)....Pages 127-148 WLLN, SLLN, LIL, and Series (Galen R. Shorack)....Pages 149-191 Characteristic Functions and Determining Classes (Galen R. Shorack)....Pages 193-223 CLTs via Characteristic Functions (Galen R. Shorack)....Pages 225-270 Infinitely Divisible and Stable Distributions \(^o\) (Galen R. Shorack)....Pages 271-296 Brownian Motion and Empirical Processes (Galen R. Shorack)....Pages 297-343 Martingales (Galen R. Shorack)....Pages 345-385 Convergence in Law on Metric Spaces \(^o\) (Galen R. Shorack)....Pages 387-399 Asymptotics via Empirical Processes (Galen R. Shorack)....Pages 401-415 Back Matter ....Pages 417-510