Shorack G.R. Probability for statisticians

2nd edition. — New York: Springer, 2017. — 519 p.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.Measures
Measurable Functions and Convergence
Integration
Derivatives via Signed Measures
Measures and Processes on Products
Distribution and Quantile Functions
Independence and Conditional Distributions
WLLN, SLLN, LIL, and Series
Characteristic Functions and Determining Classes
CLTs via Characteristic Functions
Infinitely Divisible and Stable Distributions \(^o\)
Brownian Motion and Empirical Processes
Martingales
Convergence in Law on Metric Spaces \(^o\)
Asymptotics via Empirical Processes