The choice of examples used in this text clearly illustrate its use for a one-year graduate course. The material to be presented in the classroom constitutes a little more than half the text, while the rest of the text provides background, offers different routes that could be pursued in the classroom, as well as additional material that is appropriate for self-study. 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, with both the bootstrap and trimming presented. The section on martingales covers censored data martingales.

This modern approach to probability theory will serve as a reference book for research statisticians and mathematicians.

Zusammenfassung

From the reviews:

"This book offers a rigorous introduction to measure theoretic probability with particular attention to topics of interest to mathematical statisticians. ... recommended to anyone interseted in the probability underlying modern statistics."
D.L. McLeish in "Short Book Reviews", Vol. 21/1, April 2001

"The book originated from a graduate level course given by the author on probability at the University of Washington, Seattle. It is an excellent textbook for a course in probability for students in mathematical statistics. It provides a solid grounding in the probabilistic tools and techniques that are necessary to do theoretical research in statistics.
For the teaching of probability theory to post graduate statistics students, this is certainly one of the most attractive books available and is highly recommended for that purpose. It is also an extremely good reference source of value to any research statistician.
SASA News, Dec. 2001

"This book contains a wealth of material and is very rigorous. It may serve as a good reference book and as a source for a graduate course in probability. ... The author provides detailed notes on the use of the text for a graduate course of probability. ... Overall, this is an excellent book to acquire." (Arup Bose, Sankhya, Vol. 64 (1), 2002)

"The present textbook grew out of probability lecturers given at the University of Washington in Seattle. ... is an excellent reference. The present monograph can strongly be recommended. It is a highlight in modern probability theory with strong applications to mathematical statistics. It may serve as a textbook for advanced lecturers and seminars but it is also worthwhile as reference book for modern aspects in probability theory. ... I am happy to have this book on my desk." (Arnold Janssen, Metrika, April, 2001)

"This is a textbook about probability theory, with a view towards applications in statistics. ... The book contains585 pages of text. It contains a lot of useful material which every theoretical statistician should know. ... All in all, this is an interesting book and certainly recommended." (R. Helmers, Kwantitatieve Methoden, Vol. 22 (68), 2001)

"This text covers a broad range of probability theory with special emphasis on fields which are useful for statistics. One characteristic unique to this text is the presentation of different approaches to the CLT. The presentation is in the style: definition - proposition/theorem - proof. More detailed motivations are concentrated in special paragraphs. Questions are included to open the view for further generalizations and motivate the reader to think themselves. Exercises are everywhere." (R. Schlittgen, Zentralblatt MATH, Vol. 951, 2001)

Inhalt
Measures.- Measurable Functions and Convergence.- Integration.- Derivatives via Signed Measures.- Measures and Processes on Products.- General Topology and Hilbert Space.- Distribution and Quantile Functions.- Independence and Conditional Distributions.- Special Distributions.- WLLN, SLLN, LIL, and Series.- Convergence in Distribution.- Brownian Motion and Empirical Processes.- Characteristic Functions.- CLTs via Characteristic Functions.- Infinitely Divisible and Stable Distributions.- Asymptotics via Empirical Proceses.- Asymptotics via Stein's Approach.- Martingales.- Convergence in Law on Metric Spaces.