Probability for Statisticians

1. Front Matter

   Pages i-xviii

   PDF

2. Measures

   Pages 1-20

3. Measurable Functions and Convergence

   Pages 21-36

4. Integration

   Pages 37-60

5. Derivatives via Signed Measures

   Pages 61-78

6. Measures and Processes on Products

   Pages 79-93

7. General Topology and Hilbert Space

   Pages 95-106

8. Distribution and Quantile Functions

   Pages 107-150

9. Independence and Conditional Distributions

   Pages 151-178

10. Special Distributions

    Pages 179-202

11. WLLN, SLLN, LIL, and Series

    Pages 203-254

12. Convergence in Distribution

    Pages 255-294

13. Brownian Motion and Empirical Processes

    Pages 295-340

14. Characteristic Functions

    Pages 341-363

15. CLTs via Characteristic Functions

    Pages 365-397

16. Infinitely Divisible and Stable Distributions

    Pages 399-413

17. Asymptotics via Empirical Proceses

    Pages 415-447

18. Asymptotics via Stein’s Approach

    Pages 449-465

19. Martingales

    Pages 467-530

20. Convergence in Law on Metric Spaces

    Pages 531-544

Probability for Statisticians is intended as a text for a one year graduate course aimed especially at students in statistics. The choice of examples illustrates this intention clearly. The material to be presented in the classroom constitutes a bit more than half the text, and the choices the author makes at the University of Washington in Seattle are spelled out. The rest of the text provides background, offers different routes that could be pursued in the classroom, ad offers 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 funcion presentation. Additionally, there is considerable emphasis placed on the quantile function as well as the distribution function. The bootstrap and trimming are both presented. The martingale coverage includes coverage of censored data martingales. The text includes measure theoretic preliminaries, from which the authors own course typically includes selected coverage. The author is a professor of Statistics and adjunct professor of Mathematics at the University of Washington in Seattle. He served as chair of the Department of Statistics 1986-- 1989. He received his PhD in Statistics from Stanford University. He is a fellow of the Institute of Mathematical Statistics, and is a former associate editor of the Annals of Statistics.

• Brownian motion
• Law of large numbers
• Martingale
• mathematical statistics
• statistics

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)

• Department of Statistics, University of Washington, Seattle, USA

  Galen R. Shorack

Policies and ethics