Probability for Statisticians is an introductory textbook in probability theory at the graduate level that for some reasons is not very well-known, not even among statisticians. The "for Statisticians" part of the title should not be taken to mean that this is not a serious textbook in probability. As a matter of fact all relevant topics for an introductory course in probability are there and, in addition, there are many applications of those concepts. These are clearly aimed at the use of probability theory in mathematical statistics.
But, in my opinion, this is good: there are too many textbooks with poor choice of examples and applications. This textbook delivers a lot of solid examples and is not shy about details. The book lacks some introductory discussions on topics like Percolation Theory, Renewal Processes and even some basic Stochastic Integration (except a small section within Martingales) . However, considering the way many textbooks deal with these topics, it is a blessing as Shorack won't scar his readers for life.

The author: Galen R. Shorack (University of Washington, Dept. of Statistics) is arguably best-known for his work in the field of empirical processes, a fairly technical area of Statistics/Probability with important applications. Something that is reflected in this book where one chapter is devoted to Asymptotics via Empirical Processes. I cannot recall a single other introductory textbook devoting any space to a very technical topic like Empirical Processes.


The book:
Contents:
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: General Topology and Hilbert Space
Chapter 7: Distribution and Quantile Functions
Chapter 8: Independence and Conditional Distributions
Chapter 9: Special Distributions
Chapter 10: WLLN, SLLN, LIL, and Series
Chapter 11: Convergence in Distribution
Chapter 12: Brownian Motion and Empirical Processes
Chapter 13: Characteristic Functions
Chapter 14: CLT's via Characteristic Functions
Chapter 15: Infinitely Divisible and Stable Distributions
Chapter 16: Asymptotics via Empirical Processes
Chapter 17: Asymptotics via Stein's Approach
Chapter 18: Martingales
Chapter 19: Convergence in Law on Metric Spaces
Appendix A: Distribution Summmaries

Total pages: 585

Is the book well-written? Overall yes, although here and there  I find section that are not exactly the easiest to follow. Also, the notation is heavy. The text is packed tight and there is plenty of symbols and formulae and sometimes the result is not the most pleasant to the eye. Also, in places, the notations is not the most standard and I found myself struggling to keep track of things.
In some cases, important results are buried in a lot of text and only a lot of perseverance will allow a reader not to miss them or, more precisely, not to miss their importance.
I read this book out of curiosity already knowing what most important results to look for are. Had I been an honest novice on the subject, I fear that here and there I might have missed a few.
The format of the book, almost a square, is the most unpleasant feature to me personally.


Are many proofs left to the reader as exercises? Thankfully not. Students can be students. Occasionally, some interesting results or special cases are left to the reader to prove, but it is not a systematic issue like in another textbook I reviewed before this one.
In general it might not be a bad idea to leave some serious work to a student. But graduate students' life is often too busy and time to devote to individual subjects scarce. So authors, please, do not  go overboard with problems. We pay good money for your textbooks, so give us the solutions (maybe even as a companion volume). Thank you! Shorack is found innocent of this charge, however.

What are the most "unusual" features? Well, first of all let me say that the adjective unusual should not be meant in a derogatory way, quite the opposite, indeed. We already mentioned the chapter on Empirical Processes. Most textbooks do not go further than mentioning the Kolmogorov Smirnov distance or little more. Here you have a full chapter with many results about convergence, Brownian Motion and its relation to empirical processes. The chapter on Martingales contains also an introduction to counting processes (the branch of mathematical statistics behind Survival Analysis). While these features are of most interest for a statistician, their applications are no minor feat and make the book much less theoretical (and vague, in some cases) than most standard textbooks and are good for mathematicians as well. Distribution functions and quantiles receive an extensive treatment, way more than in more orthodox textbooks on the topic written by mathematicians.

Any relevant topic missing? I would have to say Levy Processes. While there is a short chaper on Infinitely Divisible and Stable Distributions, Levy Processes are the most obvious topic missing. In general, stochastic processes are a little bit dispersed among several topics. That is justified since Shorack is writing for an audience that uses stochastic processes as tools and is not, in general, keen to study them as an abstract idea only. Perhaps, given Shorack's background I would have expected to find some coverage of statistics for stochastic processes applied to diffusion models. However, even as it is, the book is quite large.

Ideal audience?  Graduate students in Statistics are the most likely to benefit from this book written by a statistician for statisticians. However, its unusual balance of theory and applications (albeit theoretical applications) makes it very ideal even for students in Economics/Econometrics, Biostatistics, and  Mathematics.
Some mathematicians might act snotty towards a book that in the title says "  ..  for Statisticians". However, I have seen some books on probability theory written by mathematicians that are definitely not worth the paper they are printed on. So mathematicians should give this book a serious chance. They might actually even end up liking it, quite frankly.

Is the book good for self-study? I would say that it is. Here and there, unfortunately, some concepts are left to problems which could make life miserable for those who buy the book for self study. However, so many other details are in the books that the pluses by far outweight the few minuses.

What kind of mathematical background does one need for this book? It is not a basic book. You need what I would define as a solid master's level analysis (honors' course in analysis) course with some knowledge of general topology. While measure theory is developed in the textbook, previous knowledge would be useful. This is a textbook aimed at graduate students working towards a Ph.D. degree in Statistics/Biostatistics or Economics/Econometrics, so rigorous mathematical thinking is expected.

Is the book good as a reference for practitioners? Most definitely. Besides I cannot think of any other textbook that packs so many  elements of theory used in professions that use a lot of probability theory. There are many Ph.D. programs in Statistics in the US and outside where, for example, there is hardly any specialist in Empirical Processes and this textbook, while it will not replace one, it will give at least some decent background to anybody with a need or just an interest/curiosity.

Recommended: Yes