Download Probability for Statisticians by Galen R. Shorack (2000) Springer

Sankhyā : The Indian Journal of Statistics
2002, Volume 64, Series A, Pt. 1, pp 180-181
Probability for Statisticians
by Galen R. Shorack
(2000) Springer-Verlag, New York,
Price $79.95 xviii+585 ISBN 0-387-98953-6
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.
Chapters 1–6 provide the mathematical foundation for the rest of the
text. These chapters essentially cover, the definition and construction of
measures (Chapter 1), different notions of convergence (Chapter 2), Lebesgue
integral and fundamental properties of integral (Chapter 3), Radon-Nikodym
theorem and related topics (Chapter 4), countably infinite product probability spaces and basics of random elements and processes (Chapter 5) and,
elements of general topology, metric spaces and Hilbert spaces (Chapter 6).
Chapters 7–8 present the basic tools needed for the probability theory
to follow. The most important topics of Chapter 7 are the concepts of
distribution functions, quantile transformation and slowly varying functions.
Chapter 8 focusses on introducing independence, tail σ field, and regular
conditional probability.
Chapter 9 is a brief introduction to elementary probability theory covering distribution theory and the multivariate normal distribution.
The heart of the book is Chapters 10–19. Chapter 10 provides a particularly complete picture of the weak and the strong law of large numbers.
Some other topics of interest here are the maximal inequalities, the law of the
iterated logarithm and the strong Markov property for sums of independent
and identically distributed random variables.
Chapter 11 contains the classical central limit theorem and some of its
generalisations with proofs based on Stein’s method. These proofs are efficient enough to yield bootstrap limit theorems and can also deal with
scenarios involving trimming.
Chapter 12 is on Brownian motion covering its existence, strong Markov
property and embedding of partial sums.
Chapter 13 covers the basic properties of characteristic function and
proves Esseen’s lemma.
Chapter 14 provides the characteristic function proof of several limit
theorems, including the local limit theorem. It also covers normal approxi-
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mations, Edgeworth expansions and gamma approximation.
Chapter 15 is an introduction to infinitely divisible distributions and
stable laws.
Chapter 16 introduces empirical processes and uses quantile function
inequalities to derive limit results on linear rank statistics, finite population
sampling and the bootstrap.
Chapter 17 uses Stein’s method to prove the central limit theorem for U
statistics and Hoeffding’s combinatorial central limit theorem.
Chapter 18 is an introduction to martingales with the usual topics optional sampling theorem, submartingale convergence theorem, Doob-Meyer
decomposition receiving adequate coverage.
Chapter 19 covers distributional convergence in metric spaces.
The author provides detailed notes on the use of the text for a graduate
course on probability. He describes it as a “..thin self-contained textbook
within this larger presentation”. Surely there is lot more interesting material
beyond that and even those well versed in probability may pick up some new
things of interest.
This book may seem too technical and dry and at times difficult to read
but one of the reasons is the generality with which some of the topics have
been presented. There are a few typographical errors and at places terms
have been used which do not seem to have been defined earlier. As an
example of the latter, the covariance matrix is defined on page 191 but has
been used on page 185. However, none of these errors are serious enough to
affect the flow of reading and are easily corrected by the instructor.
There are a few other points which perhaps need attention in the next
edition. As examples, (i) the use of the indicator function simplifies the
proof of many inequalities (pages 49, 50), (ii) a formula for the density
when the transformation is not necessarily 1-1 may be given (page 78), (iii)
in without replacement sampling that the covariance is independent of the
indices needs to be explained (page 180).
At the same time there are quite a few aspects which sets it apart from
other books on probability. Two examples which immediately come to mind
are (i) good treatment of Stein’s method (ii) good bootstrap results. Overall,
this is an excellent book to acquire.
Arup Bose
Indian Statistical Institute
203 B T Road, Kolkata 700108, India
E-mail: [email protected]