This hefty (500-page) volume covers weak convergence, empirical processes, and their statistical applications. The first topic occupies 100 pages and gives a concise introduction to the theory of weak convergence for ranodm elements that are not necessarily measurable functions. The basic idea is derived from Hoffmann-Jorgensen. Such a problem arises in a natural way when one is interested in the weak limit behavior of the sample distributon function of an iid sequence. Indeed, the empirical distribution function is not (Borel) measurable as a random element with values in the Skorokhod space D of cadlag functions, provided that the latter is endowed with the supremum norm. This has led to different approaches to this tedious problem. There are two well-known solutions. The first consists of avoiding the uniform topology in D and instead using a so-called Skorokhod metric and the corresponding Borel sigma field. This is a method for people with good intuition. (A masterly presentation can be found in Billingsley 1968.) The other well-known approach is to equip D with the sup-norm, but take a sigma field smaller than the Borel field -- namely the one generated by the open balls. The empirical distribution function is then a measurable function with respect to this sigma field. This method was very elegantly described by Pollard (1984). Hoffmann-Jorgensen's idea involves not requiring measurability of the weakly converging random elements, only of the limiting random element. Nonmeasurability necessarily leads to problems with integrals and probabilities. Hoffmann-Jorgensen's answer was to work with outer probability measures and outer integrals, a concept that has been used since the begining of measure theory.
As may have become clear from the foregoing outline, this is not an easy matter, and therefore an up-to-date textbook treatment was called for, and indeed is provided by the two authors. Anyone with some background knowledge in measure theory and functional analysis can understand the underlying theory.
Throughout the rest of the book, the power of the new concept of weak convergence is demonstrated. Naturally, empirical processes are the principle objects of study; this topic occupies 200 pages in Chapter 2. Here the reader will find a systematic readable approach to the modern theory of empirical processes -- namely empirical processes indexed by a class of functions. Notions like Donsker class, Vapnik-Chervonenkis class, covering number, entropy, and bracketing arise, all of which are well-known to the specialist working in empirical process theory or in probability theory on abstract spaces. Anyone who applies asymptotic statistical methods makes frequent implicit use of this theory -- for example when proving consistency or weak convergence results of estimators that depend on an estimated parameter, or when estimating a density using kernels that depend on a bandwidth. Then convergence results uniform over a class of functions are needed. Chapter provides the correct theory to use. Along with the general theory, various useful tools and techniques are treated; for instance, exponential tail estimates, maximal inequalities, and symmetrization techniques. The appendix contains various useful facts usually scattered throughout the literature. Chapter 3 (150 pages) gives a multitude of applications of empirical process theory. A prominent role is played by the bootstrap with its various modifications such as exchangeable, multiplier, and wild bootstrap. This chapter does not aim at completeness but rather stresses the potential use of empirical process methods in nonparametric statistics; the theme "bootstrap" could (and indeed does) fill whole volumes. Further topics are (M,Z) estimation, continuity, convolution, the delta method, and minimax theorems. This chapter also contains various inequalities, estimates and techniques that are useful to anyone interested in asymptotic methods in probability theory/statistics.
This long-awaited book closely follows in the tradition of its predecessors (Billingsley 1968 and Pollard 1984). It can serve as a standard text for anyone interested in modern empirical process theory. I highly recommend it to researchers, as it provides a concise account of the existing theory. Lecturers in probability theory/statistics should consider it as a sound basis for a variety of graduate courses. Graduate students will enjoy it, but also should be urged to look at the "classical" monographs mentioned herein to see how the theory has developed.
The references (20 pp.) cover the relevant literature. The subject index (13 pp.) describes the content of the book quite adequately. Some of the key words are quite surprising (e.g. "alternative proof of Corollary 2.9.9" and "E^*T is not always ET^*"). The book contains many exercises and problems that aim to help the reader gain a better understanding of the theory presented. Notes in every chapter give a bibliographic guide through the jungle of the empirical process literature and to the historical development of the ideas.
For such a sizeable volume, the price is reasonable.
REFERENCES
Billingsley, P. (1968). Convergence of Probability Measures. New York: Wiley.
Pollard, D. (1984). Convergence of Stochastic Processes. Berlin: Springer.