Uniform central limit theorems / Richard M. Dudley

This book is very well structured, excellently written and remarkably self-contained. The material that is given without proofs is placed in doubly starred sections and is not used in the rest of the book; and sections with one star contain material that is not used in the sequel, except perhaps for other starred sections. Each chapter ends with a set of exercises and with historical notes. These qualities make the book very well suited for graduate courses, and it is also an excellent reference book. Among the subjects only mentioned, but not developed, perhaps we should mention Talagrand’s law of large numbers, the uniform (in P) central limit theorem, Gaussian and random entropy characterizations of Donsker classes, the law of the iterated logarithm, and deviation and concentration exponential inequalities. But, developing these subjects in the self-contained style and at the level of rigor and detail of this book would require another one of at least the same size. (Zentralblatt)

Notes bibliogr. Index

This monograph is a well-written treatise on functional central limit theorems. The material is presented in an abstract framework, but examples discussed focus more or less on abstract empirical processes and their limits. After a brief introduction to the subject, in Chapter 2 the monograph presents a detailed discussion of Gaussian processes (inequalities, majorizing measures, compactness, and sample continuity). Chapters 3–10 then discuss the by now classical ingredients of abstract weak convergence theory and their application to empirical processes: Donsker classes, Vapnik-Chervonenkis combinatorics, measurability, limit theorems for Vapnik-Chervonenkis and related classes, metric entropy with inclusion and bracketing, approximations of functions and sets. Finally, Chapters 9–12 deal with sums in general Banach spaces, universal and uniform central limit theorems, the two-sample case and the bootstrap, and some examples where the general theory does not apply.
For most of the results detailed proofs are given. The list of references is fairly complete so that this book is recommended for all who want to know more about a subject which by now is considered a must in abstract large-sample theory. (MathSciNet)