This book shows how the central limit theorem for independent, identically distributed random variables with values in general, multidimensional spaces, holds uniformly over some large classes of functions. The author, an acknowledged expert, gives a thorough treatment of the subject, including several topics not found in any previous book, such as the Fernique-Talagrand majorizing measure theorem for Gaussian processes, an extended treatment of Vapnik-Chervonenkis combinatorics, the Ossiander L2 bracketing central limit theorem, the Giné-Zinn bootstrap central limit theorem in probability, the Bronstein theorem on approximation of convex sets, and the Shor theorem on rates of convergence over lower layers. Other results of Talagrand and others are surveyed without proofs in separate sections. Problems are included at the end of each chapter so the book can be used as an advanced text. The book will interest mathematicians working in probability, mathematical statisticians and computer scientists working in computer learning theory.

‘It is for certain that this will soon be a classic piece of work in the empirical process literature.’

N. D. C. Veraberbeke Source: ISI Short Book Reviews

‘The material of the book is very well organized, and incorporates recent developments … an invaluable reference to researchers in this field … it has been written in a style that makes it accessible to students; proofs have been carried out with meticulous care, and definitions are well motivated.’

Erich Berger Source: Bulletin of the London Mathematical Society