Preface to the Second Edition

Summary

This book developed out of some topics courses given at M.I.T. and my lectures at the St.-Flour probability summer school in 1982. The material of the book has been expanded and extended considerably since then. At the end of each chapter are some problems and notes on that chapter.

Starred sections are not cited later in the book except perhaps in other starred sections. The first edition had several double-starred sections in which facts were stated without proofs. This edition has no such sections.

The following, not proved in the first edition, now are: (i) for Donsker's theorem on the classical empirical process α_n := √n(F_n − F), and the Komlós–Major–Tusnády strengthening to give a rate of convergence, the Bretagnolle–Massart proof with specified constants; (ii) Massart's form of the Dvoretzky–Kiefer–Wolfowitz inequality for α_n with optimal constant; (iii) Talagrand's generic chaining approach to boundedness of Gaussian processes, which replaces the previous treatment of majorizing measures; (iv) characterization of uniform Glivenko–Cantelli classes of functions (from a paper by Dudley, Giné, and Zinn, but here with a self-contained proof); (v) Giné and Zinn's characterization of uniform Donsker classes of functions; (vi) its consequence that uniformly bounded, suitably measurable classes of functions satisfying Pollard's entropy condition are uniformly Donsker; and (vii) Bousquet, Koltchinskii, and Panchenko's theorem that a convex hull preserves the uniform Donsker property.