This classic work on empirical processes has been considerably expanded and revised from the original edition. When samples become large, the probability laws of large numbers and central limit theorems are guaranteed to hold uniformly over wide domains. The author, an acknowledged expert, gives a thorough treatment of the subject, including 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. This new edition contains several proved theorems not included in the first edition, including the Bretagnolle-Massart theorem giving constants in the Komlos-Major-Tusnady rate of convergence for the classical empirical process, Massart’s form of the Dvoretzky-Kiefer-Wolfowitz inequality with precise constant, Talagrand’s generic chaining approach to boundedness of Gaussian processes, a characterization of uniform Glivenko-Cantelli classes of functions, Giné and Zinn’s characterization of uniform Donsker classes (i.e., classing Donsker uniformly over all probability measures P), and the Bousquet-Koltchinskii-Panchenko theorem that the convex hull of a uniform Donsker class is uniform Donsker. The book will be an essential reference for mathematicians working in infinite-dimensional central limit theorems, mathematical statisticians, and computer scientists working in computer learning theory. Problems are included at the end of each chapter so the book can also be used as an advanced text.

- Cover
- Half title
- Series
- Title
- Copyright
- Dedication
- Contents
- Preface to the Second Edition
- 1 Donskers Theorem, Metric Entropy, and Inequalities
- 2 Gaussian Processes; Sample Continuity
- 3 Foundations of Uniform Central Limit Theorems; Donsker Classes
- 4 Vapnik–Červonenkis Combinatorics
- 5 Measurability
- 6 Limit Theorems for Vapnik–Červonenkis and Related Classes
- 7 Metric Entropy, with Inclusion and Bracketing
- 8 Approximation of Functions and Sets
- 9 The Two-Sample Case, the Bootstrap, and Confidence Sets
- 10 Uniform and Universal Limit Theorems
- 11 Classes of Sets or Functions Too Large for Central Limit Theorems
- Appendix A Differentiating under an Integral Sign
- Appendix B Multinomial Distributions
- Appendix C Measures on Nonseparable Metric Spaces
- Appendix D An Extension of Lusins Theorem
- Appendix E Bochner and Pettis Integrals
- Appendix F Nonexistence of Some Linear Forms
- Appendix G Separation of Analytic Sets; Borel Injections
- Appendix H Young–Orlicz Spaces
- Appendix I Modifications and Versions of Isonormal Processes
- Bibliography
- Notation Index
- Author Index
- Subject Index