Weak convergence and empirical processes [electronic resource] : with applications to statistics / A.W. van der Vaart, Jon A. Wellner.

Intro
Preface
Preface to the Second Edition
Reading Guide
Contents
PART 1 Stochastic Convergence
1.1 Introduction
1.2 Outer Integrals and Measurable Majorants
Problems and Complements
1.3 Weak Convergence
Problems and Complements
1.4 Product Spaces
1.5 Spaces of Bounded Functions
Problems and Complements
1.6 Spaces of Locally Bounded Functions
1.6.1 Convex processes
Problems and Complements
1.7 Ball Sigma-Field and Measurability of Suprema
Problems and Complements
1.8 Hilbert Spaces
Problems and Complements

1.9 Almost Sure and in Probability Convergence
Problems and Complements
1.10 Weak, Almost Uniform and in Probability Convergence
Problems and Complements
1.11 Refinements
Problems and Complements
1.12 Uniformity and Metrization
Problems and Complements
1.13 Conditional Weak Convergence
Problems and Complements
1.14 Skorokhod Space
1.14.1 Compact Domain
1.14.2 Unbounded Domain
1.14.2.1 Conditional Distribution Bounds
1.14.2.2 Mixed Moment Bounds
1.14.2.3 Counting Processes
1.14.2.4 Semi-Martingales
Problems and Complements

1.15 Stable Convergence
Problems and Complements
1 Notes
PART 2 Empirical Processes
2.1 Introduction
2.1.1 Overview
2.1.2 Asymptotic Equicontinuity
2.1.3 Maximal Inequalities
2.1.4 Central Limit Theorem in Banach Spaces
Problems and Complements
2.2 Maximal Inequalities
2.2.1 Orlicz Norms
2.2.2 Covering Numbers and Entropy
2.2.3 Sub-Gaussian Inequalities
2.2.4 Mixed Sub-Gaussian-Exponential Inequalities
2.2.5 Generic Chaining
2.2.6 Majorizing Measures
Problems and Complements
2.3 Symmetrization and Measurability
2.3.1 Symmetrization

2.3.2 More Symmetrization
2.3.3 Separable Versions
Problems and Complements
2.4 Glivenko-Cantelli Theorems
Problems and Complements
2.5 Donsker Theorems
2.5.1 Uniform Entropy
2.5.2 Bracketing
Problems and Complements
2.6 Uniform Entropy Numbers
2.6.1 VC-Classes of Sets
2.6.2 VC-Classes of Functions
2.6.3 Convex Hulls and VC-Hull Classes
2.6.4 VC-Major Classes
2.6.5 Examples
2.6.6 Permanence Properties
2.6.7 Fat Shattering
Problems and Complements
2.7 Entropies of Function Classes
2.7.1 Hl̲der Classes
2.7.2 Sobolev and Besov Spaces

2.7.2.1 Besov Bodies
2.7.3 Monotone Functions
2.7.4 Convex Sets and Functions
2.7.5 Analytic Functions
2.7.6 Parametrized Classes
2.7.7 Gaussian Mixtures
2.7.8 Sets with Smooth Boundaries
Problems and Complements
2.8 Uniformity in the Underlying Distribution
2.8.1 Glivenko-Cantelli Theorems
2.8.2 Donsker Theorems
2.8.3 Central Limit Theorem Under Sequences
Problems and Complements
2.9 Multiplier Central Limit Theorems
Problems and Complements
2.10 Permanence under Transformation
2.10.1 Closures and Convex Hulls