Weak Convergence and Empirical Processes

1. Front Matter

   Pages i-xvi

   PDF

2. Stochastic Convergence

   1. Front Matter

      Pages 1-1

      PDF

   2. Introduction

      • Aad W. van der Vaart, Jon A. Wellner

      Pages 2-5

   3. Outer Integrals and Measurable Majorants

      • Aad W. van der Vaart, Jon A. Wellner

      Pages 6-15

   4. Weak Convergence

      • Aad W. van der Vaart, Jon A. Wellner

      Pages 16-28

   5. Product Spaces

      • Aad W. van der Vaart, Jon A. Wellner

      Pages 29-33

   6. Spaces of Bounded Functions

      • Aad W. van der Vaart, Jon A. Wellner

      Pages 34-42

   7. Spaces of Locally Bounded Functions

      • Aad W. van der Vaart, Jon A. Wellner

      Pages 43-44

   8. The Ball Sigma-Field and Measurability of Suprema

      • Aad W. van der Vaart, Jon A. Wellner

      Pages 45-48

   9. Hilbert Spaces

      • Aad W. van der Vaart, Jon A. Wellner

      Pages 49-51

   10. Convergence: Almost Surely and in Probability

       • Aad W. van der Vaart, Jon A. Wellner

       Pages 52-56

   11. Convergence: Weak, Almost Uniform, and in Probability

       • Aad W. van der Vaart, Jon A. Wellner

       Pages 57-66

   12. Refinements

       • Aad W. van der Vaart, Jon A. Wellner

       Pages 67-70

   13. Uniformity and Metrization

       • Aad W. van der Vaart, Jon A. Wellner

       Pages 71-74

3. Empirical Processes

   1. Front Matter

      Pages 79-79

      PDF

   2. Introduction

      • Aad W. van der Vaart, Jon A. Wellner

      Pages 80-94

   3. Maximal Inequalities and Covering Numbers

      • Aad W. van der Vaart, Jon A. Wellner

      Pages 95-106

   4. Symmetrization and Measurability

      • Aad W. van der Vaart, Jon A. Wellner

      Pages 107-121

   5. Glivenko-Cantelli Theorems

      • Aad W. van der Vaart, Jon A. Wellner

      Pages 122-126

   6. Donsker Theorems

      • Aad W. van der Vaart, Jon A. Wellner

      Pages 127-133

This book tries to do three things. The first goal is to give an exposition of certain modes of stochastic convergence, in particular convergence in distribution. The classical theory of this subject was developed mostly in the 1950s and is well summarized in Billingsley (1968). During the last 15 years, the need for a more general theory allowing random elements that are not Borel measurable has become well established, particularly in developing the theory of empirical processes. Part 1 of the book, Stochastic Convergence, gives an exposition of such a theory following the ideas of J. Hoffmann-J!1Jrgensen and R. M. Dudley. A second goal is to use the weak convergence theory background devel­ oped in Part 1 to present an account of major components of the modern theory of empirical processes indexed by classes of sets and functions. The weak convergence theory developed in Part 1 is important for this, simply because the empirical processes studied in Part 2, Empirical Processes, arenaturally viewed as taking values in nonseparable Banach spaces, even in the most elementary cases, and are typically not Borel measurable. Much of the theory presented in Part 2 has previously been scattered in the journal literature and has, as a result, been accessible only to a relatively small number of specialists. In view of the importance of this theory for statis­ tics, we hope that the presentation given here will make this theory more accessible to statisticians as well as to probabilists interested in statistical applications.

• Estimator
• Mathematics
• Maxima
• Maximum
• Minimum
• Random variable
• Statistics

"...succeeds and complements Billingsleys classic work and will become the standard source of study and reference for students and researchers...." The Statistician

• Department of Mathematics and Computer Science, Free University, Amsterdam, The Netherlands

  Aad W. Vaart

• Statistics, University of Washington, Seattle, USA

  Jon A. Wellner

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