Weak Convergence and Empirical Processes

With Applications to Statistics

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

Part of the Springer Series in Statistics book series (SSS)

Table of contents

  1. Front Matter
    Pages i-xvi
  2. Stochastic Convergence

    1. Front Matter
      Pages 1-1
    2. Aad W. van der Vaart, Jon A. Wellner
      Pages 2-5
    3. Aad W. van der Vaart, Jon A. Wellner
      Pages 6-15
    4. Aad W. van der Vaart, Jon A. Wellner
      Pages 16-28
    5. Aad W. van der Vaart, Jon A. Wellner
      Pages 29-33
    6. Aad W. van der Vaart, Jon A. Wellner
      Pages 34-42
    7. Aad W. van der Vaart, Jon A. Wellner
      Pages 43-44
    8. Aad W. van der Vaart, Jon A. Wellner
      Pages 45-48
    9. Aad W. van der Vaart, Jon A. Wellner
      Pages 49-51
    10. Aad W. van der Vaart, Jon A. Wellner
      Pages 52-56
    11. Aad W. van der Vaart, Jon A. Wellner
      Pages 57-66
    12. Aad W. van der Vaart, Jon A. Wellner
      Pages 67-70
    13. Aad W. van der Vaart, Jon A. Wellner
      Pages 71-74
  3. Empirical Processes

    1. Front Matter
      Pages 79-79
    2. Aad W. van der Vaart, Jon A. Wellner
      Pages 80-94
    3. Aad W. van der Vaart, Jon A. Wellner
      Pages 95-106
    4. Aad W. van der Vaart, Jon A. Wellner
      Pages 107-121
    5. Aad W. van der Vaart, Jon A. Wellner
      Pages 122-126
    6. Aad W. van der Vaart, Jon A. Wellner
      Pages 127-133

About this book

Introduction

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, are naturally 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.

Keywords

Estimator Mathematics Maxima Maximum Minimum Random variable Statistics

Authors and affiliations

  • Aad W. van der Vaart
    • 1
  • Jon A. Wellner
    • 2
  1. 1.Department of Mathematics and Computer ScienceFree UniversityAmsterdamThe Netherlands
  2. 2.StatisticsUniversity of WashingtonSeattleUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4757-2545-2
  • Copyright Information Springer-Verlag New York 1996
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-1-4757-2547-6
  • Online ISBN 978-1-4757-2545-2
  • Series Print ISSN 0172-7397
  • About this book