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Convergence of Stochastic Processes

  • David Pollard

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

Table of contents

  1. Front Matter
    Pages i-xiv
  2. David Pollard
    Pages 1-5
  3. David Pollard
    Pages 122-137
  4. David Pollard
    Pages 138-169
  5. David Pollard
    Pages 170-187
  6. Back Matter
    Pages 189-218

About this book

Introduction

A more accurate title for this book might be: An Exposition of Selected Parts of Empirical Process Theory, With Related Interesting Facts About Weak Convergence, and Applications to Mathematical Statistics. The high points are Chapters II and VII, which describe some of the developments inspired by Richard Dudley's 1978 paper. There I explain the combinatorial ideas and approximation methods that are needed to prove maximal inequalities for empirical processes indexed by classes of sets or classes of functions. The material is somewhat arbitrarily divided into results used to prove consistency theorems and results used to prove central limit theorems. This has allowed me to put the easier material in Chapter II, with the hope of enticing the casual reader to delve deeper. Chapters III through VI deal with more classical material, as seen from a different perspective. The novelties are: convergence for measures that don't live on borel a-fields; the joys of working with the uniform metric on D[O, IJ; and finite-dimensional approximation as the unifying idea behind weak convergence. Uniform tightness reappears in disguise as a condition that justifies the finite-dimensional approximation. Only later is it exploited as a method for proving the existence of limit distributions. The last chapter has a heuristic flavor. I didn't want to confuse the martingale issues with the martingale facts.

Keywords

Brownian bridge Brownian motion Convergence Gaussian process Martingale Maxima Stochastic processes mathematical statistics random function statistics stochastic process

Authors and affiliations

  • David Pollard
    • 1
  1. 1.Department of StatisticsYale UniversityNew HavenUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4612-5254-2
  • Copyright Information Springer-Verlag New York 1984
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-1-4612-9758-1
  • Online ISBN 978-1-4612-5254-2
  • Series Print ISSN 0172-7397
  • Buy this book on publisher's site
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