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High Dimensional Probability II

  • Evarist Giné
  • David M. Mason
  • Jon A. Wellner
Conference proceedings

Part of the Progress in Probability book series (PRPR, volume 47)

Table of contents

  1. Front Matter
    Pages i-xi
  2. Inequalities

    1. Front Matter
      Pages 1-1
    2. Victor H. de la Peña, Michael J. Klass, Tze Leung Lai
      Pages 3-11
    3. Evarist Giné, Rafał Latała, Joel Zinn
      Pages 13-38
    4. Iosif Pinelis
      Pages 49-63
  3. General Empirical Process Theory

    1. Front Matter
      Pages 77-77
    2. Peter Gaenssler, Daniel Rost
      Pages 79-87
    3. Dragan Radulović, Marten Wegkamp
      Pages 89-105
  4. Gaussian Processes

  5. Strong Approximation and Embedding

    1. Front Matter
      Pages 181-181
    2. Paul Deheuvels, Uwe Einmahl, David M. Mason
      Pages 183-205
    3. Alexander I. Sakhanenko
      Pages 223-245
  6. The Law of the Integrated Logarithm

  7. Large Deviations

    1. Front Matter
      Pages 279-279
    2. Ludovic Menneteau
      Pages 293-312
  8. Sums of Independent Random Variables in High Dimensions

    1. Front Matter
      Pages 313-313
    2. Davar Khoshnevisan, Yimin Xiao
      Pages 329-345
  9. Random Vectors and Processes

    1. Front Matter
      Pages 347-347
    2. Joël De Connick, Zbigniew J. Jurek
      Pages 349-357
    3. Jolanta K. Misiewicz, Krzysztof Tabisz
      Pages 359-366
    4. Jerzy Szulga, Fred Molz
      Pages 377-387
  10. Function Estimation

    1. Front Matter
      Pages 389-389
    2. Eduard Belitser, Sara van de Geer
      Pages 391-403
    3. Vladimir Koltchinskii, Dmitriy Panchenko
      Pages 443-457
  11. Statistics in a Multidimensional Setting

  12. Back Matter
    Pages 511-512

About these proceedings

Introduction

High dimensional probability, in the sense that encompasses the topics rep­ resented in this volume, began about thirty years ago with research in two related areas: limit theorems for sums of independent Banach space valued random vectors and general Gaussian processes. An important feature in these past research studies has been the fact that they highlighted the es­ sential probabilistic nature of the problems considered. In part, this was because, by working on a general Banach space, one had to discard the extra, and often extraneous, structure imposed by random variables taking values in a Euclidean space, or by processes being indexed by sets in R or Rd. Doing this led to striking advances, particularly in Gaussian process theory. It also led to the creation or introduction of powerful new tools, such as randomization, decoupling, moment and exponential inequalities, chaining, isoperimetry and concentration of measure, which apply to areas well beyond those for which they were created. The general theory of em­ pirical processes, with its vast applications in statistics, the study of local times of Markov processes, certain problems in harmonic analysis, and the general theory of stochastic processes are just several of the broad areas in which Gaussian process techniques and techniques from probability in Banach spaces have made a substantial impact. Parallel to this work on probability in Banach spaces, classical proba­ bility and empirical process theory were enriched by the development of powerful results in strong approximations.

Keywords

Bootstrapping Gaussian process Markov process Martingale Maxima Random variable Stochastic processes linear optimization local time probability statistics stochastic process

Editors and affiliations

  • Evarist Giné
    • 1
  • David M. Mason
    • 3
  • Jon A. Wellner
    • 2
  1. 1.Departments of Mathematics & StatisticsUniversity of ConnecticutStorrsUSA
  2. 2.Department of StatisticsUniversity of WashingtonSeattleUSA
  3. 3.Department of Food and Resource Econ.University of DelawareNewarkUSA

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