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Gaussian Random Functions

  • Book
  • © 1995

Overview

Authors:
  1. M. A. Lifshits

    1. St. Petersburg University of Finance and Economics, St. Petersburg, Russia

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Part of the book series: Mathematics and Its Applications (MAIA, volume 322)

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Table of contents (19 chapters)

  1. Front Matter

    Pages i-xi
    Download chapter PDF

  2. Gaussian Distributions and Random Variables

    • M. A. Lifshits
    Pages 1-7

  3. Multi-Dimensional Gaussian Distributions

    • M. A. Lifshits
    Pages 8-15

  4. Covariances

    • M. A. Lifshits
    Pages 16-21

  5. Random Functions

    • M. A. Lifshits
    Pages 22-29

  6. Examples of Gaussian Random Functions

    • M. A. Lifshits
    Pages 30-40

  7. Modelling the Covariances

    • M. A. Lifshits
    Pages 41-52

  8. Oscillations

    • M. A. Lifshits
    Pages 53-67

  9. Infinite-Dimensional Gaussian Distributions

    • M. A. Lifshits
    Pages 68-83

  10. Linear Functionals, Admissible Shifts, and the Kernel

    • M. A. Lifshits
    Pages 84-100

  11. The Most Important Gaussian Distributions

    • M. A. Lifshits
    Pages 101-107

  12. Convexity and the Isoperimetric Property

    • M. A. Lifshits
    Pages 108-138

  13. The Large Deviations Principle

    • M. A. Lifshits
    Pages 139-155

  14. Exact Asymptotics of Large Deviations

    • M. A. Lifshits
    Pages 156-176

  15. Metric Entropy and the Comparison Principle

    • M. A. Lifshits
    Pages 177-210

  16. Continuity and Boundedness

    • M. A. Lifshits
    Pages 211-229

  17. Majorizing Measures

    • M. A. Lifshits
    Pages 230-245

  18. The Functional Law of the Iterated Logarithm

    • M. A. Lifshits
    Pages 246-257

  19. Small Deviations

    • M. A. Lifshits
    Pages 258-275

  20. Several Open Problems

    • M. A. Lifshits
    Pages 276-281
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Keywords

About this book

It is well known that the normal distribution is the most pleasant, one can even say, an exemplary object in the probability theory. It combines almost all conceivable nice properties that a distribution may ever have: symmetry, stability, indecomposability, a regular tail behavior, etc. Gaussian measures (the distributions of Gaussian random functions), as infinite-dimensional analogues of tht< classical normal distribution, go to work as such exemplary objects in the theory of Gaussian random functions. When one switches to the infinite dimension, some "one-dimensional" properties are extended almost literally, while some others should be profoundly justified, or even must be reconsidered. What is more, the infinite-dimensional situation reveals important links and structures, which either have looked trivial or have not played an independent role in the classical case. The complex of concepts and problems emerging here has become a subject of the theory of Gaussian random functions and their distributions, one of the most advanced fields of the probability science. Although the basic elements in this field were formed in the sixties-seventies, it has been still until recently when a substantial part of the corresponding material has either existed in the form of odd articles in various journals, or has served only as a background for considering some special issues in monographs.

Authors and Affiliations

  • St. Petersburg University of Finance and Economics, St. Petersburg, Russia

    M. A. Lifshits

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