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Multiple Wiener-Itô Integrals

With Applications to Limit Theorems

  • Book
  • © 2014

Overview

Authors:
  1. Péter Major

    1. Hungarian Academy of Sciences Alfréd Rényi Mathematical Institute, Budapest, Hungary

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Part of the book series: Lecture Notes in Mathematics (LNM, volume 849)

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Table of contents (9 chapters)

  1. Front Matter

    Pages i-xiii
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  2. On a Limit Problem

    • Péter Major
    Pages 1-8

  3. Wick Polynomials

    • Péter Major
    Pages 9-14

  4. Random Spectral Measures

    • Péter Major
    Pages 15-26

  5. Multiple Wiener–Itô Integrals

    • Péter Major
    Pages 27-42

  6. The Proof of Itô’s Formula: The Diagram Formula and Some of Its Consequences

    • Péter Major
    Pages 43-64

  7. Subordinated Random Fields: Construction of Self-similar Fields

    • Péter Major
    Pages 65-79

  8. On the Original Wiener–Itô Integral

    • Péter Major
    Pages 81-86

  9. Non-central Limit Theorems

    • Péter Major
    Pages 87-112

  10. History of the Problems: Comments

    • Péter Major
    Pages 113-122

  11. Back Matter

    Pages 123-128
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Keywords

About this book

The goal of this Lecture Note is to prove a new type of limit theorems for normalized sums of strongly dependent random variables that play an important role in probability theory or in statistical physics. Here non-linear functionals of stationary Gaussian fields are considered, and it is shown that the theory of Wiener–Itô integrals provides a valuable tool in their study. More precisely, a version of these random integrals is introduced that enables us to combine the technique of random integrals and Fourier analysis. The most important results of this theory are presented together with some non-trivial limit theorems proved with their help.

This work is a new, revised version of a previous volume written with the goal of giving a better explanation of some of the details and the motivation behind the proofs. It does not contain essentially new results; it was written to give a better insight to the old ones. In particular, a more detailed explanation of generalized fields is included to show that what is at the first sight a rather formal object is actually a useful tool for carrying out heuristic arguments.

Authors and Affiliations

  • Hungarian Academy of Sciences Alfréd Rényi Mathematical Institute, Budapest, Hungary

    Péter Major

Bibliographic Information

  • Book Title: Multiple Wiener-Itô Integrals

  • Book Subtitle: With Applications to Limit Theorems

  • Authors: Péter Major

  • Series Title: Lecture Notes in Mathematics

  • DOI: https://doi.org/10.1007/978-3-319-02642-8

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer International Publishing Switzerland 2014

  • Softcover ISBN: 978-3-319-02641-1Published: 16 December 2013

  • eBook ISBN: 978-3-319-02642-8Published: 02 December 2013

  • Series ISSN: 0075-8434

  • Series E-ISSN: 1617-9692

  • Edition Number: 2

  • Number of Pages: XIII, 126

  • Number of Illustrations: 4 b/w illustrations

  • Topics: Probability Theory and Stochastic Processes

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