Skip to main content
SpringerLink
Log in
Book cover

Wiener Chaos: Moments, Cumulants and Diagrams

A survey with Computer Implementation

  • Book
  • © 2011

Overview

Authors:
  1. Giovanni Peccati

    1. Mathematics Research Unit, University of Luxembourg, Luxembourg

    View author publications

    You can also search for this author in PubMed   Google Scholar

  2. Murad S. Taqqu

    1. Department of Mathematics and Statistics, Boston University, Boston

    View author publications

    You can also search for this author in PubMed   Google Scholar

  • A self-contained and probability-oriented introduction to the theory of lattice of partitions, with a unique software implementation that makes our book an ideal introduction to the field
  • A complete and self-contained combinatorial analysis of cumulants and diagram formulae, unique in its genre
  • An introduction to Wiener chaos, and a new combinatorial interpretation of recently proved limit theorems
  • Includes supplementary material: sn.pub/extras

Part of the book series: Bocconi & Springer Series (BS, volume 1)

This is a preview of subscription content, log in via an institution to check access.

Access this book

Log in via an institution
eBook USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Other ways to access

Licence this eBook for your library

Institutional subscriptions

Table of contents (12 chapters)

  1. Front Matter

    Pages I-XIII
    Download chapter PDF

  2. Introduction

    • Giovanni Peccati, Murad S. Taqqu
    Pages 1-6

  3. The lattice of partitions of a finite set

    • Giovanni Peccati, Murad S. Taqqu
    Pages 7-29

  4. Combinatorial expressions of cumulants and moments

    • Giovanni Peccati, Murad S. Taqqu
    Pages 31-44

  5. Diagrams and multigraphs

    • Giovanni Peccati, Murad S. Taqqu
    Pages 45-56

  6. Wiener-Itô integrals and Wiener chaos

    • Giovanni Peccati, Murad S. Taqqu
    Pages 57-108

  7. Multiplication formulae

    • Giovanni Peccati, Murad S. Taqqu
    Pages 109-125

  8. Diagram formulae

    • Giovanni Peccati, Murad S. Taqqu
    Pages 127-144

  9. From Gaussian measures to isonormal Gaussian processes

    • Giovanni Peccati, Murad S. Taqqu
    Pages 145-157

  10. Hermitian random measures and spectral representations

    • Giovanni Peccati, Murad S. Taqqu
    Pages 159-169

  11. Some facts about Charlier polynomials

    • Giovanni Peccati, Murad S. Taqqu
    Pages 171-175

  12. Limit theorems on the Gaussian Wiener chaos

    • Giovanni Peccati, Murad S. Taqqu
    Pages 177-202

  13. CLTs in the Poisson case: the case of double integrals

    • Giovanni Peccati, Murad S. Taqqu
    Pages 203-205

  14. Back Matter

    Pages 207-270
    Download chapter PDF
Back to top

Keywords

About this book

The concept of Wiener chaos generalizes to an infinite-dimensional setting the properties of orthogonal polynomials associated with probability distributions on the real line. It plays a crucial role in modern probability theory, with applications ranging from Malliavin calculus to stochastic differential equations and from probabilistic approximations to mathematical finance. This book is concerned with combinatorial structures arising from the study of chaotic random variables related to infinitely divisible random measures. The combinatorial structures involved are those of partitions of finite sets, over which Möbius functions and related inversion formulae are defined. This combinatorial standpoint (which is originally due to Rota and Wallstrom) provides an ideal framework for diagrams, which are graphical devices used to compute moments and cumulants of random variables. Several applications are described, in particular, recent limit theorems for chaotic random variables. An Appendix presents a computer implementation in MATHEMATICA for many of the formulae.

Reviews

From the book reviews:

“The objective of this book is to provide a detailed account of the combinatorial structures arising from the study of multiple stochastic integrals. … the presentation is very clear, with all the necessary proofs and examples. The authors clearly accomplish the three goals they list in the introduction (to provide a unified approach to the diagram method using set partition, to give a combinatorial analysis of multiple stochastic integrals in the most general setting, and to discuss chaotic limit theorems).” (Sergey V. Lototsky, Mathematical Reviews, Issue 2012 d)

“The book provides a comprehensive and detailed introduction to the theory of multiple stochastic integrals and some results for the Wiener chaos representation of random variables. … The book is recommended for anyone who needs a precise guidance to the theory.” (Gábor Szűcs, Acta Scientiarum Mathematicarum (Szeged), Vol. 77 (3-4), 2011)

Authors and Affiliations

  • Mathematics Research Unit, University of Luxembourg, Luxembourg

    Giovanni Peccati

  • Department of Mathematics and Statistics, Boston University, Boston

    Murad S. Taqqu

About the authors

Giovanni Peccati is a Professor of Stochastic Analysis and Mathematical Finance at Luxembourg University. Murad S. Taqqu is a Professor of Mathematics and Statistics at Boston University.

Bibliographic Information

Publish with us

Policies and ethics

Back to top

Search

Navigation