Bocconi & Springer Series

Wiener Chaos: Moments, Cumulants and Diagrams

A survey with Computer Implementation

Authors: Peccati, Giovanni, Taqqu, Murad

  • 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
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eBook $109.00
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  • ISBN 978-88-470-1679-8
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  • Immediate eBook download after purchase
Hardcover $139.00
price for USA
  • ISBN 978-88-470-1678-1
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Softcover $139.00
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  • ISBN 978-88-470-5604-6
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Rent the ebook  
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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.

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.

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)


Table of contents (12 chapters)

  • Introduction

    Peccati, Giovanni (et al.)

    Pages 1-6

  • The lattice of partitions of a finite set

    Peccati, Giovanni (et al.)

    Pages 7-29

  • Combinatorial expressions of cumulants and moments

    Peccati, Giovanni (et al.)

    Pages 31-44

  • Diagrams and multigraphs

    Peccati, Giovanni (et al.)

    Pages 45-56

  • Wiener-Itô integrals and Wiener chaos

    Peccati, Giovanni (et al.)

    Pages 57-108

Buy this book

eBook $109.00
price for USA (gross)
  • ISBN 978-88-470-1679-8
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $139.00
price for USA
  • ISBN 978-88-470-1678-1
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Softcover $139.00
price for USA
  • ISBN 978-88-470-5604-6
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Rent the ebook  
  • Rental duration: 1 or 6 month
  • low-cost access
  • online reader with highlighting and note-making option
  • can be used across all devices
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Bibliographic Information

Bibliographic Information
Book Title
Wiener Chaos: Moments, Cumulants and Diagrams
Book Subtitle
A survey with Computer Implementation
Authors
Series Title
Bocconi & Springer Series
Series Volume
1
Copyright
2011
Publisher
Springer-Verlag Mailand
Copyright Holder
Springer Milan
eBook ISBN
978-88-470-1679-8
DOI
10.1007/978-88-470-1679-8
Hardcover ISBN
978-88-470-1678-1
Softcover ISBN
978-88-470-5604-6
Series ISSN
2039-1471
Edition Number
1
Number of Pages
XIII, 274
Topics