Wiener Chaos: Moments, Cumulants and Diagrams

A survey with computer implementation

  • Giovanni Peccati
  • Murad S. Taqqu
Part of the Bocconi & Springer Series book series (BS, volume 1)

Table of contents

  1. Front Matter
    Pages I-XIII
  2. Giovanni Peccati, Murad S. Taqqu
    Pages 1-6
  3. Giovanni Peccati, Murad S. Taqqu
    Pages 7-29
  4. Giovanni Peccati, Murad S. Taqqu
    Pages 31-44
  5. Giovanni Peccati, Murad S. Taqqu
    Pages 45-56
  6. Giovanni Peccati, Murad S. Taqqu
    Pages 57-108
  7. Giovanni Peccati, Murad S. Taqqu
    Pages 109-125
  8. Giovanni Peccati, Murad S. Taqqu
    Pages 127-144
  9. Giovanni Peccati, Murad S. Taqqu
    Pages 145-157
  10. Giovanni Peccati, Murad S. Taqqu
    Pages 159-169
  11. Giovanni Peccati, Murad S. Taqqu
    Pages 171-175
  12. Giovanni Peccati, Murad S. Taqqu
    Pages 177-202
  13. Giovanni Peccati, Murad S. Taqqu
    Pages 203-205
  14. Back Matter
    Pages 207-270

About this book

Introduction

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.

Keywords

Diagram formulae Lattices of partitions Limit theorems Moments and cumulants Wiener chaos

Authors and affiliations

  • Giovanni Peccati
    • 1
  • Murad S. Taqqu
    • 2
  1. 1.Mathematics Research UnitUniversity of LuxembourgLuxembourg
  2. 2.Department of Mathematics and StatisticsBoston UniversityBoston

Bibliographic information

  • DOI https://doi.org/10.1007/978-88-470-1679-8
  • Copyright Information Springer Milan 2011
  • Publisher Name Springer, Milano
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-88-470-1678-1
  • Online ISBN 978-88-470-1679-8
  • Series Print ISSN 2039-1471
  • About this book