Wiener Chaos: Moments, Cumulants and Diagrams: A survey with Computer Implementation

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Giovanni Peccati Murad S. Taqqu

Wiener Chaos

Moments, Cumulants and Diagrams: A survey with Computer Implementation

Giovanni Peccati, Murad S. Taqqu, Peccati

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English

Springer Verlag
28 December 2010

Economics & Business; Discrete mathematics; Probability & statistics; Combinatorics & graph theory; Applied mathematics; Stochastics

Series: Bocconi & Springer Series

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 Moebius 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.

By: Giovanni Peccati, Murad S. Taqqu, Peccati
Imprint: Springer Verlag
Country of Publication: Italy
Edition: 2011
Volume: 1
Dimensions: Height: 235mm, Width: 155mm, Spine: 17mm
Weight: 606g
ISBN: 9788847016781
ISBN 10: 8847016789
Series: Bocconi & Springer Series
Pages: 274
Publication Date: 28 December 2010
Audience: Professional and scholarly , Undergraduate
Format: Hardback
Publisher's Status: Active

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 for Wiener Chaos: Moments, Cumulants and Diagrams: A survey with Computer Implementation

From the 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)