Wiener Chaos: Moments, Cumulants and Diagrams

1. Front Matter

   Pages I-XIII

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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

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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.

• Diagram formulae
• Lattices of partitions
• Limit theorems
• Moments and cumulants
• Wiener chaos
• quantitative finance
• combinatorics

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)

• Mathematics Research Unit, University of Luxembourg, Luxembourg

  Giovanni Peccati

• Department of Mathematics and Statistics, Boston University, Boston

  Murad S. Taqqu

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.

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