Peccati G., Taqqu M.S. Wiener Chaos: Moments, Cumulants and Diagrams: A Survey with Computer Implementation

Mailand: Springer, 2011. — 290 p.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.The lattice of partitions of a finite set
Combinatorial expressions of cumulants and moments
Diagrams and multigraphs
Wiener-Itô integrals and Wiener chaos
Multiplication formulae
Diagram formulae
From Gaussian measures to isonormal Gaussian processes
Hermitian random measures and spectral representations
Some facts about Charlier polynomials
Limit theorems on the Gaussian Wiener chaos
CLTs in the Poisson case: the case of double integrals