Wiener Chaos: Moments, Cumulants and Diagrams

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.

The concept of Wiener chaos generalizes to an infinite-dimensional setting theproperties of orthogonal polynomials associated with probability distributionson the real line. It plays a crucial role in modern probability theory, with applicationsranging from Malliavin calculus to stochastic differential equations and fromprobabilistic approximations to mathematical finance.This book is concerned with combinatorial structures arising from the studyof 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 usedto 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.

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