Wiener chaos: Moments, cumulants and diagrams. A survey with computer implementation. (English) Zbl 1231.60003

Peccati, Giovanni; Taqqu, Murad S.

Bocconi & Springer Series 1. Milano: Bocconi University Press; Milano: Springer (ISBN 978-88-470-1678-1/hbk). xiii, 274 p. (2011).

This book is a survey which is devoted to the analysis of combinatorial structures in the Wiener chaos. Actually its central concern are the so-called “completely random measures” which generalize Gaussian and Poisson measures (and also contain Gamma measures, for example), with the aim of providing a unified treatment of moments and cumulants associated with multiple stochastic integrals (constructed from such a random measure).
In such \(d\)-dimensional multiple integrals, diagonal subsets of \(\mathbb{R}^d\) play an important role, since the product measure of a non-atomic random measure generally does not vanish on lower-dimensional linear subspaces. These significant diagonal subsets (of \(\mathbb{R}^d\)) are most conveniently described by partitions of \(\{1,\dots, d\}\), because of which they are used in order to provide a unified and systematic treatment of multiple stochastic integrals.
The combinatorial structures that are involved are those of lattices of partitions of \(\{1,\dots, d\}\), over which Möbius functions and associated inversion formulae are discussed. Moreover, this combinatorial standpoint, which is due to Rota and Wallstrom, is most convenient when dealing with diagram formulae.
Therefore, this book starts with basic properties of partition lattices and diagrams and develops from there a big quantity of formulae associated with products, moments and cumulants of multiple stochastic integrals with respect to a completely random measure.
Of course, the basic, main and recurrent examples are the Poisson and particularly the Gaussian measures, for which statements and effective computations are most numerous.
Several applications are given, in particular, to recent central limit theorems for random variables within a given chaos.
An explicit implementation in ‘Mathematica’ completes the text.

Reviewer: Jacques Franchi (Strasbourg)

MSC:

60-02	Research exposition (monographs, survey articles) pertaining to probability theory
60-04	Software, source code, etc. for problems pertaining to probability theory
60G99	Stochastic processes
60F05	Central limit and other weak theorems
60G57	Random measures
65H05	Numerical computation of solutions to single equations
60G15	Gaussian processes
05A18	Partitions of sets

Keywords:

random measure; multiple stochastic integral; Itô-Wiener chaos; partitions of a finite set; Möbius inversion formula; moments and cumulants; Gaussian and Poisson multiple integral; central limit theorems; Hermite and Charlier polynomials

Software:

Mathematica