This paper deals with Poisson processes on an arbitrary measurable space. Using a direct approach, we derive formulae for moments and cumulants of a vector of multiple Wiener-Ito integrals with respect to the compensated Poisson process. Second, a multivariate central limit theorem is shown for a vector whose components admit a finite chaos expansion of the type of a Poisson U-statistic. The approach is based on recent results of Peccati et al. (2010),combining Malliavin calculus and Stein's method, and also yields Berry-Esseen type bounds. As applications, moment formulae and central limit theorems for general geometric functionals of intersection processes associated with a stationary Poisson process of k-dimensional flats in Rd are discussed.
- Berry-Esseen type bounds; central limit theorem; intersection process; multiple Wiener-Ito integral; Poisson process; Poisson flat process; product formula; stochastic geometry; Wiener-Ito chaos expansion