Moments and central limit theorems for some multivariate Poisson functionals

TL;DR

This paper develops formulas for moments and cumulants of multivariate Poisson functionals, proves a multivariate central limit theorem, and provides Berry-Esseen bounds, with applications to geometric functionals of intersection processes.

Contribution

It introduces a direct approach to derive moments and cumulants for multivariate Poisson functionals and establishes a multivariate CLT with bounds, extending previous results.

Findings

Derived explicit moment and cumulant formulas for Poisson functionals

Proved a multivariate central limit theorem with Berry-Esseen bounds

Applied results to geometric functionals of intersection processes in stochastic geometry

Abstract

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-It\^o 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.\ 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 $\R^d$ are discussed.