The Chen-Stein method for Poisson functionals

TL;DR

This paper introduces a new inequality on the Poisson space that provides bounds for the total variation distance between a given integer-valued random variable and a Poisson distribution, with applications to Wiener-Itô integrals and geometric graph statistics.

Contribution

It combines Malliavin calculus and the Chen-Stein method to study Poisson approximations on configuration spaces, offering new bounds and convergence conditions.

Findings

Derived a general inequality for Poisson approximation on the Poisson space.

Provided explicit rates of convergence for geometric graph statistics.

Established conditions for convergence of Wiener-Itô integrals to Poisson distributions.

Abstract

We establish a general inequality on the Poisson space, yielding an upper bound for the distance in total variation between the law of a regular random variable with values in the integers and a Poisson distribution. Several applications are provided, in particular: (i) to deduce a set of sufficient conditions implying that a sequence of (suitably shifted) multiple Wiener-It\^o integrals converges in distribution to a Poisson random variable, and (ii) to compute explicit rates of convergence for the Poisson approximation of statistics associated with geometric random graphs with sparse connections (thus refining some findings by Lachi\`eze-Rey and Peccati (2011)). This is the first paper studying Poisson approximations on configuration spaces by combining the Malliavin calculus of variations and the Chen-Stein method.