The Chen-Stein method for Poisson functionals

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.