The Chen-Stein method for Poisson functionals

Giovanni Peccati (1)

1 FSTC - Faculté des Sciences, de la Technologie et de la Communication

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-Itô 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 Lachièze-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.

Mots clés: Chen-Stein Method; Contractions; Malliavin Calculus; Poisson Limit Theorems; Poisson Space; Random Graphs; Stein's Method; Total Variation Distance; Wiener Chaos

Domaines: Probabilités [math.PR]

Giovanni Peccati. The Chen-Stein method for Poisson functionals. 2012. ⟨hal-00654235v3⟩