Functional Poisson approximation in Rubinstein distance

Laurent Decreusefond (1, 2) , Matthias Schülte (3) , Christoph Thäle (4)

1 LTCI - Laboratoire Traitement et Communication de l'Information
2 Télécom ParisTech
3 KIT - Karlsruhe Institute of Technology
4 Faculty of Mathematics [Bochum]

A Poisson or a binomial process on an abstract state space and a symmetric function $f$ acting on $k$-tuples of its points are considered. They induce a point process on the target space of $f$. The main result is a functional limit theorem which provides an upper bound for an optimal transportation distance between the image process and a Poisson process on the target space. The technical background are a version of Stein's method for Poisson process approximation, a Glauber dynamic representation for the Poisson process and the Malliavin formalism. As applications of the main result, error bounds for approximations of U-statistics by Poisson, compound Poisson and stable random variables are derived and examples from stochastic geometry are investigated.

Mots clés: Binomial process; configuration space; functional limit theorem; Glauber dynamics; Poisson process; Rubinstein distance; Stein's method; stochastic geometry; U-statistics

Domaines: Probabilités [math.PR]

Laurent Decreusefond, Matthias Schülte, Christoph Thäle. Functional Poisson approximation in Rubinstein distance. 2014. ⟨hal-01010967v1⟩