Functional Poisson approximation in Kantorovich-Rubinstein distance with applications to U-statistics and stochastic geometry

Abstract

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 dynamics 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.

arXiv commentPublished at http://dx.doi.org/10.1214/15-AOP1020 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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• Stein's method, Palm theory and Poisson process approximation
• The Chen-Stein method for Poisson functionals
• Gibbs point process approximation: Total variation bounds using Stein's method
• Upper bounds on Rubinstein distances on configuration spaces and applications
• Upper bounds for spatial point process approximations
• Stein's method and Poisson process approximation for a class of Wasserstein metrics

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