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May 2016 Functional Poisson approximation in Kantorovich–Rubinstein distance with applications to U-statistics and stochastic geometry
Laurent Decreusefond, Matthias Schulte, Christoph Thäle
Ann. Probab. 44(3): 2147-2197 (May 2016). DOI: 10.1214/15-AOP1020

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.

Citation

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Laurent Decreusefond. Matthias Schulte. Christoph Thäle. "Functional Poisson approximation in Kantorovich–Rubinstein distance with applications to U-statistics and stochastic geometry." Ann. Probab. 44 (3) 2147 - 2197, May 2016. https://doi.org/10.1214/15-AOP1020

Information

Received: 1 June 2014; Revised: 1 March 2015; Published: May 2016
First available in Project Euclid: 16 May 2016

zbMATH: 1347.60027
MathSciNet: MR3502603
Digital Object Identifier: 10.1214/15-AOP1020

Subjects:
Primary: 60F17, 60G55
Secondary: 60D05, 60E07, 60H07

Rights: Copyright © 2016 Institute of Mathematical Statistics

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Vol.44 • No. 3 • May 2016
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