The Annals of Probability

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

Laurent Decreusefond, Matthias Schulte, and Christoph Thäle

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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.

Article information

Source
Ann. Probab., Volume 44, Number 3 (2016), 2147-2197.

Dates
Received: June 2014
Revised: March 2015
First available in Project Euclid: 16 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.aop/1463410041

Digital Object Identifier
doi:10.1214/15-AOP1020

Mathematical Reviews number (MathSciNet)
MR3502603

Zentralblatt MATH identifier
1347.60027

Subjects
Primary: 60F17: Functional limit theorems; invariance principles 60G55: Point processes
Secondary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60E07: Infinitely divisible distributions; stable distributions 60H07: Stochastic calculus of variations and the Malliavin calculus

Keywords
Binomial process configuration space functional limit theorem Glauber dynamics Kantorovich–Rubinstein distance Malliavin formalism Poisson process Stein’s method stochastic geometry U-statistics

Citation

Decreusefond, Laurent; Schulte, Matthias; Thäle, Christoph. Functional Poisson approximation in Kantorovich–Rubinstein distance with applications to U-statistics and stochastic geometry. Ann. Probab. 44 (2016), no. 3, 2147--2197. doi:10.1214/15-AOP1020. https://projecteuclid.org/euclid.aop/1463410041


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