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

TL;DR

This paper develops a functional limit theorem for Poisson process approximation using Stein's method, with applications to U-statistics and stochastic geometry, providing explicit error bounds and new insights into process convergence.

Contribution

It introduces a novel functional limit theorem with explicit bounds for Poisson process approximation in the Kantorovich-Rubinstein distance, combining Stein's method, Glauber dynamics, and Malliavin calculus.

Findings

Provides explicit error bounds for Poisson approximation of U-statistics.

Applies the main theorem to stochastic geometry examples.

Establishes a new framework for process convergence analysis.

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.