Stein's method and Poisson process approximation for a class of Wasserstein metrics

TL;DR

This paper develops bounds for Poisson process approximation using Stein's method in Wasserstein metrics, extending previous work to a broader class of metrics and demonstrating applications to specific point process models.

Contribution

It introduces upper bounds for Poisson process approximation in a generalized Wasserstein metric $d_2^{(p)}$, extending existing results for $p=1$ to general $p$ and illustrating their usefulness.

Findings

Bounds control differences in expectations of $p$th order statistics

Application to 2-runs and hard core models

Extension of Wasserstein metrics for point process approximation

Abstract

Based on Stein's method, we derive upper bounds for Poisson process approximation in the $L_1$-Wasserstein metric $d_2^{(p)}$, which is based on a slightly adapted $L_p$-Wasserstein metric between point measures. For the case $p=1$, this construction yields the metric $d_2$ introduced in [Barbour and Brown Stochastic Process. Appl. 43 (1992) 9--31], for which Poisson process approximation is well studied in the literature. We demonstrate the usefulness of the extension to general $p$ by showing that $d_2^{(p)}$-bounds control differences between expectations of certain $p$th order average statistics of point processes. To illustrate the bounds obtained for Poisson process approximation, we consider the structure of 2-runs and the hard core model as concrete examples.