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

Abstract

Based on Stein’s method, we derive upper bounds for Poisson process approximation in the L₁-Wasserstein metric d₂^(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 d₂ 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 d₂^(p)-bounds control differences between expectations of certain pth 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.