Gaussian approximation for sums of region-stabilizing scores

TL;DR

This paper extends Gaussian approximation results for sums of region-stabilizing scores of Poisson processes, allowing for more general stabilization regions, non-diffuse intensities, and unbounded scores, with applications to minimal points.

Contribution

It relaxes previous assumptions on stabilization regions and intensity measures, broadening the applicability of Gaussian approximation for Poisson functionals.

Findings

Provides bounds on convergence rates in Wasserstein and Kolmogorov distances.

Extends results to non-ball stabilization regions and non-diffuse intensities.

Applies to counting minimal points in high-dimensional Poisson processes.

Abstract

We consider the Gaussian approximation for functionals of a Poisson process that are expressible as sums of region-stabilizing (determined by the points of the process within some specified regions) score functions and provide a bound on the rate of convergence in the Wasserstein and the Kolmogorov distances. While such results have previously been shown in Lachi\`eze-Rey, Schulte and Yukich (2019), we extend the applicability by relaxing some conditions assumed there and provide further insight into the results. This is achieved by working with stabilization regions that may differ from balls of random radii commonly used in the literature concerning stabilizing functionals. We also allow for non-diffuse intensity measures and unbounded scores, which are useful in some applications. As our main application, we consider the Gaussian approximation of number of minimal points in a homogeneous Poisson process in $[0,1]^d$ with $d \ge 2$, and provide a presumably optimal rate of convergence.