Poisson process approximation under stabilization and Palm coupling

TL;DR

This paper develops new bounds for Poisson process approximation of stabilizing functionals in stochastic geometry, using Stein's method and Palm coupling, applicable to unbounded interaction ranges.

Contribution

It introduces novel Poisson approximation bounds for stabilizing functionals with unbounded interaction, extending previous results with new coupling and stabilization techniques.

Findings

Bounds derived using the generator approach to Stein's method.

Applicable to functionals with unbounded interaction ranges.

Extends previous Poisson approximation results in stochastic geometry.

Abstract

We present new Poisson process approximation results for stabilizing functionals of Poisson and binomial point processes. These functionals are allowed to have an unbounded range of interaction and encompass many examples in stochastic geometry. Our bounds are derived for the Kantorovich-Rubinstein distance using the generator approach to Stein's method. We give different types of bounds for different point processes. While some of our bounds are given in terms of coupling of the point process with its Palm version, the others are in terms of the local dependence structure formalized via the notion of stabilization. We provide two supporting examples for our new framework - one is for Morse critical points of the distance function, and the other is for large k-nearest neighbor balls. Our bounds considerably extend the results in Barbour and Brown (1992), Decreusefond, Schulte and Thale (2016) and Otto (2020).