Poisson process approximation under stabilization and Palm coupling
Annales Henri Lebesgue, Volume 5 (2022), pp. 1489-1534.

KeywordsFunctional limit theorems, Poisson process approximation, Kantorovich-Rubinstein distance, Point processes, Stein’s method, Glauber dynamics, Palm coupling, Stabilizing statistics, $k$-nearest neighbor balls, Morse critical points, Binomial point processes

### 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 Thäle (2016) and Otto (2020).

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