Compound Poisson process approximation under β-mixing and stabilization

The Annals of Applied Probability, vol. 35, pp. 2544–2569

Abstract

We establish Poisson and compound Poisson approximations for stabilizing statistics of β-mixing point processes and give explicit rates of convergence. Our findings are based on a general estimate of the total variation distance of a stationary β-mixing process and its Palm version. As main contributions, this article (i) extends recent results on Poisson process approximation to non-Poisson/binomial input, (ii) gives concrete bounds for compound Poisson process approximation in a Wasserstein distance and (iii) illustrates the applicability of the general result in an example on minimal angles in the stationary Poisson–Delaunay tessellation. The latter is among the first (nontrivial) situations in stochastic geometry, where compound Poisson approximation can be established with explicit extremal index and cluster size distribution.

Authors 2

1. Nicolas Chenavier

   Université du littoral côte d'opale

   Affiliation as printed

   Department of Mathematics, Université du Littoral Côte d’Opale

2. Moritz Otto Aachen Mathematical Institute

   Leiden University

   Affiliation as printed

   Mathematical Institute, Leiden University