Publisher DOI: 10.1214/21-EJP723
Title: Poisson approximation with applications to stochastic geometry
Language: English
Authors: Pianoforte, Federico 
Schulte, Matthias 
Keywords: Chen-Stein method; Exponential approximation; Extremes; Poisson approximation; Poisson-Voronoi tessellations; Runs; Size-bias coupling; Stochastic geometry; U-statistics
Issue Date: 2021
Source: Electronic Journal of Probability 26 : 149 (2021)
Abstract (english): 
This article compares the distributions of integer-valued random variables and Poisson random variables. It considers the total variation and the Wasserstein distance and provides, in particular, explicit bounds on the pointwise difference between the cumulative distribution functions. Special attention is dedicated to estimating the difference when the cumulative distribution functions are evaluated at 0. This permits to approximate the minimum (or maximum) of a collection of random variables by a suitable random variable in the Kolmogorov distance. The main theoretical results are obtained by combining the Chen-Stein method with size-bias coupling and a generalization of size-bias coupling for integer-valued random variables developed herein. A wide variety of applications are then discussed with a focus on stochastic geometry. In particular, transforms of the minimal circumscribed radius and the maximal inradius of Poisson-Voronoi tessellations as well as the minimal inter-point distance of the points of a Poisson process are considered and bounds for their Kolmogorov distances to extreme value distributions are derived.
ISSN: 1083-6489
Journal: Electronic journal of probability 
Institute: Mathematik E-10 
Document Type: Article
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