Poisson approximation with applications to stochastic geometry

TL;DR

This paper develops bounds for approximating distributions of integer-valued variables with Poisson distributions, using advanced coupling methods, and applies these results to problems in stochastic geometry.

Contribution

It introduces a generalized size-bias coupling technique and explicit bounds for distribution approximation, with a focus on stochastic geometry applications.

Findings

Explicit bounds on distribution differences in total variation and Wasserstein distances.

Effective approximation of minima and maxima of random variables using Poisson models.

Application of Chen-Stein and size-bias coupling methods to stochastic geometry problems.

Abstract

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.