Poisson approximation with applications to stochastic geometry

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. 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.

Multivariate Poisson and Poisson process approximations with applications to Bernoulli sums and -statistics

“…Note that, since the indicator functions defined on N d 0 are Lipschitz continuous, for random vectors in N d 0 the Wasserstein distance dominates the total variation distance, and it is not hard to find sequences that converge in total variation distance but not in Wasserstein distance. Our goal is to extend the approach developed in [25] for the Poisson approximation of random variables to the multivariate case.…”

“…It should be noted that a bound that slightly improves (1.2) can easily be obtained as shown in the following section in Remark 2.1, which corresponds to (1.8) in [25,Theorem 1.3] when d = 1.…”

“…The proof of Theorem 3.1 is obtained by applying Theorem 1.1. When , Equation (1.1) corresponds to the condition required in [25, Theorem 1.2], which establishes sharper Poisson approximation results than the one obtained in the univariate case from Theorem 1.1. Therefore, for the sum of dependent Bernoulli random variables, a sharper bound for the Wasserstein distance can be derived from [25, Theorem 1.2], while for the total variation distance a bound may be deduced from [1, Theorem 1], [25, Theorem 1.2], or [32, Theorem 1].…”