  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.
