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This article compares the distributions of integer-valued random variablesand Poisson random variables. It considers the total variation and theWasserstein distance and provides, in particular, explicit bounds on thepointwise difference between the cumulative distribution functions. Specialattention is dedicated to estimating the difference when the cumulativedistribution functions are evaluated at 0. This permits to approximate theminimum (or maximum) of a collection of random variables by a suitable randomvariable in the Kolmogorov distance. The main theoretical results are obtainedby combining the Chen-Stein method with size-bias coupling and a generalizationof size-bias coupling for integer-valued random variables developed herein. Awide variety of applications are then discussed with a focus on stochasticgeometry.

Poisson approximation with applications to stochastic geometry