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This article derives quantitative limit theorems for multivariate Poisson andPoisson process approximations. Employing the solution of Stein's equation forPoisson random variables, we obtain an explicit bound for the multivariatePoisson approximation of random vectors in the Wasserstein distance. The boundis then utilized in the context of point processes to provide a Poisson processapproximation result in terms of a new metric called $d_\pi$, stronger than thetotal variation distance, defined as the supremum over all Wassersteindistances between random vectors obtained by evaluating the point processes onarbitrary collections of disjoint sets. As applications, the multivariatePoisson approximation of the sum of $m$-dependent Bernoulli random vectors, thePoisson process approximation of point processes of $U$-statistic structure andthe Poisson process approximation of point processes with Papangelou intensityare considered. Our bounds in $d_\pi$ are as good as those already available inthe literature.

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