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

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