Poisson and Gaussian approximation of weighted local empirical processes

We consider the local empirical process indexed by sets, a substantial generalization of the well-studied uniform tail empirical process. We show that the weak limit of weighted versions of this process is Poisson under certain conditions, whereas it is Gaussian in other situations. Our main theorems provide many new results as well as a unified approach to a number of asymptotic distributional results for weighted empirical processes, which up to now appeared to be isolated facts. Our results have applications in 'local' statistical procedures; we will, in particular, show their usefulness in multivariate extreme value theory.