Poisson limits for empirical point processes

Define the scaled empirical point process on an independent and identically distributed sequence {Yi : i ≤ n} as the random point measure with masses at a n Yi. For suitable an we obtain the weak limit of these point processes through a novel use of a dimension-free method based on the convergence of compensators of multiparameter martingales. The method extends previous results in several directions. We obtain limits at points where the density of Yi may be zero, but has regular variation. The joint limit of the empirical process evaluated at distinct points is given by independent Poisson processes. These results also hold for multivariate Yi with little additional effort. Applications are provided both to nearestneighbour density estimation in high dimensions, and to the asymptotic behaviour of multivariate extremes such as those arising from bivariate normal copulas.