Some notes on Poisson limits for empirical point processes

Define the scaled empirical point process on an independent and iden-tically distributed sequence {Yi: i ≤ n} on IRd as the random point measure with masses at a−1n 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 lim-its at points where the density of Yi may be zero, but has regular varia-tion. The joint limit of the empirical process evaluated at distinct points is given by independent Poisson processes. Applications are provided both to nearest-neighbour density estimation in high dimensions, and to the asymptotic behaviour of multivariate extremes such as those arising from bivariate normal copulas