Some notes on Poisson limits for empirical point processes

The authors define the scaled empirical point process. They 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. They obtain limits at points where the density may be zero, but has regular variation. The joint limit of the empirical process evaluated at distinct points is given by independent Poisson processes. They provide applications 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. The Canadian Journal of Statistics 37: 347-360; 2009 (C) 2009 Statistical Society of Canada