  Define the scaled empirical point process on an independent and identically
distributed sequence $\{Y_i: i\le n\}$ as the random point measure with masses
at $a_n^{-1} Y_i$. For suitable $a_n$ 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 $Y_i$ 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 $Y_i$ with little
additional effort. 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.
