Poisson approximation of empirical processes

This paper concerns the statistical modelling of Poisson processes. It consists of an intermediate step of replacing truncated empirical point processes by Poisson processes having identical intensity measures and, secondly, the building of models of limiting Poisson processes. The second step is exemplified with models of extreme value processes. In that context, it is shown that point processes of exceedances converge to some extreme value process in terms of the variational distance if, and only if, the underlying distribution belongs to the strong domain of attraction of an extreme value distribution. AMS 1980 Subject Classifications: Primary 60G55, 62B15