An Upper Bound on the Binomial Process Approximation to the Exceedance Process

In this paper we deal with an approximation to the point process of exceedances over a higher threshold. We compute sharp bounds on the remainder terms when actual distributions of exceedances are replaced by appropriate generalized Pareto distributions (GPD). The bound will be formulated in terms of the von Mises function. The ultimate as well as the penultimate approximations are considered; in the latter case the shape parameter of the approximating GPD depends on the threshold.