Von Mises conditions, [delta]-neighborhoods and rates of convergence for maxima

It is well known that the rate of convergence, w.r.t. the variational distance, of normalized maxima to an extreme value distribution is of order O(n-[delta]), if the underlying distribution function F belongs to a certain [delta]-neighborhood of a generalized Pareto distribution. In this paper, we prove that the converse is true under mild monotonicity conditions on a certain von Mises term.