Best Attainable Rate of Joint Convergence of Extremes

Abstract

Suppose that the underlying distribution function is ultimately continuous and strictly increasing. In this case the rate of joint convergence of the k largest order statistics, equally standardized, in a sample of n i.i.d. random variables is O(k/n), uniformly in k if, and only if, the underlying distribution function is ultimately a generalized Pareto distribution. Thus, generalized Pareto distributions yield the best rate of joint convergence of extremes.