Representations and limit theorems for extreme value distributions

Abstract

Let Xn1 ≦ Xn2 ≦ ··· ≦ Xnn denote the order statistics from a sample of n independent, identically distributed random variables, and suppose that the variables Xnn, Xn, n–1, ···, when suitably normalized, have a non-trivial limiting joint distribution ξ1, ξ2, ···, as n → ∞. It is well known that the limiting distribution must be one of just three types. We provide a canonical representation of the stochastic process {ξn, n ≧ 1} in terms of exponential variables, and use this representation to obtain limit theorems for ξ n as n →∞.