Central limit theorems for sums of extreme values

Abstract

Abstract Given a sequence of non-negative independent and identically distributed random variables, we determine conditions on the common distribution such that the sum of appropriately normalized and centred upper k n extreme values based on the first n random variables converges in distribution to a normal random variable, where k n → ∞ and k n/ n → 0 as n → ∞. The probabilistic problem is motivated by recent statistical work on the estimation of the exponent of a regularly varying distribution function. Our main tool is a new Brownian bridge approximation to the uniform empirical and quantile processes in weighted supremum norms.