A note on the asymptotic normality of sums of extreme values

TL;DR

This paper proves that sums of the largest extreme values from a sequence of variables in the Gumbel domain, when properly normalized, follow a normal distribution asymptotically, extending previous results.

Contribution

It extends existing results by showing the asymptotic normality of sums of upper extreme values for variables in the Gumbel domain of attraction.

Findings

Sum of upper extreme values converges to normal distribution after normalization.

Results apply to sequences where the number of extremes grows but remains small relative to total.

Extends prior work by Csörgo and Mason (1985).

Abstract

Let $X_1$, $X_2$,... be a sequence of independent random variables with common distribution function $F$ in the domain of attraction of a Gumbel extreme value distribution and for each integer $n\geq 1$, let $X_{1,n} \leq ... X_{n,n}$ denote the order statistics based on the first $n$ of these random variables. Along with related results it is shown that for any sequence of positive integers $k_n \rightarrow +\infty$ and $k_{n}/n \rightarrow 0$ as $n \rightarrow 0$ the sum of the upper $k_n$ extreme values $X_{n-k_{n},n}+...+X_{n,n}$, when properly centered and normalized, converges in distribution to a standard normal random variable $N(0, 1)$. These results constitute an extension of results by S. Cs\"{o}rg\H{o} and D.M. Mason (1985).