A note on the asymptotic normality of sums of extreme values

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 $n\ge 1$, let $X_{1,n}\le ...X_{n,n}$ 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 $k_n\to +\infty$ and $k_n/n\to 0$ as $n\to 0$ 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. Csörgő and D.M. Mason (1985).

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