A note on the asymptotic normality of sums of extreme values

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).