Abstract. Let $X_{1,n}\le\ldots \le X_{n,n}$ be the order statistics of $n$ independent random variables with a common distribution function $F$ and let $k_n$ be positive integers such that $k_n\to\infty $ and $k_n/n\to\alpha $ as $n\to\infty $, where $0\le\alpha < 1$. Given a known function $f$ and known constants $d_{i,n}$, $1\le i\le n$, that are all specified by the statistician, consider the linear combinations $T_n(k,k_n)=\sum_{i=k+1}^{k_n} d_{n+1-i,n}f(X_{n+1-i,n})$ of extreme values, where $k\ge0$ is any fixed integer. We find necessary and sufficient conditions for the existence of normalizing and centering constants $A_n>0$ and $C_n$ such that the sequence $E_{n}=\{T_n(k,k_n)-C_n\} /A_n$ converges in distribution along subsequences of the integers $\{n\} $ to non-degenerate limits and completely describe the possible subsequential limiting distributions. We also give a necessary and sufficient condition for the existence of $A_n$ and $C_n$ such that $E_n$ be asymptotically normal along a given subsequence.

AMS Subject Classification (1991): 60F05; 62E20; 62G30

Received September 14, 1992. (Registered under 5544/2009.)