Let $X_{1,n}\leq \cdots \leq X_{n,n}$ be the order statistics of $n$ independent random variables with a common distribution function $F$ and let $k_n$ be positive integers such that $k_n \rightarrow \infty$ and $k_n/n \rightarrow \alpha$ as $n \rightarrow \infty$, where $0 \leq \alpha < 1$. We find necessary and sufficient conditions for the existence of normalizing and centering constants $A_n > 0$ and $C_n$ such that the sequence $E_n = \frac{1}{A_n}\bigg\{\sum^{k_n}_{i=1} X_{n+1-i,n} - C_n\bigg\}$ converges in distribution along subsequences of the integers $\{n\}$ to nondegenerate 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, and with suitable $A_n$ and $C_n$ determine the limiting distributions of $E_n$ along the whole sequence $\{n\}$ when $F$ is in the domain of attraction of an extreme value distribution.