The Asymptotic Distribution of Sums of Extreme Values from a Regularly Varying Distribution

Abstract

Let $X_{1,n} \leqq \cdots \leqq X_{n,n}$ be the order statistics of $n$ independent and identically distributed positive random variables with common distribution function $F$ satisfying $1 - F(x) = x^{-\alpha}L^\ast(x), x > 0$, where $\alpha$ is any positive number and $L^\ast$ is any function slowly varying at infinity. We give a complete description of the asymptotic distribution of the sum of the top $k_n$ extreme values $X_{n+1-k_n,n}, X_{n+2-k_n,n}, \ldots, X_{n,n}$ for any sequence of positive integers $k_n$ such that $k_n \rightarrow \infty$ and $k_n/n \rightarrow 0$ as $n \rightarrow \infty$.