The Asymptotic Distribution of Intermediate Sums

Abstract

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 numbers such that $k_n \rightarrow \infty$ and $k_n/n \rightarrow 0$ as $n \rightarrow \infty$, and consider the sums $I_n(a, b) = \sum^{\lbrack bk_n\rbrack}_{i=\lbrack ak_n\rbrack+1} X_{n+1-i,n}$ of intermediate order statistics, where $0 < a < b$. We find necessary and sufficient conditions for the existence of constants $A_n > 0$ and $C_n$ such that $A^{-1}_n(I_n(a,b) - C_n)$ converges in distribution along subsequences of the positive integers $\{n\}$ to nondegenerate limits and completely describe the possible subsequential limiting distributions.