Laws of Large Numbers for Sums of Extreme Values

Abstract

Let $X_1, X_2, \cdots$, be a sequence of nonnegative i.i.d. random variables with common distribution $F$, and for each $n \geq 1$ let $X_{1n} \leq \cdots \leq X_{nn}$ denote the order statistics based on $X_1, \cdots, X_n$. Necessary and sufficient conditions are obtained for averages of the extreme values $X_{n+1-i, n}i = 1, \cdots, k_n + 1$ of the form: $k^{-1}_n \sum^{k_n}_{i = 1} (X_{n+1-i, n} - X_{n-k_n,n})$, where $k_n \rightarrow\infty$ and $n^{-1}k_n \rightarrow 0$, to converge in probability or almost surely to a finite positive constant. In the process, characterizations are given of the classes of distributions with regularly varying upper tails and of distributions with "exponential-like" upper tails.