A Law of the Iterated Logarithm for Sums of Extreme Values from a Distribution with a Regularly Varying Upper Tail

Abstract

Let $X_1, X_2,\ldots$ be independent observations from a distribution with a regularly varying upper tail with index $a$ greater than 2. For each $n \geq 1$, let $X_{1,n} \leq \cdots \leq X_{n,n}$ denote the order statistics based on $X_1,\ldots, X_n$. Choose any sequence of integers $(k_n)_{n\geq 1}$ such that $1 \leq k_n \leq n, k_n \rightarrow \infty$, and $k_n/n \rightarrow 0$. It has been recently shown by S. Csorgo and Mason (1986) that the sum of the extreme values $X_{n,n} + \cdots + X_{n-k_n,n}$, when properly centered and normalized, converges in distribution to a standard normal random variable. In this paper, we completely characterize such sequences $(k_n)_{n\geq 1}$ for which the corresponding law of the iterated logarithm holds.