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July, 1987 A Law of the Iterated Logarithm for Sums of Extreme Values from a Distribution with a Regularly Varying Upper Tail
Erich Haeusler, David M. Mason
Ann. Probab. 15(3): 932-953 (July, 1987). DOI: 10.1214/aop/1176992074

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.

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Erich Haeusler. David M. Mason. "A Law of the Iterated Logarithm for Sums of Extreme Values from a Distribution with a Regularly Varying Upper Tail." Ann. Probab. 15 (3) 932 - 953, July, 1987. https://doi.org/10.1214/aop/1176992074

Information

Published: July, 1987
First available in Project Euclid: 19 April 2007

zbMATH: 0646.60034
MathSciNet: MR893907
Digital Object Identifier: 10.1214/aop/1176992074

Subjects:
Primary: 62G30
Secondary: 60F15

Keywords: Empirical processes , Law of the iterated logarithm , order statistics , sums of extreme values

Rights: Copyright © 1987 Institute of Mathematical Statistics

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Vol.15 • No. 3 • July, 1987
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