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April, 1991 The Asymptotic Distribution of Extreme Sums
Sandor Csorgo, Erich Haeusler, David M. Mason
Ann. Probab. 19(2): 783-811 (April, 1991). DOI: 10.1214/aop/1176990451


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 integers such that $k_n \rightarrow \infty$ and $k_n/n \rightarrow \alpha$ as $n \rightarrow \infty$, where $0 \leq \alpha < 1$. We find necessary and sufficient conditions for the existence of normalizing and centering constants $A_n > 0$ and $C_n$ such that the sequence $E_n = \frac{1}{A_n}\bigg\{\sum^{k_n}_{i=1} X_{n+1-i,n} - C_n\bigg\}$ converges in distribution along subsequences of the integers $\{n\}$ to nondegenerate limits and completely describe the possible subsequential limiting distributions. We also give a necessary and sufficient condition for the existence of $A_n$ and $C_n$ such that $E_n$ be asymptotically normal along a given subsequence, and with suitable $A_n$ and $C_n$ determine the limiting distributions of $E_n$ along the whole sequence $\{n\}$ when $F$ is in the domain of attraction of an extreme value distribution.


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Sandor Csorgo. Erich Haeusler. David M. Mason. "The Asymptotic Distribution of Extreme Sums." Ann. Probab. 19 (2) 783 - 811, April, 1991.


Published: April, 1991
First available in Project Euclid: 19 April 2007

zbMATH: 0736.62015
MathSciNet: MR1106286
Digital Object Identifier: 10.1214/aop/1176990451

Primary: 60F05

Keywords: asymptotic distribution , sums of extreme values

Rights: Copyright © 1991 Institute of Mathematical Statistics


Vol.19 • No. 2 • April, 1991
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