The Annals of Probability

The Asymptotic Distribution of Extreme Sums

Sandor Csorgo, Erich Haeusler, and David M. Mason

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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.

Article information

Ann. Probab., Volume 19, Number 2 (1991), 783-811.

First available in Project Euclid: 19 April 2007

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Primary: 60F05: Central limit and other weak theorems

Sums of extreme values asymptotic distribution


Csorgo, Sandor; Haeusler, Erich; Mason, David M. The Asymptotic Distribution of Extreme Sums. Ann. Probab. 19 (1991), no. 2, 783--811. doi:10.1214/aop/1176990451.

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