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January, 1994 The Asymptotic Distribution of Intermediate Sums
Sandor Csorgo, David M. Mason
Ann. Probab. 22(1): 145-159 (January, 1994). DOI: 10.1214/aop/1176988852

Abstract

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 numbers such that $k_n \rightarrow \infty$ and $k_n/n \rightarrow 0$ as $n \rightarrow \infty$, and consider the sums $I_n(a, b) = \sum^{\lbrack bk_n\rbrack}_{i=\lbrack ak_n\rbrack+1} X_{n+1-i,n}$ of intermediate order statistics, where $0 < a < b$. We find necessary and sufficient conditions for the existence of constants $A_n > 0$ and $C_n$ such that $A^{-1}_n(I_n(a,b) - C_n)$ converges in distribution along subsequences of the positive integers $\{n\}$ to nondegenerate limits and completely describe the possible subsequential limiting distributions.

Citation

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Sandor Csorgo. David M. Mason. "The Asymptotic Distribution of Intermediate Sums." Ann. Probab. 22 (1) 145 - 159, January, 1994. https://doi.org/10.1214/aop/1176988852

Information

Published: January, 1994
First available in Project Euclid: 19 April 2007

zbMATH: 0793.60020
MathSciNet: MR1258870
Digital Object Identifier: 10.1214/aop/1176988852

Subjects:
Primary: 60F05

Keywords: asymptotic distribution , stochastic compactness of maxima , Sums of intermediate order statistics

Rights: Copyright © 1994 Institute of Mathematical Statistics

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Vol.22 • No. 1 • January, 1994
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