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August, 1982 Laws of Large Numbers for Sums of Extreme Values
David M. Mason
Ann. Probab. 10(3): 754-764 (August, 1982). DOI: 10.1214/aop/1176993783


Let $X_1, X_2, \cdots$, be a sequence of nonnegative i.i.d. random variables with common distribution $F$, and for each $n \geq 1$ let $X_{1n} \leq \cdots \leq X_{nn}$ denote the order statistics based on $X_1, \cdots, X_n$. Necessary and sufficient conditions are obtained for averages of the extreme values $X_{n+1-i, n}i = 1, \cdots, k_n + 1$ of the form: $k^{-1}_n \sum^{k_n}_{i = 1} (X_{n+1-i, n} - X_{n-k_n,n})$, where $k_n \rightarrow\infty$ and $n^{-1}k_n \rightarrow 0$, to converge in probability or almost surely to a finite positive constant. In the process, characterizations are given of the classes of distributions with regularly varying upper tails and of distributions with "exponential-like" upper tails.


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David M. Mason. "Laws of Large Numbers for Sums of Extreme Values." Ann. Probab. 10 (3) 754 - 764, August, 1982.


Published: August, 1982
First available in Project Euclid: 19 April 2007

zbMATH: 0493.60039
MathSciNet: MR659544
Digital Object Identifier: 10.1214/aop/1176993783

Primary: 60F15

Keywords: Extreme values , G2G30 , order statistics , regular variation

Rights: Copyright © 1982 Institute of Mathematical Statistics


Vol.10 • No. 3 • August, 1982
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