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August, 1982 On Limiting Distributions of Intermediate Order Statistics from Stationary Sequences
Vernon Watts, Holger Rootzen, M. R. Leadbetter
Ann. Probab. 10(3): 653-662 (August, 1982). DOI: 10.1214/aop/1176993774


Let $X_1, X_2, \cdots$, be a sequence of random variables and write $X^{(n)}_k$ for the $k$th largest among $X_1, X_2, \cdots, X_n$. If $\{k_n\}$ is a sequence of integers such that $k_n \rightarrow \infty, k_n/n \rightarrow 0$, the sequence $\{X^{(n)}_{k_n}\}$ is referred to as the sequence of intermediate order statistics corresponding to the intermediate rank sequence $\{k_n\}$. The possible limiting distributions for $X^{(n)}_{k_n}$ have been characterized (under mild restrictions) by various authors when the random variables $X_1, X_2, \cdots$ are independent and identically distributed. In this paper we consider the case when the $\{X_n\}$ form a stationary sequence and obtain a natural dependence restriction under which the "classical" limits still apply. It is shown in particular that the general dependence restriction applies to normal sequences when the covariance sequence $\{r_n\}$ converges to zero as fast as an appropriate power $n^{-\rho}$ as $n \rightarrow \infty$.


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Vernon Watts. Holger Rootzen. M. R. Leadbetter. "On Limiting Distributions of Intermediate Order Statistics from Stationary Sequences." Ann. Probab. 10 (3) 653 - 662, August, 1982.


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

zbMATH: 0487.62015
MathSciNet: MR659535
Digital Object Identifier: 10.1214/aop/1176993774

Primary: 60F05
Secondary: 60G10 , 60G15

Keywords: intermediate ranks , order statistics , ranks , Stationary processes

Rights: Copyright © 1982 Institute of Mathematical Statistics

Vol.10 • No. 3 • August, 1982
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