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December, 1954 Extreme Values in Samples from $m$-Dependent Stationary Stochastic Processes
G. S. Watson
Ann. Math. Statist. 25(4): 798-800 (December, 1954). DOI: 10.1214/aoms/1177728670


The limiting distributions for the order statistics of $n$ successive observations in a sequence of independent and identically distributed random variables are shown to hold also when the sequence is generated by a stationary stochastic process of a certain moving average type. A sequence of random variables $\{x_i\}$ has been called $m$-dependent [3] if $| i - j | > m$ implies that $x_i$ and $x_j$ are independent. If the variables in a strictly stationary sequence are $m$-dependent and have a finite upper bound to their range of variation, the largest in a sample of $n$ successive members tends with probability one to this upper bound. This is a simple extension of Dodd's results [1] for the case of independence. The following theorem shows that when this upper bound is infinite, the asymptotic distribution of the largest in such a sample is the same as in the case of independence.


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G. S. Watson. "Extreme Values in Samples from $m$-Dependent Stationary Stochastic Processes." Ann. Math. Statist. 25 (4) 798 - 800, December, 1954.


Published: December, 1954
First available in Project Euclid: 28 April 2007

zbMATH: 0056.36204
MathSciNet: MR65122
Digital Object Identifier: 10.1214/aoms/1177728670

Rights: Copyright © 1954 Institute of Mathematical Statistics

Vol.25 • No. 4 • December, 1954
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