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January, 1987 Extreme Values for Stationary and Markov Sequences
George L. O'Brien
Ann. Probab. 15(1): 281-291 (January, 1987). DOI: 10.1214/aop/1176992270


Let $(X_n)_{n=1,2,\ldots}$ be a strictly stationary sequence of real-valued random variables. Let $M_{i,j} = \max(X_{i+1},\ldots, X_j)$ and let $M_n = M_{0,n}$. Let $(c_n)$ be a sequence of real numbers. It is shown under general circumstances that $P\lbrack M_n \leq c_n\rbrack - (P\lbrack X_1 \leq c_n\rbrack)^{nP\lbrack M_{1,p_n}\leq c_n\mid X_1>c_n\rbrack} \rightarrow 0$, for any sequence $(p_n)$ satisfying certain growth-rate conditions. Under suitable mixing conditions, there exists a distribution function $G$ such that $P\lbrack M_n \leq c_n\rbrack - (G(c_n))^n \rightarrow 0$ for all sequences $(c_n)$. These theorems hold in particular if $(X_n)$ is a function of a positive Harris Markov sequence. Some examples are included.


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George L. O'Brien. "Extreme Values for Stationary and Markov Sequences." Ann. Probab. 15 (1) 281 - 291, January, 1987.


Published: January, 1987
First available in Project Euclid: 19 April 2007

zbMATH: 0619.60025
MathSciNet: MR877604
Digital Object Identifier: 10.1214/aop/1176992270

Primary: 60F05
Secondary: 60G10 , 60J05

Keywords: extremal index , extreme value , function of a Markov sequence , Maximum , minimum , Mixing , phantom distribution function , stationary sequence , weak limit

Rights: Copyright © 1987 Institute of Mathematical Statistics

Vol.15 • No. 1 • January, 1987
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