The Annals of Probability

Extreme Values for Stationary and Markov Sequences

George L. O'Brien

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Abstract

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.

Article information

Source
Ann. Probab., Volume 15, Number 1 (1987), 281-291.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176992270

Digital Object Identifier
doi:10.1214/aop/1176992270

Mathematical Reviews number (MathSciNet)
MR877604

Zentralblatt MATH identifier
0619.60025

JSTOR
links.jstor.org

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 60G10: Stationary processes 60J05: Discrete-time Markov processes on general state spaces

Keywords
Extreme value stationary sequence maximum minimum weak limit mixing extremal index phantom distribution function function of a Markov sequence

Citation

O'Brien, George L. Extreme Values for Stationary and Markov Sequences. Ann. Probab. 15 (1987), no. 1, 281--291. doi:10.1214/aop/1176992270. https://projecteuclid.org/euclid.aop/1176992270


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