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.
Publié le : 1987-01-14
Classification:
Extreme value,
stationary sequence,
maximum,
minimum,
weak limit,
mixing,
extremal index,
phantom distribution function,
function of a Markov sequence,
60F05,
60G10,
60J05
@article{1176992270,
author = {O'Brien, George L.},
title = {Extreme Values for Stationary and Markov Sequences},
journal = {Ann. Probab.},
volume = {15},
number = {4},
year = {1987},
pages = { 281-291},
language = {en},
url = {http://dml.mathdoc.fr/item/1176992270}
}
O'Brien, George L. Extreme Values for Stationary and Markov Sequences. Ann. Probab., Tome 15 (1987) no. 4, pp. 281-291. http://gdmltest.u-ga.fr/item/1176992270/