Extreme Values for Stationary and Markov Sequences

Abstract

Let (Xn)n=1,2,... be a strictly stationary sequence of real-valued random variables. Let Mi,j = max(Xi+1,..., Xj) and let Mn = M0,n. Let (cn) 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 (pn) satisfying certain growth-rate conditions. Under suitable mixing conditions, there exists a distribution function G such that P[ Mn ≤ cn] - (G(cn))n → 0 for all sequences (cn). These theorems hold in particular if (Xn) is a function of a positive Harris Markov sequence. Some examples are included.