Extreme Values for Stationary and Markov Sequences

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.