Tail and dependence behaviour of levels that persist for a fixed period of time

Abstract

This work emerges from a study of the extremal behavior of a daily maximum sea water levels series, $\{X_i\}$ , presented in Draisma \cite{drai}. In its approach, a new series, $\{Y_i\}$ , is defined, consisting of water levels that persist for a fixed period of time. In this paper, we study the tail behavior of $\{Y_i\}$ , in case $\{X_i\}$ is independent and identically distributed (i.i.d.) and in case $\{X_i\}$ is a max-autoregressive sequence (we will consider two different max-autoregressive processes), whose distribution function is in the Fr\'echet domain of attraction. We also determine Ledford and Tawn tail dependence index (\cite{tawn1}, \cite{tawn2}) and we analyze the asymptotic tail dependence of the random pair $(Y_i,Y_{i+m})$, in all considered cases. According to Drees \cite{drees1}, we obtain the limit behavior of the tail empirical quantile function associated with a random sample $(Y_1,Y_2,...Y_n)$ and hence the asymptotic normality of a class of estimators of the tail index that includes Hill estimator.