The Tail Empirical Process for Stationary Sequences

Diagnostic plots are an important part of extreme value statistics. This paper provides a theoretical basis for such plots by proving weak convergence of the tail empirical process for a large class of stationary processes. The conditions needed for convergence are (i) restrictions on the long-range dependence (mixing), (ii) moment restrictions on the amount of clustering of extremes, and (iii) convergence of the covariance function. We further establish how the limit process is changed if exceedances of a nonrandom level are replaced by exceedances of a high quantile of the observations. Two simple examples, k-dependent sequences and stable AR(1)-processes are considered.