The distributions of cluster functionals of extreme events in a dth-order Markov chain

Abstract

The paper concerns the asymptotic distributions of cluster functionals of extreme events in a dth-order stationary Markov chain {Xn, n = 1,2,…} for which the joint distribution of (X1,…,Xd+1) is absolutely continuous. Under some distributional assumptions for {Xn}, we establish weak convergence for a class of cluster functionals and obtain representations for the asymptotic distributions which are well suited for simulation. A number of examples important in applications are presented to demonstrate the usefulness of the results.