The extremal index of a higher-order stationary Markov chain

Abstract

The paper presents a method of computing the extremal index of a real-valued, higher-order (kth-order, $k \geq 1$) stationary Markov chain ${X_n}$. The method is based on the assumption that the joint distribution of $k +1$ consecutive variables is in the domain of attraction of some multivariate extreme value distribution. We introduce limiting distributions of some rescaled stationary transition kernels, which are used to define a new $k -1$th-order Markov chain ${Y_n}$, say. Then, the kth-order Markov chain ${Z_n}$ defined by $Z_n = Y_1 + \dots + Y_n$ is used to derive a representation for the extremal index of ${X_n}$. We further establish convergence in distribution of multilevel exceedance point processes for ${X_n}$ in terms of ${Z_n}$. The representations for the extremal index and for quantities characterizing the distributional limits are well suited for Monte Carlo simulation.