The extremal index of a higher-order stationary Markov chain

The paper presents a method of computing the extremal index of a real-valued, higher-order (kth-order, k.1) stationary Markov chain Xn. 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.1th-order Markov chain Yn, say. Then, the kth-order Markov chain Zn defined by Zn=Y1+.+Yn is used to derive a representation for the extremal index of Xn. We further establish convergence in distribution of multilevel exceedance point processes for Xn in terms of Zn. The representations for the extremal index and for quantities characterizing the distributional limits are well suited for Monte Carlo simulation.