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 greater than or equal to 1) station ary 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 introd uce limiting distributions of some rescaled stationary transition kern els, 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+. ..+ Y-n is used to derive a representation for the extremal index of ( X-n). We further establish convergence in distribution of multilevel e xceedance point processes for (X-n) in terms of (Z(n)). The representa tions for the extremal index and for quantities characterizing the dis tributional limits are well suited for Monte Carlo simulation.