Open Access
May 1998 The extremal index of a higher-order stationary Markov chain
Seokhoon Yun
Ann. Appl. Probab. 8(2): 408-437 (May 1998). DOI: 10.1214/aoap/1028903534

Abstract

The paper presents a method of computing the extremal index of a real-valued, higher-order (kth-order, k1) 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 k1th-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.

Citation

Download Citation

Seokhoon Yun. "The extremal index of a higher-order stationary Markov chain." Ann. Appl. Probab. 8 (2) 408 - 437, May 1998. https://doi.org/10.1214/aoap/1028903534

Information

Published: May 1998
First available in Project Euclid: 9 August 2002

zbMATH: 0942.60038
MathSciNet: MR1624945
Digital Object Identifier: 10.1214/aoap/1028903534

Subjects:
Primary: 60G70 , 60J05
Secondary: 60G10 , 60G55

Keywords: exceedance point processes , extremal index , multivariate extreme value distributions , stationary Markov chains

Rights: Copyright © 1998 Institute of Mathematical Statistics

Vol.8 • No. 2 • May 1998
Back to Top