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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, $k \geq 1$) 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.

Citation

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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

Rights: Copyright © 1998 Institute of Mathematical Statistics

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Vol.8 • No. 2 • May 1998
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