The Annals of Applied Probability

The extremal index of a higher-order stationary Markov chain

Seokhoon Yun

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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.

Article information

Ann. Appl. Probab., Volume 8, Number 2 (1998), 408-437.

First available in Project Euclid: 9 August 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G70: Extreme value theory; extremal processes 60J05: Discrete-time Markov processes on general state spaces
Secondary: 60G10: Stationary processes 60G55: Point processes

Extremal index multivariate extreme value distributions exceedance point processes stationary Markov chains


Yun, Seokhoon. The extremal index of a higher-order stationary Markov chain. Ann. Appl. Probab. 8 (1998), no. 2, 408--437. doi:10.1214/aoap/1028903534.

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