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Extreme Value Theory for a Class of Markov Chains with Values in ℝd

Part of:Markov processesStochastic processes

Published online by Cambridge University Press:  01 July 2016

Roland Perfekt
Roland Perfekt*
Affiliation:
University of Lund
*
∗ Postal address: Department of Mathematical Statistics, University of Lund, Box 118, S-22100, Sweden.
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Abstract

We consider extreme value theory for a class of stationary Markov chains with values in ℝd. The asymptotic distribution of M n , the vector of componentwise maxima, is determined under mild dependence restrictions and suitable assumptions on the marginal distribution and the transition probabilities of the chain. This is achieved through computation of a multivariate extremal index of the sequence, extending results of Smith [26] and Perfekt [21] to a multivariate setting. As a by-product, we obtain results on extremes of higher-order, real-valued Markov chains. The results are applied to a frequently studied random difference equation.

Keywords

MULTIVARIATE EXTREME VALUESMULTIVARIATE EXTREMAL INDEXSTATIONARY MARKOV CHAIN

MSC classification

Primary:60G70: Extreme value theory; extremal processes60J05: Discrete-time Markov processes on general state spaces
Secondary:60G10: Stationary processes

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 29 , Issue 1 , March 1997, pp. 138 - 164
Copyright
Copyright © Applied Probability Trust 1997 

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