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Markov Tail Chains

Part of: Applications Markov processes Stochastic processes Stochastic analysis

Published online by Cambridge University Press:  30 January 2018

A. Janssen  and
J. Segers
A. Janssen*
Affiliation:
University of Hamburg
J. Segers*
Affiliation:
Université Catholique de Louvain
*
∗ Postal address: Department of Mathematics, University of Hamburg, Bundesstrasse 55, 20146 Hamburg, Germany. Email address: [email protected]
∗∗ Postal address: Institut de Statistique, Université Catholique de Louvain, Voie du Roman Pays 20, B-1348 Louvain-la-Neuve, Belgium. Email address: [email protected]
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Abstract

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The extremes of a univariate Markov chain with regularly varying stationary marginal distribution and asymptotically linear behavior are known to exhibit a multiplicative random walk structure called the tail chain. In this paper we extend this fact to Markov chains with multivariate regularly varying marginal distributions in Rd. We analyze both the forward and the backward tail process and show that they mutually determine each other through a kind of adjoint relation. In a broader setting, we will show that even for non-Markovian underlying processes a Markovian forward tail chain always implies that the backward tail chain is also Markovian. We analyze the resulting class of limiting processes in detail. Applications of the theory yield the asymptotic distribution of both the past and the future of univariate and multivariate stochastic difference equations conditioned on an extreme event.

Keywords

Autoregressive conditional heteroskedasticityextreme value distribution(multivariate) Markov chainmultivariate regular variationrandom walkstochastic difference equationtail chaintail-switching potential

MSC classification

Primary: 60G70: Extreme value theory; extremal processes 60J05: Discrete-time Markov processes on general state spaces
Secondary: 60G10: Stationary processes 60H25: Random operators and equations 62P05: Applications to actuarial sciences and financial mathematics
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 4 , December 2014 , pp. 1133 - 1153
Copyright
© Applied Probability Trust 

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