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Extreme events of Markov chains

Part of: Stochastic processes Markov processes

Published online by Cambridge University Press:  17 March 2017

I. Papastathopoulos ,
K. Strokorb ,
J. A. Tawn  and
A. Butler
I. Papastathopoulos*
Affiliation:
University of Edinburgh and the Alan Turing Institute
K. Strokorb*
Affiliation:
University of Mannheim
J. A. Tawn*
Affiliation:
Lancaster University
A. Butler*
Affiliation:
Biomathematics and Statistics Scotland
*
* Postal address: School of Mathematics, University of Edinburgh, James Clerk Maxwell Building, Peter Guthrie Tait Road, Edinburgh EH9 3FD, UK. Email address: i.papastathopoulos@ed.ac.uk
** Postal address: Institute of Mathematics, University of Mannheim, 68131 Mannheim, Germany. Email address: strokorb@math.uni-mannheim.de
*** Postal address: Department of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF, UK. Email address: j.tawn@lancaster.ac.uk
**** Postal address: Biomathematics and Statistics Scotland, James Clerk Maxwell Building, Peter Guthrie Tait Road, Edinburgh EH9 3FD, UK. Email address: adam.butler@bioss.ac.uk
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Abstract

The extremal behaviour of a Markov chain is typically characterised by its tail chain. For asymptotically dependent Markov chains, existing formulations fail to capture the full evolution of the extreme event when the chain moves out of the extreme tail region, and, for asymptotically independent chains, recent results fail to cover well-known asymptotically independent processes, such as Markov processes with a Gaussian copula between consecutive values. We use more sophisticated limiting mechanisms that cover a broader class of asymptotically independent processes than current methods, including an extension of the canonical Heffernan‒Tawn normalisation scheme, and reveal features which existing methods reduce to a degenerate form associated with nonextreme states.

Keywords

Asymptotic independenceconditional extremesextreme value theoryMarkov chainhidden tail chaintail chain

MSC classification

Primary: 60G70: Extreme value theory; extremal processes 60J05: Discrete-time Markov processes on general state spaces
Secondary: 60G10: Stationary processes
Type
Research Article
Information
Advances in Applied Probability , Volume 49 , Issue 1 , March 2017 , pp. 134 - 161
Copyright
Copyright © Applied Probability Trust 2017 

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