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Extreme events of Markov chains

Published online by Cambridge University Press:  17 March 2017

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: [email protected]
** Postal address: Institute of Mathematics, University of Mannheim, 68131 Mannheim, Germany. Email address: [email protected]
*** Postal address: Department of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF, UK. Email address: [email protected]
**** Postal address: Biomathematics and Statistics Scotland, James Clerk Maxwell Building, Peter Guthrie Tait Road, Edinburgh EH9 3FD, UK. Email address: [email protected]

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.

Type
Research Article
Copyright
Copyright © Applied Probability Trust 2017 

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