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Asymptotics of Markov Kernels and the Tail Chain

Part of: Stochastic processes Applications Markov processes

Published online by Cambridge University Press:  04 January 2016

Sidney I. Resnick  and
David Zeber
Sidney I. Resnick*
Affiliation:
Cornell University
David Zeber*
Affiliation:
Cornell University
*
∗ Postal address: School of Operations Research and Industrial Engineering, Cornell University, 284 Rhodes Hall, Ithaca, NY 14853, USA. Email address: sir1@cornell.edu
∗∗ Postal address: Department of Statistical Science, Cornell University, 301 Malott Hall, Ithaca, NY 14853, USA. Email address: dsz5@cornell.edu
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Abstract

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An asymptotic model for the extreme behavior of certain Markov chains is the ‘tail chain’. Generally taking the form of a multiplicative random walk, it is useful in deriving extremal characteristics, such as point process limits. We place this model in a more general context, formulated in terms of extreme value theory for transition kernels, and extend it by formalizing the distinction between extreme and nonextreme states. We make the link between the update function and transition kernel forms considered in previous work, and we show that the tail chain model leads to a multivariate regular variation property of the finite-dimensional distributions under assumptions on the marginal tails alone.

Keywords

Extreme valuesMarkov chainmultivariate regular variationtransition kerneltail chainheavy tail

MSC classification

Primary: 60G70: Extreme value theory; extremal processes 60J05: Discrete-time Markov processes on general state spaces
Secondary: 62P05: Applications to actuarial sciences and financial mathematics
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 45 , Issue 1 , March 2013 , pp. 186 - 213
Copyright
© Applied Probability Trust 

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