Advances in Applied Probability

Asymptotics of Markov kernels and the tail chain

Sidney I. Resnick and David Zeber

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Abstract

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.

Article information

Source
Adv. in Appl. Probab., Volume 45, Number 1 (2013), 186-213.

Dates
First available in Project Euclid: 15 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.aap/1363354108

Digital Object Identifier
doi:10.1239/aap/1363354108

Mathematical Reviews number (MathSciNet)
MR3077546

Zentralblatt MATH identifier
1270.60061

Subjects
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

Keywords
Extreme values Markov chain multivariate regular variation transition kernel tail chain heavy tail

Citation

Resnick, Sidney I.; Zeber, David. Asymptotics of Markov kernels and the tail chain. Adv. in Appl. Probab. 45 (2013), no. 1, 186--213. doi:10.1239/aap/1363354108. https://projecteuclid.org/euclid.aap/1363354108


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