Advances in Applied Probability

Asymptotics of Markov kernels and the tail chain

Sidney I. Resnick and David Zeber

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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.

Article information

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

First available in Project Euclid: 15 March 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

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