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The extremal index for a Markov chain. (English) Zbl 0759.60059

Let \(\{X_ n\}\) be a stationary positive recurrent Markov chain in discrete time with continuous state space. Assume that \[ P^ n\{X_ 1\leq x_ 1+\log n,\;X_ 2\leq x_ 2+\log n\}\to G(x_ 1,x_ 2)\quad\text{as } n\to\infty, \] where \(G\) is a bivariate extreme distribution function with Gumbel margins. Under mild additional conditions the author shows that the Markov chain \(\{X_ n\}\) in the tails looks like a random walk and he presents a method of computing the extremal index of \(\{X_ n\}\) in terms of the fluctuation properties of that random walk.

MSC:

60G70 Extreme value theory; extremal stochastic processes
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
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