## 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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