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

Smith, Richard L.

J. Appl. Probab. 29, No. 1, 37-45 (1992).

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.

Reviewer: W.Dziubdziela (Kielce)

MSC:

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

Keywords:

positive recurrent Markov chain; extreme distribution function; fluctuation properties; random walk