Computing the extremal index of special Markov chains and queues

AbstractWe consider extremal properties of Markov chains. Rootzén (1988) gives conditions for stationary, regenerative sequences so that the normalized process of level exceedances converges in distribution to a compound Poisson process. He also provides expressions for the extremal index and the compounding probabilities; in general it is not easy to evaluate these.We show how in a number of instances Markov chains can be coupled with two random walks which, in terms of extremal behaviour, bound the chain from above and below. Using a limiting argument it is shown that the lower bound converges to the upper one, yielding the extremal index and the compounding probabilities of the Markov chain. An FFT algorithm by Grübel (1991) for the stationary distribution of a G/G/1 queue is adapted for the extremal index; it yields approximate, but very accurate results. Compounding probabilities are calculated explicitly in a similar fashion.The technique is applied to the G/G/1 queue, G/M/c queues and ARCH processes, whose extremal behaviour de Haan et al. (1989) characterized using simulation