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Computing the extremal index of special Markov chains and queues

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  • Hooghiemstra, Gerard
  • Meester, Ludolf E.

Abstract

We 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.

Suggested Citation

  • Hooghiemstra, Gerard & Meester, Ludolf E., 1996. "Computing the extremal index of special Markov chains and queues," Stochastic Processes and their Applications, Elsevier, vol. 65(2), pages 171-185, December.
    • Handle: RePEc:eee:spapps:v:65:y:1996:i:2:p:171-185
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    References listed on IDEAS

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    1. Engle, Robert F, 1982. "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation," Econometrica, Econometric Society, vol. 50(4), pages 987-1007, July.
    2. de Haan, Laurens & Resnick, Sidney I. & Rootzén, Holger & de Vries, Casper G., 1989. "Extremal behaviour of solutions to a stochastic difference equation with applications to arch processes," Stochastic Processes and their Applications, Elsevier, vol. 32(2), pages 213-224, August.
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    1. Evsey V. Morozov & Irina V. Peshkova & Alexander S. Rumyantsev, 2023. "Bounds and Maxima for the Workload in a Multiclass Orbit Queue," Mathematics, MDPI, vol. 11(3), pages 1-15, January.

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