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Extremes and clustering of nonstationary max-AR(1) sequences

Author

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  • Alpuim, M. T.
  • Catkan, N. A.
  • Hüsler, J.

Abstract

We consider general nonstationary max-autoregressive sequences Xi, i [greater-or-equal, slanted] 1, with Xi = Zimax(Xi - 1, Yi) where Yi, i [greater-or-equal, slanted] 1 is a sequence of i.i.d. random variables and Zi, i [greater-or-equal, slanted] 1 is a sequence of independent random variables (0 [less-than-or-equals, slant] Zi [less-than-or-equals, slant] 1), independent of Yi. We deal with the limit law of extreme values Mn = maxXi, i [less-than-or-equals, slant] n (as n --> [infinity]) and evaluate the extremal index for the case where the marginal distribution of Yi is regularly varying at [infinity]. The limit of the point process of exceedances of a boundary un by Xi, i [less-than-or-equals, slant] n, is derived (as n --> [infinity]) by analysing the convergence of the cluster distribution and of the intensity measure.

Suggested Citation

  • Alpuim, M. T. & Catkan, N. A. & Hüsler, J., 1995. "Extremes and clustering of nonstationary max-AR(1) sequences," Stochastic Processes and their Applications, Elsevier, vol. 56(1), pages 171-184, March.
    • Handle: RePEc:eee:spapps:v:56:y:1995:i:1:p:171-184
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    References listed on IDEAS

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    1. Hooghiemstra, G. & Scheffer, C. L., 1986. "Some limit theorems for an energy storage model," Stochastic Processes and their Applications, Elsevier, vol. 22(1), pages 121-127, May.
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    Cited by:

    1. Kühne Robert & Rüschendorf Ludger, 2003. "Optimal stopping and cluster point processes," Statistics & Risk Modeling, De Gruyter, vol. 21(3/2003), pages 261-282, March.

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