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Calculating the extremal index for a class of stationary sequences

  • Michael R. Chernick (a1), Tailen Hsing (a2) and William P. McCormick (a3)
Abstract

A local mixing condition D (k) is introduced for stationary sequences satisfying Leadbetter's condition D. Under the local mixing condition, the asymptotic distribution of the sample maximum can be calculated with the knowledge of the joint distribution of k consecutive terms. Some examples are given to illustrate the notion.

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Present address: Risk Data Corporation, Two Venture Plaza, Suite 400, Irvine, CA 92718-3331, USA.
∗∗ Postal address: Department of Statistics, Texas A & M University, College Station, TX 77843-3143, USA.
∗∗∗ Postal address: Department of Statistics, University of Georgia, Athens, GA 30602, USA.
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Research supported by NSF Grant 8814006.

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References
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Chernick, M. R. (1978) Mixing Conditions and Limit Theorems for Maxima of Some Stationary Sequences. Ph.D. Dissertation, Stanford University.
Chernick, M. R. (1981) A limit theorem for the maximum of autoregressive processes with uniform marginal distributions. Ann. Prob. 9, 145149.
Chernick, M. R. and Davis, R. (1982) Extremes in autoregressive processes with uniform marginal distributions. Statist. Prob. Lett. 1, 8588.
Davis, R. A. and Resnick, S. I. (1985) Limit theory for moving averages of random variables with regularly varying tail probabilities. Ann. Prob. 13, 179195.
Hsing, T. (1989) On a loss of memory property of the maximum. Statist. Prob. Lett. 8, 493496.
Kallenberg, O. (1983) Random Measures , 2nd edn. Academic Press, London.
Leadbetter, M. R. and Nandagopalan, L. (1989) On exceedance point processes for stationary sequences under mild oscillation restrictions. In Extreme Value Theory: Proceedings, Oberwolfach 1987, ed. Hüsler, J. and Reiss, R. D., Lecture Notes in Statistics 51, 6980. Springer-Verlag, Berlin.
Leadbetter, M. R., Lindgren, G., and Rootzén, H. (1983) Extremes and Related Properties of Random Sequences and Processes. Springer-Verlag, New York.
O'Brien, G. L. (1987) Extreme values for stationary Markov sequences. Ann. Prob. 15, 281289.
Resnick, S. I. (1987) Extreme Values, Regular Variation, and Point Processes. Springer-Verlag, New York.
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Advances in Applied Probability
  • ISSN: 0001-8678
  • EISSN: 1475-6064
  • URL: /core/journals/advances-in-applied-probability
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