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Calculating the extremal index for a class of stationary sequences

Published online by Cambridge University Press:  01 July 2016

Michael R. Chernick ,
Tailen Hsing  and
William P. McCormick
Michael R. Chernick*
Affiliation:
Nichols Research Corporation
Tailen Hsing*
Affiliation:
Texas A & M University
William P. McCormick*
Affiliation:
University of Georgia
*
∗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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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.

Keywords

ARMA PROCESSEXTREME VALUEMIXING CONDITION
Type
Research Article
Information
Advances in Applied Probability , Volume 23 , Issue 4 , December 1991 , pp. 835 - 850
Copyright
Copyright © Applied Probability Trust 1991 

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Footnotes

Research supported by NSF Grant 8814006.

References

Chernick, M. R. (1978) Mixing Conditions and Limit Theorems for Maxima of Some Stationary Sequences. Ph.D. Dissertation, Stanford University.Google Scholar
Chernick, M. R. (1981) A limit theorem for the maximum of autoregressive processes with uniform marginal distributions. Ann. Prob. 9, 145–149.Google Scholar
Chernick, M. R. and Davis, R. (1982) Extremes in autoregressive processes with uniform marginal distributions. Statist. Prob. Lett. 1, 85–88.Google Scholar
Davis, R. A. and Resnick, S. I. (1985) Limit theory for moving averages of random variables with regularly varying tail probabilities. Ann. Prob. 13, 179–195.Google Scholar
Hsing, T. (1989) On a loss of memory property of the maximum. Statist. Prob. Lett. 8, 493–496.Google Scholar
Kallenberg, O. (1983) Random Measures , 2nd edn. Academic Press, London.CrossRefGoogle Scholar
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, 69–80. Springer-Verlag, Berlin.CrossRefGoogle Scholar
Leadbetter, M. R., Lindgren, G., and Rootzén, H. (1983) Extremes and Related Properties of Random Sequences and Processes. Springer-Verlag, New York.Google Scholar
O'Brien, G. L. (1987) Extreme values for stationary Markov sequences. Ann. Prob. 15, 281–289.Google Scholar
Resnick, S. I. (1987) Extreme Values, Regular Variation, and Point Processes. Springer-Verlag, New York.Google Scholar