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A counterexample concerning the extremal index

Published online by Cambridge University Press:  01 July 2016

Richard L. Smith
Richard L. Smith*
Affiliation:
University of Surrey
*
∗ Postal address: Department of Mathematics, University of Surrey, Guildford, GU2 5XH, UK.
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Abstract

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The concept of an extremal index, which is a measure of local dependence amongst the exceedances over a high threshold by a stationary sequence, has a natural interpretation as the reciprocal of mean cluster size. We exhibit a counterexample which shows that this interpretation is not necessarily correct.

Information

Type
Letters to the Editor
Information
Advances in Applied Probability , Volume 20 , Issue 3 , September 1988 , pp. 681 - 683
Copyright
Copyright © Applied Probability Trust 1988 

Footnotes

Supported by Air Force Office of Scientific Research at University of North Carolina; Grant Number F 49620 85C 0144.

References

Alpuim, M. T. (1987) High-level exceedances in stationary processes with index. Preprint, University of Lisbon.Google Scholar
Davis, R. A. (1982) Limit laws for the maximum and minimum of stationary sequences. Z. Wahrscheinlichkeitsth. 61, 31–42.Google Scholar
Hsing, T., Hüsler, J. and Leadbetter, M. R. (1988) Limits for exceedance point processes. Prob. Thy. Rel. Fields. To appear.Google Scholar
Leadbetter, M. R. (1983) Extremes and local dependence in stationary sequences. Z. Wahrscheinlichkeitsth. 65, 291–306.Google Scholar
Leadbetter, M. R., Lindgren, G. and Rootzen, H. (1983) Extremes and Related Properties of Random Sequences and Series. Springer-Verlag, New York.Google Scholar
Loynes, R. M. (1965) Extreme values in uniformly mixing stationary stochastic processes. Ann. Math. Statist. 36, 993–999.Google Scholar
Newell, G. F. (1964) Asymptotic extremes for m-dependent random variables. Ann. Math. Statist. 35, 1322–1325.CrossRefGoogle Scholar
O'Brien, G. L. (1974a) Limit theorems for the maximum term of a stationary process. Ann. Prob. 2, 540–545.Google Scholar
O'Brien, G. L. (1974b) The maximum term of uniformly mixing stationary processes. Z. Wahrschlichkeitsth. 30, 57–63.Google Scholar
O'Brien, G. L. (1987) Extreme values for stationary and Markov sequences. Ann. Prob. 15, 281–291.CrossRefGoogle Scholar
Rootzén, H. (1988) maxima and exceedances of stationary Markov chains. Adv. Appl. Prob. 20, 371–390.CrossRefGoogle Scholar
Smith, R. L. (1984) Threshold methods for samples extremes. In Statistical Extremes and Applications, ed. Tiago de Oliveira, J., Reidel, Dordrecht, 621–638.Google Scholar