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The upcrossings index and the extremal index

Part of: Stochastic processes Limit theorems

Published online by Cambridge University Press:  14 July 2016

H. Ferreira
H. Ferreira*
Affiliation:
University of Beira Interior
*
∗Postal address: Department of Mathematics, University of Beira Interior, 6200 Covilhã, Portugal. Email address: [email protected]
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Abstract

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For stationary sequences X = {Xn}n≥1 we relate τ, the limiting mean number of exceedances of high levels un by X1,…,Xn, and ν, the limiting mean number of upcrossings of the same level, through the expression θ = (ν/τ)η, where θ is the extremal index of X and η is a new parameter here called the upcrossings index. The upcrossings index is a measure of the clustering of upcrossings of u by variables in X, and the above relation extends the known relation θ = ν/τ, which holds under the mild-oscillation local restriction D″(u) on X. We present a new family of local mixing conditions D̃(k)(u) under which we prove that (a) the intensity of the limiting point process of upcrossings and η can both be computed from the k-variate distributions of X; and (b) the cluster size distributions for the exceedances are asymptotically equivalent to those for the lengths of one run of exceedances or the lengths of several consecutive runs which are separated by at most k − 2 nonexceedances and, except for the last one, each contain at most k − 2 exceedances.

Keywords

Point process of exceedancespoint process of upcrossingsextremal indexupcrossings indexmixing condition

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60G55: Point processes 60G70: Extreme value theory; extremal processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 43 , Issue 4 , December 2006 , pp. 927 - 937
Copyright
© Applied Probability Trust 2006 

References

Ancona-Navarrete, M. A. and Tawn, J. A. (2000). A comparison of methods for estimating the extremal index. Extremes 3, 5–38.Google Scholar
Chernick, M. R., Hsing, T. and McCormick, W. P. (1991). Calculating the extremal index for a class of stationary sequences. Adv. Appl. Prob. 23, 835–850.Google Scholar
Hsing, T., Hüsler, J. and Leadbetter, M. R. (1988). On the exceedance point process for a stationary sequence. Prob. Theory Relat. Fields 78, 97–112.Google Scholar
Kallenberg, O. (1976). Random Measures. Academic Press, London.Google Scholar
Leadbetter, M. R. (1974). On extreme values in stationary sequences. Z. Wahrscheinlichkeitsth. 28, 289–303.Google Scholar
Leadbetter, M. R. (1983). Extremes and local dependence in stationary sequences. Z. Wahrscheinlichkeitsth. 65, 291–306.Google Scholar
Leadbetter, M. R. and Nandagopalan, S. (1989). On exceedance point processes for stationary sequences under mild oscillation restrictions. In Extreme Value Theory (Oberwolfach 1987; Lecture Notes Statist. 51), eds Hüsler, J. and Reiss, D., Springer, Berlin, pp. 69–80.CrossRefGoogle Scholar
O'Brien, G. L. (1987). Extreme values for stationary and Markov sequences. Ann. Prob. 15, 281–291.Google Scholar