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Crossings of max-stable processes

Part of: Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Richard A. Davis  and
Sidney I. Resnick
Richard A. Davis*
Affiliation:
Colorado State University
Sidney I. Resnick*
Affiliation:
Cornell University
*
∗ Postal address: Department of Statistics, Colorado State University, Fort Collins, CO 80523, USA.
∗∗ Postal address: School of Operations Research and Industrial Engineering, Cornell University, ETC Building, Ithaca, NY 14853, USA.
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Abstract

The expected number of upcrossings for a max-stable process is computed and compared with known results for stable processes. Asymptotically the formulas are of the same order.

Keywords

EXTREME VALUE THEORYPOISSON PROCESSESSTATIONARY PROCESSESLEVEL CROSSINGSSTABLE PROCESSES

MSC classification

Primary: 60G17: Sample path properties
Secondary: 60G10: Stationary processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 31 , Issue 1 , March 1994 , pp. 130 - 138
Copyright
Copyright © Applied Probability Trust 1994 

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Footnotes

Research partially supported by NSF Grant DMS-9100392 at Colorado State University.

Research partially supported by NSF Grant DMS–9100027 at Cornell University.

References

Adler, R., Samorodnitsky, G. and Gadrich, T. (1993) The expected number of level crossings for stationary, harmonisable, symmetric stable processes. Ann. Appl. Prob. 3, 553–575.Google Scholar
Albin, P. (1992) Extremes and crossings for differentiable stationary processes with application to Gaussian processes in Rm and Hilbert space. Stoch. Proc. Appl. 42, 119–147.Google Scholar
Balkema, A. and Haan, L. De, (1988) Almost sure continuity of stable moving average processes with index less than one. Ann. Prob. 16, 333–343.CrossRefGoogle Scholar
Cramer, H. and Leadbetter, M. R. (1967) Stationary and Related Stochastic Processes. Wiley, New York.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.CrossRefGoogle Scholar
Davis, R. A. and Resnick, S. I. (1989) Basic properties and prediction of max-ARMA processes. J. Appl. Prob. 21, 781–803.Google Scholar
Giné, E., Hahn, M. and Vatan, P. (1990) Max-infinitely divisible and max-stable sample continuous processes. Prob. Theory Rel. Fields 87, 139–165.Google Scholar
Haan, L. De (1984) A spectral representation for max-stable processes. Ann. Prob. 12, 1194–1204.Google Scholar
Haan, L. De and Pickands, J. (1986) Stationary min-stable processes. Prob. Theory Rel. Fields 72, 477–492.Google Scholar
Hsing, T. (1986) Extreme value theory for suprema of random variables with regularly varying tail probabilities. Stoch. Proc. Appl. 22, 51–57.Google Scholar
Leadbetter, ?. (1973) Point processes generated by level crossings. In Stochastic Point Processes, ed. Lewis, P. A. W. Wiley, New York.Google Scholar
Leadbetter, M., Lindgren, G. and Rootzen, H. (1983) Extremes and Related Properties of Random Sequences and Processes. Springer-Verlag, New York.CrossRefGoogle Scholar
Marcus, M. (1989) Some bounds for the expected number of level crossings of symmetric harmonizable p-stable processes, Stoch. Proc. Appl. 33, 217–231.Google Scholar
Resnick, S. (1987) Extreme Values, Regular Variation, and Point Processes. Springer-Verlag, New York.CrossRefGoogle Scholar
Resnick, S. and Roy, R. (1991) Random usc functions, max-stable and continuous choice. Ann. Appl. Prob. 1, 267–292.Google Scholar