## Electronic Journal of Probability

### On Clusters of High Extremes of Gaussian Stationary Processes with $\varepsilon$-Separation

#### Abstract

The clustering of extremes values of a stationary Gaussian process $X(t),t\in[0,T]$ is considered, where at least two time points of extreme values above a high threshold are separated by at least a small positive value $\varepsilon$. Under certain assumptions on the correlation function of the process, the asymptotic behavior of the probability of such a pattern of clusters of exceedances is derived exactly where the level to be exceeded by the extreme values, tends to $\infty$. The excursion behaviour of the paths in such an event is almost deterministic and does not depend on the high level $u$. We discuss the pattern and the asymptotic probabilities of such clusters of exceedances.

#### Article information

Source
Electron. J. Probab., Volume 15 (2010), paper no. 59, 1825-1862.

Dates
Accepted: 14 November 2010
First available in Project Euclid: 1 June 2016

https://projecteuclid.org/euclid.ejp/1464819844

Digital Object Identifier
doi:10.1214/EJP.v15-828

Mathematical Reviews number (MathSciNet)
MR2738340

Zentralblatt MATH identifier
1226.60082

Subjects
Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60G15: Gaussian processes 60G10: Stationary processes

Rights

#### Citation

Huesler, Juerg; Ladneva, Anna; Piterbarg, Vladimir. On Clusters of High Extremes of Gaussian Stationary Processes with $\varepsilon$-Separation. Electron. J. Probab. 15 (2010), paper no. 59, 1825--1862. doi:10.1214/EJP.v15-828. https://projecteuclid.org/euclid.ejp/1464819844

#### References

• Albin, P., and Piterbarg, V. I. (2000) On extremes of the minima of a stationary Gaussian process and one of its translates. Unpublished Manuscript.
• Anshin, A. B. On the probability of simultaneous extrema of two Gaussian nonstationary processes.(Russian) Teor. Veroyatn. Primen. 50 (2005), no. 3, 417–432; translation in Theory Probab. Appl. 50 (2006), no. 3, 353–366
• Berman, Simeon M. Sojourns and extremes of stationary processes. Ann. Probab. 10 (1982), no. 1, 1–46.
• Ladneva, A. and Piterbarg, V. I. (2000) On double extremes of Gaussian stationary processes. EURANDOM Technical report 2000-027, 1-18, available at http://www.eurandom.tue.nl/reports/2000/027vp.tex.
• bibitem Leadbetter, M. R., Weisman, I., de Haan, L., Rootz'en, H. (1989) On clustering of high values in statistically stationary series. In: textitProc. 4th Int. Meet. Statistical Climatology, (ed: J. Sanson). Wellington: New Zealand Meteorological Service.
• Linnik, Yu. V. Linear forms and statistical criteria. I, II.(Russian) Ukrain. Mat. Å½urnal 5, (1953). 207–243, 247–290.
• Pickands, James, III. Upcrossing probabilities for stationary Gaussian processes. Trans. Amer. Math. Soc. 145 1969 51–73.
• Piterbarg, Vladimir I. Asymptotic methods in the theory of Gaussian processes and fields.Translated from the Russian by V. V. Piterbarg.Revised by the author.Translations of Mathematical Monographs, 148. American Mathematical Society, Providence, RI, 1996. xii+206 pp. ISBN: 0-8218-0423-5
• Piterbarg, V. I.; Stamatovich, B. Rough asymptotics of the probability of simultaneous high extrema of two Gaussian processes: the dual action functional.(Russian) Uspekhi Mat. Nauk 60 (2005), no. 1(361), 171–172; translation in Russian Math. Surveys 60 (2005), no. 1, 167–168