## On clusters of high extremes of Gaussian stationary processes with $$\epsilon$$-separation.(English)Zbl 1226.60082

Summary: The clustering of extreme 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 $$\epsilon$$. 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 $$n$$ 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.

### MSC:

 60G70 Extreme value theory; extremal stochastic processes 60G15 Gaussian processes 60G10 Stationary stochastic processes
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