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.