High and Massive Excursions of Gaussian Processes

The clustering of extremes values of a stationary Gaussian process X(t), with 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 e. Under certain assumptions on the correlation function of the process, the asymptotically exact behavior of the probability of such a pattern of separated double clusters of exceedances is derived where the level to be exceeded by the extreme values, tends to ∞. The excursion behavior of the paths in-between the clusters is investigated also. If such a separated double cluster of exceedances occurs, the path is almost deterministic and follows strongly a conditional mean function, which does not depend on the high level u. We discuss the possible patterns and the asymptotic probabilities of such clusters of exceedances. The possible pattern depends on the behavior of the correlation function of the Gaussian process.