Extremes and upcrossing intensities for \(P\)-differentiable stationary processes.
Let \(\{\xi (t)\}\) be a stationary process differentiable in probability. It is known that under some technical conditions, which may be hard to verify, the asymptotic behaviour of \(P\{\sup _ {t\in [0,1]} \xi (t)>u\}\) for large \(u\) is related with the expected number of upcrossings \(\mu (u)\) of the level \(u\) by \(\{\xi (t)\}_ {t\in [0,1]}\). Non-upcrossing-based approaches to local extremes rely on weak convergence of the finite-dimensional distributions of a special process with a certain type of tightness; moreover, excursions above high levels are not allowed to last too long. The author provides a criterion for verifying a tightness-type condition. A new set of conditions ensure the validity of Rice's formula \(\mu (u)=\int _ 0^ {\infty } z f_ {\xi (1),\xi '(1)}(u,z)\, dz\). The results are applied to extremes of \(R^ n\)-valued Gaussian processes with strongly dependent component processes, and of totally skewed moving averages of \(\alpha \)-stable motions. Finally, the author proves Belyaev's multidimensional version of Rice's formula for outcrossing through smooth surfaces of \(R^ n\)-valued \(\alpha \)-stable processes.
scientific article; zbMATH DE number 2098076
scientific article