Extremes and upcrossing intensities for \(P\)-differentiable stationary processes.
type: Item
instance of: scholarly article
title: Extremes and upcrossing intensities for \(P\)-differentiable stationary processes.
zbMATH Open document ID: 1045.60049
DOI: 10.1016/S0304-4149(99)00110-6
author: J. M. P. Albin
published in: Stochastic Processes and their Applications
publication date: 2004-09-07T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 60G70
Mathematics Subject Classification ID: 60E07
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 60G10
Mathematics Subject Classification ID: 60G15
zbMATH DE Number: 2098076
zbMATH Keywords: extrema
zbMATH Keywords: local extrema
zbMATH Keywords: sojourn
zbMATH Keywords: upcrossing
zbMATH Keywords: Rice's formula
zbMATH Keywords: stationary process
zbMATH Keywords: Belyaev's formula
zbMATH Keywords: Gaussian process
zbMATH Keywords: \(\alpha \)-stable process
reviewed by: Jiří Anděl
MaRDI profile type: Publication
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full work available at URL: s0304-4149(99)00110-6
OpenAlex ID: W2129663101
description: scientific article; zbMATH DE number 2098076