On extremes and streams of upcrossing.
type: Item
instance of: scholarly article
title: On extremes and streams of upcrossing.
zbMATH Open document ID: 1053.60052
DOI: 10.1016/S0304-4149(01)00084-9
author: J. M. P. Albin
published in: Stochastic Processes and their Applications
publication date: 2004-11-26T00:00:00Z
review text: Let \(\{\xi (t)\}_{t\in [0,h]}\) be a real stationary process. Assume that the process is separable and continuous in probability. Define \(M(h)=\sup _{t\in [0,h]} \xi (t)\), \(J(q,u)=q^{-1} P\{\xi (0) >u \geq \xi (q)\}\), \(J(u)=\underline {\lim }_{n\to \infty } J(2^{-n}, u)\). The author studies relations between \(P\{M(h)>u\}\) and \(P\{\xi (0)>u\}+h J(2^n,u)\). It is proved that under some general conditions the inequality \(P\{M(h)>u\} \leq P\{\xi (0)>u\}+h J(u)\) holds. Applications of results include argmax process of Brownian motion minus parabolic drift, \(P\)-differentiable \(\alpha \)-stable processes, Markov jump processes, differentiable \(n\)-dimensional Gaussian processes, and moving \(L^2\)-norms of differentiable Gaussian processes.
Mathematics Subject Classification ID: 60G70
Mathematics Subject Classification ID: 60G10
zbMATH DE Number: 2119516
zbMATH Keywords: extrema
zbMATH Keywords: upcrossing
zbMATH Keywords: jump process
zbMATH Keywords: stable process
zbMATH Keywords: Gaussian process
reviewed by: Jiří Anděl
MaRDI profile type: Publication
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full work available at URL: s0304-4149(01)00084-9
OpenAlex ID: W2170089348
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description: scientific article; zbMATH DE number 2119516