Level crossings of absolutely continuous stationary symmetric \(\alpha\)-stable processes
type: Item
instance of: scholarly article
title: Level crossings of absolutely continuous stationary symmetric \(\alpha\)-stable processes
zbMATH Open document ID: 0883.60026
author: Gennady Samorodnitsky
author: Robert J. Adler
published in: The Annals of Applied Probability
publication date: 1998-03-04T00:00:00Z
review text: The mean rate is described at which a general absolutely continuous stationary symmetric \(\alpha\)-stable (S\(\alpha\)S) process \((0<\alpha<2)\) crosses a high level. More precisely, let this process be representable as \(X(t)= \int_S f(t,x)M(dx)\). Under smooth assumptions (notably \(f(0,x)\) absolutely continuous) it is proved that if \(C_\lambda(1)\) is the number of upcrossings of the level \(\lambda\) in the interval \([0,1]\), we have \[ \begin{multlined}\lim_{\lambda\to+\infty} \lambda^\alpha E(C_\lambda(1))= \alpha 2^{\alpha-1/2}\pi^{-(1+1/\alpha)/2}\Biggl(\Gamma\Biggl({\alpha+1\over 2}\Biggr)\Biggr)^{1+1/\alpha} C^{-1/\alpha}_\alpha\times\\ \int_s|\dot f(0,x)||f(0,x)|^{\alpha-1}\text{\textbf{1}}_{(f(0,x)\dot f(0,x)> 0)}m(dx)\end{multlined} \] with \(C_\alpha= \left(\int^{+\infty}_0 x^{-\alpha}\sin x dx\right)^{-1}\). As an example the result for the two-sided Ornstein-Uhlenbeck process is given. This result generalizes the previous one of the authors and \textit{T. Gadrich} [ibid. 3, No. 2, 553-575 (1993; Zbl 0779.60034)].
Mathematics Subject Classification ID: 60G10
Mathematics Subject Classification ID: 60G70
Mathematics Subject Classification ID: 60G17
Mathematics Subject Classification ID: 60G18
zbMATH DE Number: 1051772
zbMATH Keywords: stable processes
zbMATH Keywords: level crossings
zbMATH Keywords: Ornstein-Uhlenbeck process
reviewed by: J.-P. Lepeltier
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aoap/1034625340
OpenAlex ID: W2061543701
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DOI: 10.1214/AOAP/1034625340
description: scientific article; zbMATH DE number 1051772