How large must be the difference between local time and mesure du voisinage of Brownian motion?
type: Item
instance of: scholarly article
title: How large must be the difference between local time and mesure du voisinage of Brownian motion?
zbMATH Open document ID: 0602.60066
DOI: 10.1016/0167-7152(86)90059-3
author: Lajos Horváth
author: Pál Révész
author: Miklós Csörgo
published in: Statistics \& Probability Letters
publication date: 1986-01-01T00:00:00Z
review text: Let W(t), \(t\geq 0\), be a Wiener process on R and L(x,t) be the corresponding local time. The mesure du voisinage, M(x,t), of W is \(\lim_{h\downarrow 0}h^{1/2}N(h,x,t)\), where N(h,x,t) is the number of excursions of W away from x of length greater than h which are completed by time t. It is well known that \(M(x,t)=\sqrt{2/\pi}L(x,t)\) a.s. A result by two of the authors (Csörgö and Révész), unfortunately only available in a technical report, studies how quickly the limit defining M approaches \(\sqrt{2/\pi}L\). Specifically, for any fixed \(t>0\), \[ (1)\quad \lim_{h\downarrow 0}h^{-1/4}(-\log \quad h)^{-1}\sup_{(x,t)\in R\times [0,t']}| h^{1/2}N(h,x,t)-\sqrt{2/\pi}L(x,t)| =0,\quad a.s. \] In this paper the following result, related to how good an estimate (1) is for the rate of convergence, is shown. Theorem: For any \(\epsilon >0\) there exists \(\delta =\delta (\epsilon)>0\) and \(h_ 0=h_ 0(\epsilon)\) such that \[ P(h^{-1/4}| h^{1/2}N(h,0,1)-\sqrt{2/\pi}L(0,1)| >\delta)\geq 1-\epsilon \] if \(0<h<h_ 0\).
Mathematics Subject Classification ID: 60J55
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60J65
zbMATH DE Number: 3971917
zbMATH Keywords: Kiefer process
zbMATH Keywords: Wiener process
zbMATH Keywords: local time
zbMATH Keywords: mesure du voisinage
zbMATH Keywords: rate of convergence
reviewed by: Edward S. Boylan
MaRDI profile type: Publication
full work available at URL: 0167-7152(86)90059-3
OpenAlex ID: W2154045300
cites work: Q3326516
cites work: Mesure du voisinage and occupation density
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cites work: Brownian Local Times and Taboo Processes
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cites work: An approximation of partial sums of independent RV'-s, and the sample DF. I
cites work: A global intrinsic characterization of Brownian local time
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description: scientific article; zbMATH DE number 3971917