How long does it take to see a flat Brownian path on the average?
type: Item
instance of: scholarly article
title: How long does it take to see a flat Brownian path on the average?
zbMATH Open document ID: 0777.60078
DOI: 10.1016/0167-7152(93)90068-T
author: Thomas M. Lewis
author: Wenbo V. Li
published in: Statistics \& Probability Letters
publication date: 1993-05-16T00:00:00Z
review text: Consider a standard Brownian motion and define \(R(t,1)\) as its oscillation over the time interval \([t-1,1]\). Given \(\varepsilon>0\), let \(\tau(\varepsilon)\) be the smallest \(t\geq 1\) for which \(R(t,1)\leq\varepsilon\). The following results are proved: \[ \lim_{\varepsilon\to 0} \varepsilon^ 2 \log E(\tau(\varepsilon))=\pi^ 2/2, \;\liminf_{t\to\infty} \sqrt{\log t} R(t,1)=\liminf_{t\to\infty} \sqrt{\log t} \inf_{1\leq s\leq t} R(s,1)=\pi/\sqrt{2},\text{ a.s.}. \]
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 60G17
Mathematics Subject Classification ID: 60F15
zbMATH DE Number: 169649
zbMATH Keywords: range of Brownian motion
zbMATH Keywords: waiting time
zbMATH Keywords: strong limit theorems
zbMATH Keywords: Brownian motion
reviewed by: Tamás F. Móri
MaRDI profile type: Publication
cites work: The Occurrence of Sequence Patterns in Repeated Dependent Experiments
cites work: Q3326516
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full work available at URL: 0167-7152(93)90068-t
OpenAlex ID: W2033398114
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description: scientific article; zbMATH DE number 169649