The distribution of the sojourn time for the Brownian excursion
type: Item
instance of: scholarly article
title: The distribution of the sojourn time for the Brownian excursion
zbMATH Open document ID: 0978.60094
DOI: 10.1023/A:1010060107265
author: Lajos Takács
published in: Methodology and Computing in Applied Probability
publication date: 2002-01-15T00:00:00Z
review text: Let \(\{B(t),0<t<1\}\) be a standard Brownian bridge and let \(\{B^{*}(t),0<t<1\}\) be a standard Brownian excursion. Denote by \(w(\alpha)=\int_0^1 \delta(B(t))dt\) the sojourn time of \(B(t)\) spent in the set \((\alpha,\infty)\), \(\alpha>0\), in the time interval \((0,1)\) and use the notation \(w^{*}(\alpha)\) for the relevant sojourn time for the Brownian excursion \(B^{*}(t)\). The author utilizes the random walk approach to determine the distributions and moments of \(w^{*}(\alpha)\) and \(w(\alpha)\). He approximates the processes \(B(t)\) and \(B^{*}(t)\) by suitably chosen random walks and uses certain limit theorems to obtain the moments \(E\{[w^{*}(\alpha)]^r\}\) and \(E\{[w(\alpha)]^r\}\), \(r=1,2,\dots\), explicitly. These moments uniquely determine the distribution functions for corresponding sojourn times. Thus, explicit formulas which are suitable for numerical calculations are established and their applications to the problems connected with random trees, queueing and branching processes are presented.
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 60G15
Mathematics Subject Classification ID: 60J55
Mathematics Subject Classification ID: 60G50
zbMATH DE Number: 1367823
zbMATH Keywords: Brownian motion
zbMATH Keywords: Brownian bridge
zbMATH Keywords: Brownian excursion
zbMATH Keywords: sojourn times
zbMATH Keywords: distribution function
MaRDI profile type: Publication
full work available at URL: a:1010060107265
OpenAlex ID: W2473352597
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description: scientific article; zbMATH DE number 1367823