On the sojourn time of a random walk in a strip
type: Item
instance of: scholarly article
title: On the sojourn time of a random walk in a strip
zbMATH Open document ID: 1217.60034
published in: Siberian Mathematical Journal
publication date: 2011-03-17T00:00:00Z
review text: Let \(S_{n}=X_1+\dots+X_{n}\), \(n\geq1\), be a random walk with i.i.d. increments satisfying Cramér type conditions. For \(a\leq b\), let \(T_{n}=T_{n}([a,b])=\sum_{k=1}^nI\{a\leq S_{k}\leq b\}\) denote the sojourn time of the random walk in \([a,b]\) on the time interval \(\{1,\dots,n\}\). This paper is a continuation of the author's work [Sib. Math. J. 51, No. 1, 119--127 (2010; Zbl 1202.60067)], and corresponds to the second stage toward the final goal of finding the probabilities \(P(T_{n}=k)\). The main result gives asymptotic representations for the transform \(f_{a,b}(s,u,\lambda)=\sum_{n\geq1}z^nE[u^{T_{n}}e^{\lambda S_{n}}]\), where \(|z|<1\), \(|uz|<1\) and \(\operatorname{Re}\lambda>0\). The one-sided cases \(]-\infty,b]\) and \([a,\infty[\) are also considered.
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 60E10
zbMATH DE Number: 5867019
zbMATH Keywords: random walk
zbMATH Keywords: sojourn time
zbMATH Keywords: factorization identities
zbMATH Keywords: moment generating functions
author: V. I. Lotov
reviewed by: Aurel Spătaru
MaRDI profile type: Publication
full work available at URL: s11202-010-0064-x
OpenAlex ID: W2103926077
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DOI: 10.1007/S11202-010-0064-X
description: scientific article; zbMATH DE number 5867019