Some results on the joint distribution of the renewal epochs prior to a given time instant
type: Item
instance of: scholarly article
title: Some results on the joint distribution of the renewal epochs prior to a given time instant
zbMATH Open document ID: 0652.60092
DOI: 10.1016/0304-4149(88)90077-4
author: Ernst Schulte-Geers
author: Wolfgang Stadje
published in: Stochastic Processes and their Applications
publication date: 1988-01-01T00:00:00Z
review text: Let \((S_ n)_{n\geq 0}\) be a renewal process with interarrival times \(X_ 1,X_ 2,...\). Several results on the behavior of the renewal process up to a given time \(t>0\) or up to a given \(S_ n=s\) are proved. For example, \(X_ 1\) is stochastically dominated by \(X_{N(t)+1}\), and \(X_ 0=0,X_ 1,...,X_{N(t)+1}\) is a stochastically increasing sequence, where \(N(t)=\sup \{n\geq 0| S_ n\leq t\}\). Conditions are given under which the distribution of the process \((S_{[nt]})_{0\leq t\leq 1}\), given that \(S_ n=s\), converges weakly in D[0,1] to the point mass at the function \(x_ s(t)=st\). The result e.g. holds, if \(X_ 1\) has a strongly unimodal distribution or if \(E(X^ 2_ 1| S_ 2)\leq S^ 2_ 2/(2(1+c))\) a.s. for some \(c>0\). In this context some new characterizations of the gamma, Poisson, binomial and negative binomial distributions are derived.
Mathematics Subject Classification ID: 60K05
zbMATH DE Number: 4064212
zbMATH Keywords: inspection paradox
zbMATH Keywords: characterization of distributions
zbMATH Keywords: renewal process
zbMATH Keywords: binomial and negative binomial distributions
MaRDI profile type: Publication
full work available at URL: 0304-4149(88)90077-4
OpenAlex ID: W4245837864
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description: scientific article; zbMATH DE number 4064212