Renewal, regenerative, and branching processes with stable distributions
type: Item
instance of: scholarly article
title: Renewal, regenerative, and branching processes with stable distributions
zbMATH Open document ID: 1076.60076
published in: Journal of Mathematical Sciences (New York)
publication date: 2005-06-28T00:00:00Z
review text: The first part of the paper consists of a summary on earlier results of the authors. For an alternating renewal process with random waiting times following random working periods, a renewal theorem for the spent-time process is considered under the assumption that waiting and/or working periods are i.i.d.\ belonging to the domain of attraction of some stable law with index in \((\frac12,1]\). The renewal theorem is basic for the behaviour of an alternating regenerative process, where random waiting periods start whenever an evolution process hits state 0. In the second part of the paper this basic regeneration theorem is applied to branching processes of the form \[ Y_n=\Big(\sum_{i=1}^{Y_{n-1}}X_i(n)+M_n\Big)_+,\quad n\in\mathbb N, \] where \(M_n\) is additional random migration consisting of three components: family and individual emigration with probability \(p\), no migration with probability \(q\), and state-dependent immigration with probability \(r\); \(p+q+r=1\). In case \(q=1\), i.e.\ \(M_n\equiv0\) almost surely, \((Y_n)\) defines a usual Galton-Watson branching process. Particular cases of these migration components have been studied earlier. Since state 0 is a reflective barrier for the process \((Y_n)\), it can also be seen as an alternating regenerative process, allowing to consider all the migration components at the same time.
Mathematics Subject Classification ID: 60J80
Mathematics Subject Classification ID: 60K05
Mathematics Subject Classification ID: 60E07
zbMATH DE Number: 2183291
zbMATH Keywords: alternating renewal process
zbMATH Keywords: regenerative process
zbMATH Keywords: branching process with migration
author: Nikolay M. Yanev
author: K. V. Mitov
author: George P. Yanev
MaRDI profile type: Publication
full work available at URL: b:joth.0000031887.13577.cf
OpenAlex ID: W1998477555
DOI: 10.1023/B:JOTH.0000031887.13577.CF
description: scientific article; zbMATH DE number 2183291