The supercritical Galton-Watson process in varying environments -- Seneta-Heyde norming
type: Item
instance of: scholarly article
title: The supercritical Galton-Watson process in varying environments -- Seneta-Heyde norming
zbMATH Open document ID: 0789.60064
DOI: 10.1016/0304-4149(93)90046-7
author: J. D. Biggins
author: J. C. D'Souza
published in: Stochastic Processes and their Applications
publication date: 1994-06-06T00:00:00Z
review text: The authors consider a Galton-Watson process in varying environment \(\{Z_ n\}\) defined by \(Z_ 0=1\), \(Z_{n+1}=\sum^{Z_ n}_{i=1} X_{n,i}\), \(n \geq 0\), and \(\{X_{n,i;i}\}\) are independent identically distributed copies of a random variable \(X_ n\). A sufficient condition is given for \(\{Z_ n\}\) to have exactly one rate of growth, in the sense that there is a sequence of constants \(\{C_ n\}\) such that \(Z_ n/C_ n\) converges to a finite limit that is strictly positive on the survival set, \(\{Z_ n \to \infty\}\). The condition is that the process is uniformly supercritical and that \(\{X_ n/ \mu_ n\}\) is dominated by a random variable \(X\) with \(EX<\infty\), \(\mu_ n\) denoting \(EX_ n\). A construction is given for \(\{C_ n\}\).
Mathematics Subject Classification ID: 60J80
zbMATH DE Number: 493306
zbMATH Keywords: branching process
zbMATH Keywords: Galton-Watson process
zbMATH Keywords: varying environment
zbMATH Keywords: rate of growth
reviewed by: Annette Kopp-Schneider
MaRDI profile type: Publication
full work available at URL: 0304-4149(93)90046-7
OpenAlex ID: W2015794170
cites work: Q3664178
cites work: On a property related to convergence in probability and some applications to branching processes
cites work: Inhomogeneous Markov branching processes: Supercritical case
cites work: The supercritical Galton-Watson process in varying environments
cites work: Limit theorems for the supercritical Galton-Watson process in varying environments
cites work: A connection between the limit and the maximum random variable of a branching process in varying environments
cites work: A GALTON-WATSON BRANCHING PROCESS IN VARYING ENVIRONMENTS WITH ESSENTIALLY CONSTANT OFFSPRING MEANS AND TWO RATES OF GROWTH1
cites work: On the asymptotic behaviour of branching processes with infinite mean
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description: scientific article; zbMATH DE number 493306