A martingale approach to supercritical (CMJ) branching processes
type: Item
instance of: scholarly article
title: A martingale approach to supercritical (CMJ) branching processes
zbMATH Open document ID: 0587.60086
author: Harry Cohn
published in: The Annals of Probability
publication date: 1985-01-01T00:00:00Z
review text: A supercritical Crump-Mode-Jagers branching process \(\{Z_ t^{\phi}\}\) counted with a general random characteristic \(\phi\) is considered. It is shown that convergence in probability of \(\{e^{-\alpha t}Z_ t^{\phi}\}\) is equivalent to convergence in distribution, where \(\alpha\) is the Malthusian parameter. Norming constants C(t) are identified such that \(\{C^{-1}(t)Z_ t^{\phi}\}\) converges almost surely to a nondegenerate limit under mild restrictions on \(\phi\). The key idea rests upon the identification of a martingale derived from a weakly convergent subsequence of \(\{C^{-1}(t)Z_ t^{\phi}\}\).
Mathematics Subject Classification ID: 60J80
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60K99
zbMATH DE Number: 3942671
zbMATH Keywords: supercritical Crump-Mode-Jagers branching process
zbMATH Keywords: convergence in probability
zbMATH Keywords: Malthusian parameter
reviewed by: Luis G. Gorostiza
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176992803
OpenAlex ID: W1994804672
DOI: 10.1214/AOP/1176992803
description: scientific article; zbMATH DE number 3942671