On the survival probability of a branching process in a finite state i.i.d. environment
type: Item
instance of: scholarly article
title: On the survival probability of a branching process in a finite state i.i.d. environment
zbMATH Open document ID: 0634.60072
DOI: 10.1016/0304-4149(87)90011-1
published in: Stochastic Processes and their Applications
publication date: 1987-01-01T00:00:00Z
review text: A Galton-Watson process \(\{Z_ n\}_{n=0,1,...}\) is considered in an environment with randomly chosen states 0,1,...,r-1. The environmental process \(\{\zeta_ n\}\) is assumed i.i.d., and if m(\(\zeta)\) is the offspring mean corresponding to environment \(\zeta\), the classical condition for the process to be subcritical is then E[log m(\(\zeta\) \({}_ n)]<0.\) It is shown here that if the additional condition \(E[m(\zeta_ n)\log m(\zeta_ n)]<0\) holds (the process is strongly subcritical), then the survival probability \(P(Z_ n>0)\) tends to zero essentially geometrically, \(P(Z_ n>0)^{1/n}\to E Z_ 1\). For other subcritical processes the rate of decay of \(P(Z_ n>0)\) is strictly smaller than E \(Z_ 1\).
Mathematics Subject Classification ID: 60J80
zbMATH DE Number: 4032749
zbMATH Keywords: random environment
zbMATH Keywords: extinction probability
zbMATH Keywords: Chernoff's theorem
zbMATH Keywords: Galton- Watson process
zbMATH Keywords: subcritical processes
zbMATH Keywords: rate of decay
author: F. Michel Dekking
reviewed by: Søren Asmussen
MaRDI profile type: Publication
full work available at URL: 0304-4149(87)90011-1
OpenAlex ID: W1991859776
cites work: On the extinction times of varying and random environment branching processes
cites work: On Branching Processes with Random Environments: I: Extinction Probabilities
cites work: Q4197800
cites work: Subcritical branching processes in a two-state random environment, and a percolation problem on trees
cites work: On Branching Processes in Random Environments
description: scientific article; zbMATH DE number 4032749