Functional limit theorems for strongly subcritical branching processes in random environment
type: Item
instance of: scholarly article
title: Functional limit theorems for strongly subcritical branching processes in random environment
zbMATH Open document ID: 1080.60079
author: J. Geiger
author: V. I. Afanasyev
author: Götz Kersting
author: Vladimir Vatutin
published in: Stochastic Processes and their Applications
publication date: 2005-10-10T00:00:00Z
review text: Let \(Z_n\) denote the size of the \(n\)th generation of a one-type, discrete-time branching process in an i.i.d. environment, i.e., \(Z_n\) is the sum of \(Z_{n-1}\) random variables, each distributed according to \(Q_n\), where \(Q_1,Q_2,Q_3,\dots\) are i.i.d. random distributions on \(\{0,1,2,\dots\}\). For any distribution \(q\) on \(\{0,1,2,\dots\}\) define \(m(q):= \sum_{y= 0}yq(\{y\})\). Suppose \(\mathbb{E}[m(Q_1)\log m(Q_1)]< \infty\) (``strong subcriticality'') and \(\mathbb{E}[Z_1\log^+ Z_1]<\infty\). Then, conditioned on survival in a sufficiently distant future, the environments remain in the limit independent, in contrast to the situation in case of weaker forms of subcriticality. Furthermore, the conditioned process \((Z_n\mid Z_n> 0)\) converges in distribution to a positive recurrent Markov chain.
Mathematics Subject Classification ID: 60J80
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 60F17
zbMATH DE Number: 2213024
zbMATH Keywords: random walk
zbMATH Keywords: change of measure
zbMATH Keywords: positive recurrent Markov chain
reviewed by: Heinrich Hering
MaRDI profile type: Publication
full work available at URL: j.spa.2005.05.001
OpenAlex ID: W2078125418
cites work: Limit theorems for intermediately subcritical and strongly subcritical branching processes in a random environment
cites work: Criticality for branching processes in random environment
cites work: Q5674726
cites work: Q5718841
cites work: On the survival probability of a branching process in a finite state i.i.d. environment
cites work: On the Survival Probability of a Branching Process in a Random Environment
cites work: Q5538132
cites work: Reduced subcritical Galton-Watson processes in a random environment
cites work: Limit theorems for subcritical branching processes in random environment.
cites work: Asymptotic properties of branching processes in random environment
DOI: 10.1016/J.SPA.2005.05.001
description: scientific article