The survival probability of a critical multi-type branching process in i.i.d. random environment
type: Item
instance of: scholarly article
title: The survival probability of a critical multi-type branching process in i.i.d. random environment
zbMATH Open document ID: 1445.60063
DOI: 10.1214/17-AOP1243
author: Yanyan Li
published in: The Annals of Probability
publication date: 2018-10-24T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.aop/1535097643
review text: The authors study the asymptotic behaviour of the probability of non-extinction of a critical multi-type Galton-Watson process in i.i.d. random environments. Under suitable assumptions, including that the random environments admit so-called linear-fractional multidimensional generating functions, the authors obtain an optimal result, it is shown that the survival probability up to time \(n\) multiplied by \(\sqrt{n}\) converges to some positive limit as \(n\to\infty\). It is explained that the main difficulty which appears in the proofs is that the fluctuations of products of some random matrices have to be investigated, and there were no available results in this direction at that time. In case of general generating functions of the random environments, the authors also prove a weaker result.
reviewed by: Mátyás Barczy
Mathematics Subject Classification ID: 60J80
Mathematics Subject Classification ID: 60K37
Mathematics Subject Classification ID: 60F17
zbMATH DE Number: 6964352
zbMATH Keywords: multi-type branching process
zbMATH Keywords: survival probability
zbMATH Keywords: random environment
zbMATH Keywords: product of random matrices
zbMATH Keywords: critical case
MaRDI profile type: Publication
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Wikidata QID: Q129337085
description: scientific article; zbMATH DE number 6964352