On the survival probability of a branching process in a random environment
type: Item
instance of: scholarly article
title: On the survival probability of a branching process in a random environment
zbMATH Open document ID: 0840.60078
published in: Annales de l'Institut Henri Poincaré. Probabilités et Statistiques
publication date: 1996-03-20T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/77530
review text: Let \(\{Z_n, n \geq 0\}\) be a branching process in an independent and identically distributed random environment; see \textit{K. B. Athreya} and \textit{S. Karlin} [Ann. Math. Stat. 42, 1499-1520, 1843-1858 (1971; Zbl 0228.60032; Zbl 0228.60033)] for details. The author proves that the limit of the survival probability \(P(Z_n > 0)\) as \(n \to \infty\) is 1 if the branching process is supercritical or critical, it is \(E(Z_1)\) if the process is strongly subcritical and it is less than \(\min (1, EZ_1)\) if the process is subcritical but not strongly. The author establishes the result for the case of countable state environment under weaker conditions, thereby extending the work of \textit{F. M. Dekking} [Stochastic Processes Appl. 27, No. 1, 151-157 (1987; Zbl 0634.60072)] pertaining to a finite state environment.
reviewed by: P. R. Parthasarathy
Mathematics Subject Classification ID: 60J80
zbMATH DE Number: 847538
zbMATH Keywords: branching process
zbMATH Keywords: random environment
author: Quansheng Liu
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 847538