A necessary and sufficient condition for a branching process in a random environment to grow like the product of its means
type: Item
instance of: scholarly article
title: A necessary and sufficient condition for a branching process in a random environment to grow like the product of its means
zbMATH Open document ID: 0643.60064
DOI: 10.1016/0304-4149(88)90070-1
author: David Tanny
published in: Stochastic Processes and their Applications
publication date: 1988-01-01T00:00:00Z
review text: Let \(\{Z_ n\}\) be a branching process in a stationary ergodic sequence of environments. Let \(F_ n\) be the distribution of the family size to members of the n th generation, with mean \(m_ n\), conditional on the environmental sequence. The supercritical case, when \(E(\log (m_ 0))>0\), is mainly considered here. Let \(\pi_ n=m_ 0m_ 1...m_{n-1}\), then it is shown that the limit, W, of the martingale \(Z_ n/\pi_ n\) is non-degenerate exactly when \(\sum (m_ n)^{-1}\int x I(x>\pi_{n+1})F_ n(dx)\) is finite almost surely (provided a minor technical condition holds). As a consequence the finiteness of \(E(Z_ 1\log\) \(+(Z_ 1)/m_ 0)\) suffices for non- degeneracy, and this is also necessary if the environments are i.i.d.
Mathematics Subject Classification ID: 60J80
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60G10
zbMATH DE Number: 4048801
zbMATH Keywords: varying environment
zbMATH Keywords: random environment
zbMATH Keywords: stationary ergodic sequence
zbMATH Keywords: supercritical case
MaRDI profile type: Publication
full work available at URL: 0304-4149(88)90070-1
OpenAlex ID: W2037660536
cites work: Q3664178
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description: scientific article; zbMATH DE number 4048801