Diffusion approximation of branching migration processes
type: Item
instance of: scholarly article
title: Diffusion approximation of branching migration processes
zbMATH Open document ID: 0947.60083
DOI: 10.1007/BF02365056
published in: Journal of Mathematical Sciences (New York)
publication date: 1999-11-22T00:00:00Z
review text: Consider a branching process \(\{Z^{(k)}_0,Z^{(k)}_1,\ldots\}\) with \(Z^{(k)}_0=k\), such that for \(Z^{(k)}_n>0\), either \(j\geq 0\) particles immigrate into the population \(Z^{(k)}_n\) with probability \(q_j\) or one particle emigrates with probability \(r\) (\(\sum_{j=0}^\infty q_j+r=1\)). After this, all particles independently produce offspring according to the same probability distribution, the total number of offspring being \(Z^{(k)}_{n+1}\). Assume that the offspring mean is \(1+\alpha/k\) and the offspring second factorial moment is \(2b\). It is shown that subject to regularity conditions, for any \(t\geq 0\), \(Z_{[kt]}^{(k)}/k\) converges in distribution as \(k\to\infty\) to \(x_t\), where \(\{x_t, t\geq 0\}\), \(x_0=1\), is the diffusion with infinitesimal drift \(\alpha x\) and variance \(bx\), having zero as an absorbing barrier.
Mathematics Subject Classification ID: 60J80
Mathematics Subject Classification ID: 60F05
zbMATH DE Number: 1295565
zbMATH Keywords: diffusion approximation
zbMATH Keywords: Galton-Watson process
zbMATH Keywords: immigration
zbMATH Keywords: emigration
author: Elena E. Dyakonova
reviewed by: M. P. Quine
MaRDI profile type: Publication
cites work: Limit Theorems for a Critical Galton–Watson Branching Process with Migration
cites work: Q4031597
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full work available at URL: https://doi.org/10.1007/bf02365056
OpenAlex ID: W2085575914
description: scientific article; zbMATH DE number 1295565