Limit distributions for branching processes with immigration
type: Item
instance of: scholarly article
title: Limit distributions for branching processes with immigration
zbMATH Open document ID: 0971.60078
author: A. V. Shipileva
published in: Russian Mathematics
publication date: 2000-12-07T00:00:00Z
review text: Let \(f(z)\) be the infinitesimal generating function of the number of children of a particle of the basic process and \(g(z)\) be the infinitesimal generating function of the number of immigrants in a Markov continuous time branching process \(\mu (t)\) with immigration. It is know that if \(f'(1)<0\) and \(g'(1)<\infty,\) then \(\lim_{t\to\infty}P\{\mu (t) =k\}=P_k\) exists for any \(k=0,1,\ldots,\) and \(\sum_{k=0}^{\infty} P_k=1.\) The author solves, in a sense, an inverse problem. Namely, she shows that, for a fixed \(g(z),\) a probability distribution \(\{P_k\), \(k=0,1,\ldots\}\) may be limiting for a Markov continuous time branching process with immigration if and only if \(P_k\) obey a system of inequalities expressed in terms of the coefficients in the Taylor expansion of \(g(z).\)
Mathematics Subject Classification ID: 60J80
Mathematics Subject Classification ID: 60F05
zbMATH DE Number: 1541564
zbMATH Keywords: subcritical branching processes
zbMATH Keywords: immigration
zbMATH Keywords: limit theorems
zbMATH Keywords: interpolation problem
reviewed by: Vladimir Vatutin
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1541564