Some asymptotic results for the branching process with immigration
type: Item
instance of: scholarly article
title: Some asymptotic results for the branching process with immigration
zbMATH Open document ID: 0673.60092
DOI: 10.1016/0304-4149(89)90092-6
author: J. Winnicki
author: Ching-Zong Wei
published in: Stochastic Processes and their Applications
publication date: 1989-01-01T00:00:00Z
review text: Let \(\{X_ n\}\) be a branching process with immigration \[ X_ n=\sum^{X_{n-1}}_{j=1}Y_{n,j}+I_ n,\quad T_ n=\sum^{n}_{i=1}(1+X_{i-1})^{-1};\quad \lambda =E I_ 1,\quad m=E Y_{1,1},\quad \sigma^ 2=E(Y_{1,1}-m)^ 2. \] The authors give results on: (i) the convergence of \(X_{[nt]}/n\) in \(D^+[0,\infty];\) (ii) the convergence of \(T_ n/\log n\) if \(2\lambda /\sigma^ 2>1, m=1;\) (iii) the convergence of \(n^{\tau -1}T_ n\) if \(2\lambda /\sigma^ 2<1, m=1;\) (iv) the convergence in \(R^{\infty}\) with the metric \[ d(x,y)=\sum 2^{-i}| x_ i-y_ i| /(1+| x_ i-y_ i|)\quad if\quad m>1. \]
Mathematics Subject Classification ID: 60J80
zbMATH DE Number: 4102249
zbMATH Keywords: branching process with immigration
MaRDI profile type: Publication
full work available at URL: 0304-4149(89)90092-6
OpenAlex ID: W2009410452
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description: scientific article; zbMATH DE number 4102249