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Limit theorems for continuous-state branching processes with immigration

Part of: Limit theorems Markov processes

Published online by Cambridge University Press:  06 June 2022

Clément Foucart ,
Chunhua Ma  and
Linglong Yuan
Clément Foucart*
Affiliation:
Université SorbonneParis Nord
Chunhua Ma*
Affiliation:
Nankai University
Linglong Yuan*
Affiliation:
University of Liverpool, Xi’an Jiaotong-Liverpool University
*
*Postal address: Laboratoire Analyse, Géométrie & Applications, UMR 7539, Institut Galilée, Université Sorbonne Paris Nord, Villetaneuse, 93430, France. Email address: foucart@math.univparis13.fr
**Postal address: School of Mathematical Sciences and LPMC, Nankai University, Tianjin, 300071, P. R. China. Email address: mach@nankai.edu.cn
***Postal address: Department of Mathematical Sciences, University of Liverpool, Liverpool, L69 7ZL, UK. Email address: yuanlinglongcn@gmail.com
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Abstract

A continuous-state branching process with immigration having branching mechanism $\Psi$ and immigration mechanism $\Phi$ , a CBI $(\Psi,\Phi)$ process for short, may have either of two different asymptotic regimes, depending on whether $\int_{0}\frac{\Phi(u)}{|\Psi(u)|}\textrm{d} u<\infty$ or $\int_{0}\frac{\Phi(u)}{|\Psi(u)|}\textrm{d} u=\infty$ . When $\int_{0}\frac{\Phi(u)}{|\Psi(u)|}\textrm{d} u<\infty$ , the CBI process has either a limit distribution or a growth rate dictated by the branching dynamics. When $\scriptstyle\int_{0}\tfrac{\Phi(u)}{|\Psi(u)|}\textrm{d} u=\infty$ , immigration overwhelms branching dynamics. Asymptotics in the latter case are studied via a nonlinear time-dependent renormalization in law. Three regimes of weak convergence are exhibited. Processes with critical branching mechanisms subject to a regular variation assumption are studied. This article proves and extends results stated by M. Pinsky in ‘Limit theorems for continuous state branching processes with immigration’ (Bull. Amer. Math. Soc.78, 1972).

Keywords

Continuous-state branching processesimmigrationGrey’s martingalelimit distributionnonlinear renormalizationregularly varying functions

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60F05: Central limit and other weak theorems 60F15: Strong theorems
Type
Original Article
Information
Advances in Applied Probability , Volume 54 , Issue 2 , June 2022 , pp. 599 - 624
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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