Limit theorems for continuous-state branching processes with immigration



Foucart, Clément, Ma, Chunhua and Yuan, Linglong ORCID: 0000-0002-7851-1631
(2022) Limit theorems for continuous-state branching processes with immigration. Advances in Applied Probability, 54 (2). pp. 599-624.

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Abstract

<jats:title>Abstract</jats:title><jats:p>A continuous-state branching process with immigration having branching mechanism <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0001867821000434_inline1.png" /><jats:tex-math> $\Psi$ </jats:tex-math></jats:alternatives></jats:inline-formula> and immigration mechanism <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0001867821000434_inline2.png" /><jats:tex-math> $\Phi$ </jats:tex-math></jats:alternatives></jats:inline-formula>, a CBI<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0001867821000434_inline3.png" /><jats:tex-math> $(\Psi,\Phi)$ </jats:tex-math></jats:alternatives></jats:inline-formula> process for short, may have either of two different asymptotic regimes, depending on whether <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0001867821000434_inline4.png" /><jats:tex-math> $\int_{0}\frac{\Phi(u)}{|\Psi(u)|}\textrm{d} u&lt;\infty$ </jats:tex-math></jats:alternatives></jats:inline-formula> or <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0001867821000434_inline5.png" /><jats:tex-math> $\int_{0}\frac{\Phi(u)}{|\Psi(u)|}\textrm{d} u=\infty$ </jats:tex-math></jats:alternatives></jats:inline-formula>. When <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0001867821000434_inline6.png" /><jats:tex-math> $\int_{0}\frac{\Phi(u)}{|\Psi(u)|}\textrm{d} u&lt;\infty$ </jats:tex-math></jats:alternatives></jats:inline-formula>, the CBI process has either a limit distribution or a growth rate dictated by the branching dynamics. When <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0001867821000434_inline7.png" /><jats:tex-math> $\scriptstyle\int_{0}\tfrac{\Phi(u)}{|\Psi(u)|}\textrm{d} u=\infty$ </jats:tex-math></jats:alternatives></jats:inline-formula>, 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’ (<jats:italic>Bull. Amer. Math. Soc.</jats:italic><jats:bold>78</jats:bold>, 1972).</jats:p>

Item Type: Article
Divisions: Faculty of Science and Engineering > School of Physical Sciences
Depositing User: Symplectic Admin
Date Deposited: 07 Jul 2022 15:26
Last Modified: 18 Jan 2023 20:56
DOI: 10.1017/apr.2021.43
Related URLs:
URI: https://livrepository.liverpool.ac.uk/id/eprint/3157886