Limit theorems for continuous-state branching processes with immigration

• Published in: Advances in applied probability Vol. 54; no. 2; pp. 599 - 624
• Main Authors: Foucart, Clément, Ma, Chunhua, Yuan, Linglong
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.06.2022; Applied Probability Trust
• Subjects: Asymptotic properties; Immigration; Laplace transforms; Mathematics; Original Article; Probability; Random variables; Theorems; limit distribution; nonlinear renormalization; immigration; Continuous-state branching processes; 60J80; Grey’s martingale; 60F05; regularly varying functions; 60F15
• ISSN: 0001-8678; 1475-6064
• DOI: 10.1017/apr.2021.43

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).