Limit theorems for continuous-state branching processes with immigration

• Published in: arXiv.org
• Main Authors: Foucart, Clément, Ma, Chunhua, Yuan, Linglong
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 21.07.2021
• Subjects: Asymptotic properties; Convergence; Immigration; Random variables; Theorems; Time dependence
• ISSN: 2331-8422

We prove and extend some results stated by Mark Pinsky: Limit theorems for continuous state branching processes with immigration [Bull. Amer. Math. Soc. 78(1972), 242--244]. Consider a continuous-state branching process with immigration \((Y_t,t\geq 0)\) with branching mechanism \(\Psi\) and immigration mechanism \(\Phi\) (CBI\((\Psi,\Phi)\) for short). We shed some light on two different asymptotic regimes occurring when \(\int_{0}\frac{\Phi(u)}{|\Psi(u)|}du<\infty\) or \(\int_{0}\frac{\Phi(u)}{|\Psi(u)|}du=\infty\). We first observe that when \(\int_{0}\frac{\Phi(u)}{|\Psi(u)|}du<\infty\), supercritical CBIs have a growth rate dictated by the branching dynamics, namely there is a renormalization \(\tau(t)\), only depending on \(\Psi\), such that \((\tau(t)Y_t,t\geq 0)\) converges almost-surely to a finite random variable. When \(\int_{0}\frac{\Phi(u)}{|\Psi(u)|}du=\infty\), it is shown that the immigration overwhelms the branching dynamics and that no linear renormalization of the process can exist. Asymptotics in the second regime are studied in details for all non-critical CBI processes via a nonlinear time-dependent renormalization in law. Three regimes of weak convergence are then exhibited, where a misprint in Pinsky's paper is corrected. CBI processes with critical branching mechanisms subject to a regular variation assumption are also studied.