Abstract A continuous-state branching process with immigration having mechanism $\Psi$ and $\Phi$ , a CBI $(\Psi,\Phi)$ 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 u=\infty$ . When the has limit distribution growth rate dictated by dynamics. $\scriptstyle\int_{0}\tfrac{\Phi(u)}{|\Psi(u)|}\textr...