  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.
