Limit theorems for continuous-state branching processes with immigration

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)|}\textrm{d} overwhelms Asymptotics in latter case are studied via nonlinear time-dependent renormalization law. Three regimes weak convergence exhibited. Processes critical mechanisms subject to regular variation assumption studied. This article proves extends results stated M. Pinsky ‘Limit theorems continuous state processes immigration’ ( Bull. Amer. Math. Soc. 78 1972).