Conditional limit theorems for critical continuous-state branching processes

In this paper we study the conditional limit theorems for critical continuous-state branching processes with branching mechanism $\psi(\lambda)=\lambda^{1+\alpha}L(1/\lambda)$ where $\alpha\in [0,1]$ and $L$ is slowly varying at $\infty$. We prove that if $\alpha\in (0,1]$, there are norming constants $Q_{t}\to 0$ (as $t\uparrow +\infty$) such that for every $x>0$, $P_{x}\left(Q_{t}X_{t}\in\cdot|X_{t}>0\right)$ converges weakly to a non-degenerate limit. The converse assertion is also true provided the regularity of $\psi$ at 0. We give a conditional limit theorem for the case $\alpha=0$. The limit theorems we obtain in this paper allow infinite variance of the branching process.