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 $ψ(λ)=λ^{1+α}L(1/λ)$ where $α\in [0,1]$ and $L$ is slowly varying at $\infty$. We prove that if $α\in (0,1]$, 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 $ψ$ at 0. We give a conditional limit theorem for the case $α=0$. The limit theorems we obtain in this paper allow infinite variance of the branching process.

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