Conditional limit theorems for critical continuous-state branching processes

Abstract

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.

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