Conditional limit theorems for critical continuous-state branching processes

We study the conditional limit theorems for critical continuous-state branching processes with branching mechanism psi(lambda) = lambda L1+alpha(1/lambda), where alpha is an element of [0,1] and L is slowly varying at infinity. We prove that if alpha is an element of (0,1], there are norming constants Q(t) -> 0 (as t up arrow +infinity) such that for every x > 0, P-x(Q(t)X(t) is an element of.vertical bar X-t > 0) 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.