Conditional limit theorems for critical continuous-state branching processes

We study the conditional limit theorems for critical continuous-state branching processes with branching mechanism ψ(λ) = λ1+αL(1/λ), where α ∈ [0, 1] and L is slowly varying at ∞. We prove that if α ∈ (0, 1], there are norming constants Qt → 0 (as t ↑ +∞) such that for every x > 0, Px(QtXt ∈ · |Xt > 0) 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.