Conditional limit theorems for critical continuous-state branching processes |
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Authors: | YanXia Ren Ting Yang GuoHuan Zhao |
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Affiliation: | 1. LMAM, School of Mathematical Sciences, Peking University, Beijing, 100871, China 2. Center for Statistical Science, Peking University, Beijing, 100871, China 3. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100080, China
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Abstract: | We study the conditional limit theorems for critical continuous-state branching processes with branching mechanism ψ(λ) = λ1+α L(1/λ), where α ∈ [0, 1] and L is slowly varying at ∞. We prove that if α ∈ (0, 1], there are norming constants Q t → 0 (as t ↑ +∞) such that for every x > 0, P x (Q t X t ∈ · |X t > 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. |
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Keywords: | continuous-state branching process conditional laws regular variation |
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