Conditional limit theorems for critical continuous-state branching processes  

作　　者：REN YanXia YANG Ting ZHAO GuoHuan 

机构地区：[1]LMAM, School of Mathematical Sciences, Peking University [2]Center for Statistical Science, Peking University [3]Academy of Mathematics and Systems Science, Chinese Academy of Sciences

出　　处：《Science China Mathematics》2014年第12期2577-2588,共12页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China(Grant Nos.11271030 and 11128101);Specialized Research Fund for the Doctoral Program of Higher Education and China Postdoctoral Science Foundation(Grant No.2013M541061)

摘　　要：We study the conditional limit theorems for critical continuous-state branching processes with branching mechanism Ф（λ） = λ 1＋αL（1/λ）, where （α∈ [0, 1] and L is slowly varying at co. 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.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 at0. 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.

关 键 词：continuous-state branching process conditional laws regular variation 

分 类 号：O211.4[理学—概率论与数理统计] O213.2[理学—数学]