Galton-Watson过程中极限鞅密度函数的Lipschitz连续性  

Lipschitz Continuity of Martingale’s Limit Density Function in Galton-Watson Processes

作　　者：侯婉婷 张美娟[2] HOU Wan-ting;ZHANG Mei-juan(School of Sciences,Northeastern University,Shenyang 110819,China;School of Statistics and Mathematics,Central University of Finance and Economics,Beijing 100081,China)

机构地区：[1]东北大学理学院,辽宁沈阳110819 [2]中央财经大学统计与数学学院,北京100081

出　　处：《东北大学学报（自然科学版）》2020年第10期1517-1520,共4页Journal of Northeastern University(Natural Science)

基　　金：中央高校基本科研业务费专项资金资助项目(N180503019);国家自然科学基金资助项目(11801596);教育部人文社会科学研究规划基金资助项目(19YJA790004).

摘　　要：考虑上临界Galton-Watson过程中第n代粒子总数Z n,令W表示鞅W n=Z n/m^ n的极限.针对W的密度函数ω(x)的Lipschitz连续性问题,基于Kesten-Stigum定理,提出了更完善的证明方法和补充.同时进行了关于鞅极限性质的一系列讨论.首先修正了以往的证明方法,得到在δ≠1的情形下,ω(x)在(ε,∞)中是Lipschitz连续的,阶为δ′=min(δ,1).在δ=1的时,ω(x)的Lipschitz连续性的阶为1/2,从而保证了结论的完整性.Considering the total number Z n of the n-th generation particles in the supercritical Galton-Watson process,let W denote the limit of martingale W n=Z n/m^ n.Aiming at the Lipschitz continuity problem of the density functionω(x)of W,based on the Kesten-Stigum theorem,a more complete proof and supplement were proposed.A series of discussions on the limit properties of martingales were also conducted.First,the previous method of proof was modified,and it was obtained that in the case ofδ≠1,ω(x)is Lipschitz continuous in(ε,∞),and the order isδ′=min(δ,1).Whenδ=1,the order of Lipschitz continuity ofω(x)is 1/2,thus ensuring the completeness of the conclusion.

关 键 词：分枝过程 上临界 鞅收敛 Kesten-Stigum定理 LIPSCHITZ连续 

分 类 号：O211[理学—概率论与数理统计]