Galton-Watson过程中极限鞅密度函数的Lipschitz连续性

doi:10.12068/j.issn.1005-3026.2020.10.022

东北大学学报（自然科学版） ›› 2020, Vol. 41 ›› Issue (10): 1517-1520.DOI: 10.12068/j.issn.1005-3026.2020.10.022

• 数学 • 上一篇

Galton-Watson过程中极限鞅密度函数的Lipschitz连续性

侯婉婷¹，张美娟²

(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)

收稿日期:2019-10-15 修回日期:2019-10-15 出版日期:2020-10-15 发布日期:2020-10-20
通讯作者: 侯婉婷
作者简介:侯婉婷(1989-)，女，辽宁朝阳人，东北大学讲师，博士.
基金资助:
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).

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

HOU Wan-ting¹， ZHANG Mei-juan²

1.School of Sciences， Northeastern University， Shenyang 110819， China； 2.School of Statistics and Mathematics， Central University of Finance and Economics， Beijing 100081， China.

Received:2019-10-15 Revised:2019-10-15 Online:2020-10-15 Published:2020-10-20
Contact: ZHANG Mei-juan
About author:-
Supported by:
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摘要/Abstract

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/mⁿ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.

关键词: 分枝过程, 上临界, 鞅收敛, Kesten-Stigum定理, Lipschitz连续

Abstract: 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/mⁿ. Aiming at the Lipschitz continuity problem of the density function ω(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 of δ≠1，ω(x) is Lipschitz continuous in ［ε，∞)， and the order is δ′=min(δ，1).When δ=1， the order of Lipschitz continuity of ω(x) is 1/2， thus ensuring the completeness of the conclusion.

Key words: branching process, supercritical, martingale convergence, Kesten-Stigum theorem, Lipschitz continuous

中图分类号:

O211

侯婉婷, 张美娟. Galton-Watson过程中极限鞅密度函数的Lipschitz连续性[J]. 东北大学学报（自然科学版）, 2020, 41(10): 1517-1520.

HOU Wan-ting, ZHANG Mei-juan. Lipschitz Continuity of Martingale’s Limit Density Function in Galton-Watson Processes[J]. Journal of Northeastern University(Natural Science), 2020, 41(10): 1517-1520.

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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

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