| 张美娟,毕航达,张小玥.临界Galton-Watson过程条件极限定理的构造性证明[J].数学年刊A辑,2023,44(1):39~56 |
| 临界Galton-Watson过程条件极限定理的构造性证明 |
| Conceptual Proofs of Conditional Limit Theorems for Critical Galton-Watson Processes |
| Received:October 08, 2021 Revised:October 12, 2022 |
| DOI：10.16205/j.cnki.cama.2023.0004 |
| 中文关键词: Galton-Watson 过程, 条件极限分布, 最近共同祖先, 指数分布 |
| 英文关键词:Galton-Watson process, Conditional limit distribution, The most recent common ancestor, Exponential distribution |
| 基金项目:国家自然科学基金 (No.12271538, No.11971062, No.12101419),国家留学基金, 中央财经大学新兴交叉学科建设项目和中央财经大学学科建设项目 |
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| 中文摘要: |
| 对临界Galton-Watson 过程,本文通过精细地构造条件Galton-Watson 树的方法, 在第n代不灭绝的条件下研究第nt代粒子数Znt的构造性性质(0ntl和Zntr.本文分别给出了{Zntl/n│Zn>0 }和{Zntr/n│Zn>0} 的条件极限性质,用概率的方法部分地解释了Spitzer, Lamperti和Ney的经典条件极限的结果. 最后还给出了最近共同祖先的条件分布. |
| 英文摘要: |
| Consider critical Galton-Watson processes. The authors study the constructive properties of the number of particles in the nt-th generation, which is denoted by Znt with fixed 0 < t < 1, conditioned on non-extinction of the n-th generation. The proofs are based on constructing a conditioned Galton-Watson tree. By constructing a distinguished branch,particles in the nt-th generation of the conditioned Galton-Waltson tree can be divided into two parts, denoted by Zntl and Zntr.The conditional asymptotic properties of {Zntl/n│Zn>0 }and {Zntr/n│Zn>0} are obtained separately. This approach partly explains the classical conditional limit result of Spitzer, Lamperti and Ney by a probabilistic method. At last,the authors give the conditional distribution of the most recent common ancestor. |
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