随机环境下两个上临界分支过程的参数比较

数学物理学报 ›› 2023, Vol. 43 ›› Issue (5): 1440-1470.

随机环境下两个上临界分支过程的参数比较

范协铨^1,^*(),胡海娟¹,吴浩²,叶印娜³()

¹东北大学秦皇岛分校数学与统计学院河北秦皇岛 066003
²天津大学应用数学中心天津 300072
³西交利物浦大学数学物理学院江苏苏州 215123

收稿日期:2022-06-30 修回日期:2023-03-24 出版日期:2023-10-26 发布日期:2023-08-09
通讯作者: 范协铨 E-mail:fanxiequan@hotmail.com;yinna.ye@xjtlu.edu.cn
作者简介:叶印娜，Email: yinna.ye@xjtlu.edu.cn
基金资助:
国家自然科学基金(11971063)

Comparison on the Criticality Parameters for Two Supercritical Branching Processes in Random Environments

Fan Xiequan^1,^*(),Hu Haijuan¹,Wu Hao²,Ye Yinna³()

¹School of Mathematics and Statistics, Northeastern University at Qinhuangdao, Hebei Qinhuangdao 066003
²Center for Applied Mathematics, Tianjin University, Tianjin 300072
³School of Mathematics and Physics, Xi'an Jiaotong-Liverpool University, Jiangsu Suzhou 215123

Received:2022-06-30 Revised:2023-03-24 Online:2023-10-26 Published:2023-08-09
Contact: Xiequan Fan E-mail:fanxiequan@hotmail.com;yinna.ye@xjtlu.edu.cn
Supported by:
NSFC(11971063)

摘要/Abstract

摘要：

设 $\left(Z_{1, n}\right)_{ n \geq 0}$ 和 $\left(Z_{2, n}\right)_{ n \geq 0}$ 是两个在独立同分布随机环境下的上临界分支过程, 并且其关键参数分别为 $\mu_1$ 和 $\mu_2$. 容易知道, 在适当条件下, $\frac{1}{n} \ln Z_{1,n} $ 和 $\frac{1}{m} \ln Z_{2,m}$分别依概率收敛到 $\mu_1$ 和 $\mu_2$. 该文旨在讨论两个上临界分支过程的关键参数之差 $\mu_1-\mu_2$ 的估计问题, 它可以被看作是一类双样本 $U$ 统计量问题. 我们得到了 $\frac{1}{n} \ln Z_{1, n}-\frac{1}{m} \ln Z_{2, m}$ 的中心极限定理, 非一致性 Berry-Esseen 估计和 Cramér 型中偏差. 最后, 作为应用部分, 指出了以上的结果可用于关键参数置信区间的构造.

关键词: 分支过程, 随机环境, Berry-Esseen 估计, Cramér 型中偏差

Abstract:

Let $\{Z_{1,n}, n\geq 0\}$ and $\{Z_{2,n}, n\geq 0\}$ be two supercritical branching processes in different random environments, with criticality parameters $\mu_1$ and $\mu_2$ respectively. It is known that with certain conditions, $\frac{1}{n} \ln Z_{1,n} \rightarrow \mu_1$ and $\frac{1}{m} \ln Z_{2,m} \rightarrow \mu_2$ in probability as $m, n \rightarrow \infty.$ In this paper, we are interested in the comparison on the two criticality parameters, which can be regarded as two-sample $U$-statistic. To this end, we prove a non-uniform Berry-Esseen's bound and Cramér's moderate deviations for $\frac{1}{n} \ln Z_{1,n} - \frac{1}{m} \ln Z_{2,m}$ as $m, n \rightarrow \infty.$ An application is also given for constructing confidence intervals of $\mu_1-\mu_2$.

Key words: Branching processes, Random environments, Berry-Esseen's bound, Cramér's moderate deviations

中图分类号:

O211.65

范协铨,胡海娟,吴浩,叶印娜. 随机环境下两个上临界分支过程的参数比较[J]. 数学物理学报, 2023, 43(5): 1440-1470.

Fan Xiequan,Hu Haijuan,Wu Hao,Ye Yinna. Comparison on the Criticality Parameters for Two Supercritical Branching Processes in Random Environments[J]. Acta mathematica scientia,Series A, 2023, 43(5): 1440-1470.

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