Exact convergence rate of the central limit theorem and polynomial convergence rate for branching processes in a random environment

IF 0.9 4区数学 Q3 STATISTICS & PROBABILITY

Statistics & Probability Letters Pub Date : 2024-09-10 DOI:10.1016/j.spl.2024.110268

Yingqiu Li , Xin Zhang , Zhan Lu , Sheng Xiao

引用次数: 0

Abstract

Let $(Z_{n})$ be a supercritical branching process in an independent and identically distributed (i.i.d.) random environment. The paper studies the properties of the estimator $M_{n} = n^{- 1} \sum_{k = 0}^{n - 1} (Z_{k + 1} / Z_{k})$ introduced by Dion and Esty in 1979. We introduce a related martingale and discuss its convergence and exponential convergence rate. On this basis the exact convergence rate of the central limit theorem for normalized $M_{n}$ is given.

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随机环境中分支过程的中心极限定理精确收敛率和多项式收敛率

设 (Zn) 是独立且同分布（i.i.d. ）随机环境中的超临界分支过程。本文研究了 Dion 和 Esty 于 1979 年提出的估计器 Mn=n-1∑k=0n-1(Zk+1/Zk) 的性质。我们引入了一个相关的鞅，并讨论了它的收敛性和指数收敛率。在此基础上，给出了归一化 Mn 的中心极限定理的精确收敛率。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Statistics & Probability Letters 数学-统计学与概率论

CiteScore

1.60

自引率

0.00%

发文量

173

审稿时长

6 months

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