Scaling limits and fluctuations of a family of N-urn branching processes

IF 0.6 4区 数学 Q4 STATISTICS & PROBABILITY
Xiaofeng Xue
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引用次数: 0

Abstract

In this paper we are concerned with a family of $N$-urn branching processes, where some particles are put into $N$ urns initially and then each particle gives birth to several new particles in some urn when dies. This model includes the $N$-urn Ehrenfest model and the $N$-urn branching random walk as special cases. We show that the scaling limit of the process is driven by a $C(\mathbb{T})$-valued linear ordinary differential equation and the fluctuation of the process is driven by a generalized Ornstein-Uhlenbeck process in the dual of $C^\infty(\mathbb{T})$, where $\mathbb{T}=(0, 1]$ is the one-dimensional torus. A crucial step for proofs of above main results is to show that numbers of particles in different urns are approximately independent. As applications of our main results, limit theorems of hitting times of the process are also discussed.
一类N-urn分支过程的标度极限和涨落
本文研究了一类$N$ -瓮分支过程,其中一些粒子最初被放入$N$瓮中,每个粒子死亡后在另一个瓮中产生几个新粒子。该模型包括$N$ -urn Ehrenfest模型和$N$ -urn分支随机漫步作为特例。我们证明了该过程的标度极限是由$C(\mathbb{T})$值线性常微分方程驱动的,过程的涨落是由$C^\infty(\mathbb{T})$对偶中的广义Ornstein-Uhlenbeck过程驱动的,其中$\mathbb{T}=(0, 1]$是一维环面。证明上述主要结果的一个关键步骤是证明不同回合中的粒子数量近似独立。作为主要结果的应用,还讨论了该过程命中次数的极限定理。
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
CiteScore
1.60
自引率
10.00%
发文量
30
审稿时长
>12 weeks
期刊介绍: The Brazilian Journal of Probability and Statistics aims to publish high quality research papers in applied probability, applied statistics, computational statistics, mathematical statistics, probability theory and stochastic processes. More specifically, the following types of contributions will be considered: (i) Original articles dealing with methodological developments, comparison of competing techniques or their computational aspects. (ii) Original articles developing theoretical results. (iii) Articles that contain novel applications of existing methodologies to practical problems. For these papers the focus is in the importance and originality of the applied problem, as well as, applications of the best available methodologies to solve it. (iv) Survey articles containing a thorough coverage of topics of broad interest to probability and statistics. The journal will occasionally publish book reviews, invited papers and essays on the teaching of statistics.
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