数学物理学报 ›› 2022, Vol. 42 ›› Issue (3): 957-974.doi: 10.1007/s10473-022-0309-3

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CENTRAL LIMIT THEOREM AND CONVERGENCE RATES FOR A SUPERCRITICAL BRANCHING PROCESS WITH IMMIGRATION IN A RANDOM ENVIRONMENT

李应求1, 黄绪兰2, 彭朝晖2   

  1. 1. Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha, 410004, China;
    2. School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha, 410004, China
  • 收稿日期:2020-08-03 修回日期:2021-08-31 出版日期:2022-06-26 发布日期:2022-06-24
  • 通讯作者: Yingqiu LI,E-mail:liyq-2001@163.com E-mail:liyq-2001@163.com
  • 基金资助:
    This work was supported by the National Natural Science Foundation of China (11571052, 11731012), the Hunan Provincial Natural Science Foundation of China (2018JJ2417), and the Open Fund of Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering (2018MMAEZD02).

CENTRAL LIMIT THEOREM AND CONVERGENCE RATES FOR A SUPERCRITICAL BRANCHING PROCESS WITH IMMIGRATION IN A RANDOM ENVIRONMENT

Yingqiu LI1, Xulan HUANG2, Zhaohui PENG2   

  1. 1. Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha, 410004, China;
    2. School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha, 410004, China
  • Received:2020-08-03 Revised:2021-08-31 Online:2022-06-26 Published:2022-06-24
  • Contact: Yingqiu LI,E-mail:liyq-2001@163.com E-mail:liyq-2001@163.com
  • Supported by:
    This work was supported by the National Natural Science Foundation of China (11571052, 11731012), the Hunan Provincial Natural Science Foundation of China (2018JJ2417), and the Open Fund of Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering (2018MMAEZD02).

摘要: We are interested in the convergence rates of the submartingale ${W}_{n} =\frac{Z_{n}}{\Pi_{n}}$ to its limit ${W},$ where $(\Pi_{n})$ is the usually used norming sequence and $(Z_{n})$ is a supercritical branching process with immigration $(Y_{n})$ in a stationary and ergodic environment $\xi$. Under suitable conditions, we establish the following central limit theorems and results about the rates of convergence in probability or in law: (i) $W-W_{n}$ with suitable normalization converges to the normal law $N(0,1)$, and similar results also hold for $W_{n+k}-W_{n}$ for each fixed $k\in \mathbb{N}^{\ast};$ (ii) for a branching process with immigration in a finite state random environment, if $W_{1}$ has a finite exponential moment, then so does $W,$ and the decay rate of $\mathbb{P}(|W-W_{n}|>\varepsilon)$ is supergeometric; (iii) there are normalizing constants $a_{n}(\xi)$ (that we calculate explicitly) such that $a_{n}(\xi)(W-W_{n})$ converges in law to a mixture of the Gaussian law.

关键词: Branching process with immigration, random environment, convergence rates, central limit theorem, convergence in law, convergence in probability

Abstract: We are interested in the convergence rates of the submartingale ${W}_{n} =\frac{Z_{n}}{\Pi_{n}}$ to its limit ${W},$ where $(\Pi_{n})$ is the usually used norming sequence and $(Z_{n})$ is a supercritical branching process with immigration $(Y_{n})$ in a stationary and ergodic environment $\xi$. Under suitable conditions, we establish the following central limit theorems and results about the rates of convergence in probability or in law: (i) $W-W_{n}$ with suitable normalization converges to the normal law $N(0,1)$, and similar results also hold for $W_{n+k}-W_{n}$ for each fixed $k\in \mathbb{N}^{\ast};$ (ii) for a branching process with immigration in a finite state random environment, if $W_{1}$ has a finite exponential moment, then so does $W,$ and the decay rate of $\mathbb{P}(|W-W_{n}|>\varepsilon)$ is supergeometric; (iii) there are normalizing constants $a_{n}(\xi)$ (that we calculate explicitly) such that $a_{n}(\xi)(W-W_{n})$ converges in law to a mixture of the Gaussian law.

Key words: Branching process with immigration, random environment, convergence rates, central limit theorem, convergence in law, convergence in probability

中图分类号: 

  • 60J80