Limit properties for branching process affected by communicable diseases in random environments

doi:10.3785/j.issn.1008-9497.2022.01.008

Journal of Zhejiang University (Science Edition)

2022, Vol. 49

Issue (1): 53-59 DOI: 10.3785/j.issn.1008-9497.2022.01.008

Mathematics and Computer Science

Limit properties for branching process affected by communicable diseases in random environments

Min REN(

)

School of Mathematics and Statistics, Suzhou College, Suzhou 234000, Anhui Province, China

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Abstract

The model of branching process ${Z n, n ∈ N}$ affected by communicable diseases in independent random environment is proposed,and the limit properties of this model are discussed.Under the normalization factors ${S n, n ∈ N}$ and ${U n, n ∈ N}$ ,the normalization processes ${W ? n, n ∈ N}$ and ${W ˉ n, n ∈ N}$ are studied,the sufficient conditions of ${W ? n, n ∈ N}$ and ${W ˉ n, n ∈ N}$ a.s.and $L 1$ convergence are given; the sufficient condition of ${W ? n, n ∈ N}$ $L 2$ convergence,a sufficient condition and a necessary condition for the limit of ${W ? n, n ∈ N}$ converging to a non-degenerate at 0 random variable are obtained.

Key words： random environments communicable diseases branching process almost everywhere convergence

Received: 23 November 2020 Published: 18 January 2022

CLC:

O 211.65

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Cite this article:

Min REN. Limit properties for branching process affected by communicable diseases in random environments. Journal of Zhejiang University (Science Edition), 2022, 49(1): 53-59.

URL:

https://www.zjujournals.com/sci/EN/Y2022/V49/I1/53

随机环境中受传染性疾病影响的分枝过程的极限性质

给出了独立随机环境中受传染性疾病影响的分枝过程 ${Z n, n ∈ N}$ 的模型，讨论了该模型的极限性质，并给出了分枝过程经 ${S n, n ∈ N}$ 和 ${U n, n ∈ N}$ 规范化后 ${W ? n, n ∈ N}$ 和 ${W ˉ n, n ∈ N}$ 几乎处处收敛和 $L 1$ 收敛的充分条件，得到 ${W ? n, n ∈ N}$ $L 2$ 收敛的充分条件和 ${W ? n, n ∈ N}$ 极限非退化到0的充分条件和必要条件。

关键词： 随机环境, 传染性疾病, 分枝过程, 几乎处处收敛


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