arXiv 2006.165742020-06math.PR

具有可变生命时的Galton-Watson过程的遍历性

Ergodic property for Galton-Watson processes in which individuals have variable life times

J. R. Tan, J. P. Li

摘要：

本文研究了一个扩展的Galton-Watson过程，使个体能够在多个单位时间内生存和繁殖。我们假设每个个体可以以概率$h_k$生活在$k$季节（时间单位），并在每个季节以概率$p_m$产生$m$后代。这些可以看作是具有可数无穷多个类型的高尔顿-沃森过程，其中$i$类型的粒子只能有$i+1$和$1$类型的后代。令$\textbf{M}$为其平均子代矩阵，$γ$为其幂级数$\sum_{k\geq 0}r^k(\textbf{M}^k)_{ij}$的收敛半径。本文首先导出了计算$γ$的公式，并证明了在超临界情况下，$γ$实际上是Galton-Watson过程的灭绝概率.其次，我们给出了$\textbf{M}$是$γ$ -瞬时的，$γ$ -正的和$γ$ -零常返的明确判据，并由此讨论了该过程的遍历性. $\textbf{M}$的$γ$和$γ$ -重现性的判据依赖于寿命分布的性质，这比现有的结果更容易验证。最后，我们给出了在一定条件下每类个体的总种群规模的渐近行为，说明了具有可变寿命个体的Galton-Watson过程的演化.

英文摘要：

This paper is concerned with an extended Galton-Watson process so as to allow individuals to live and reproduce for more than one unit time. We assume that each individual can live $k$ seasons (time-units) with probability $h_k$, and produce $m$ offspring with probability $p_m$ during each season. These can be seen as Galton-Watson processes with countably infinitely many types in which particles of type $i$ may only have offspring of type $i+1$ and type $1$. Let $\textbf{M}$ be its mean progeny matrix and $γ$ be the convergence radius of the power series $\sum_{k\geq 0}r^k(\textbf{M}^k)_{ij}$. We first derive formula of calculating $γ$ and show that $γ$, in supercritical case, is actually the extinction probability of a Galton-Watson process. Next, we give clear criteria for $\textbf{M}$ to be $γ$-transient, $γ$-positive and $γ$-null recurrent from which the ergodic property of the process is discussed. The criteria for $γ$ and $γ$-recurrence of $\textbf{M}$ rely on the properties of lifetime distribution which are easier to be verified than current results. Finally, we show the asymptotic behavior of the total population size of each type of individuals under certain conditions which illustrates the evolution of Galton-Watson process in which individuals have variable lifetimes.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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