论文标题
Galton-Watson流程的Ergodic属性，其中个人具有可变的寿命时间
Ergodic property for Galton-Watson processes in which individuals have variable life times
论文作者
论文摘要
本文关注的是扩展的Galton-Watson过程，以使个人可以生活和繁殖超过一个单位时间。我们假设每个人都可以用概率$ h_k $生存$ k $季节（时间单元），并在每个季节中生产$ m $ $ $ $ p_m $。这些可以看作是Galton-Watson流程，具有无限多种类型，其中$ i $的粒子可能只有$ i+1 $ $ 1 $和类型$ 1 $的后代。令$ \ textbf {m} $为其平均后代矩阵，$γ$为功率系列$ \ sum_ {k \ geq 0} r^k（\ textbf {m}^k）_ {ij} $的收敛半径。我们首先得出计算$γ$的公式，并表明在超临界情况下，$γ$实际上是Galton-Watson过程的灭绝概率。接下来，我们给出$ \ textbf {m} $的明确标准，为$γ$ - transient，$γ$ - 阳性和$γ$ - null recurrent，从该过程中讨论了该过程的ergodic属性。 $γ$和$γ$ - $ \ textbf {m} $的$γ$ - 重点的标准取决于终身分布的属性，这些属性比当前结果更容易验证。最后，我们在某些条件下显示了每种类型个体的总人口大小的渐近行为，这说明了Galton-Watson过程的演变，其中个体具有可变的寿命。
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.
