论文标题
在不同环境中的Galton-Watson流程有缺陷
Defective Galton-Watson processes in a varying environment
论文作者
论文摘要
我们研究了通过允许后代分布在几代人的变化中获得的所谓有缺陷的Galton-Watson过程的扩展。因此，在这些过程中，个体独立于其他人的繁殖，并根据一代人的某些有缺陷的后代分布。此外，$ n $的后代分布的缺陷$ 1-f_n（1）$表示该过程在该一代人中达到吸收状态$δ$的概率。我们专注于这些过程的渐近行为。我们建立了该过程与随机变量的几乎确定的收敛性，并在$ \ mathbb {n} _0 \ cup \ {δ\} $中建立了值，我们提供了$δ$的双重灭绝 - 吸收二元性的两个特征。我们还陈述了一些结果，以及该过程以其不吸收的条件为条件，其中一些要求我们在不同环境中引入有缺陷的分支树的概念。
We study an extension of the so-called defective Galton-Watson processes obtained by allowing the offspring distribution to change over the generations. Thus, in these processes, the individuals reproduce independently of the others and in accordance to some possibly defective offspring distribution depending on the generation. Moreover, the defect $1-f_n(1)$ of the offspring distribution at generation $n$ represents the probability that the process hits an absorbing state $Δ$ at that generation. We focus on the asymptotic behaviour of these processes. We establish the almost sure convergence of the process to a random variable with values in $\mathbb{N}_0\cup\{Δ\}$ and we provide two characterisations of the duality extinction-absorption at $Δ$. We also state some results on the absorption time and the properties of the process conditioned upon its non-absorption, some of which require us to introduce the notion of defective branching trees in varying environment.
