论文标题
分支过程中具有多种类型的分支过程中的灭绝概率：一般框架
Extinction probabilities in branching processes with countably many types: a general framework
论文作者
论文摘要
我们考虑使用可计数的类型$ \ Mathcal {x} $的Galton-Watson分支过程。我们研究矢量$ {\ bf q}（a）=（q_x（a））_ {x \ in \ Mathcal {x}} $记录在类型$ a \ subseteq \ Mathcal {x} $的类型$ a \ subseteq \ mathcal {x} $的子集中灭绝的条件概率。我们首先研究矢量的位置$ {\ bf q}（a）$在后代生成向量的固定点集中，并证明$ q_x（\ {x \}）$大于或等于$ x $的$ x $ th任何固定点的输入，每当$ q_ $ q_ q_ $ q_ q_ x a）（a）（b）（b）<q_x（a）<q_x（a） $ a，b \ subseteq \ mathcal {x} $。最后，我们开发了一个一般框架来表征所有\ emph {不同的}灭绝概率向量，从而确定是否有很多有限的，数量很多，或者是无数截然不同的向量。我们以示例来说明结果，并以开放的问题结论。
We consider Galton-Watson branching processes with countable typeset $\mathcal{X}$. We study the vectors ${\bf q}(A)=(q_x(A))_{x\in\mathcal{X}}$ recording the conditional probabilities of extinction in subsets of types $A\subseteq \mathcal{X}$, given that the type of the initial individual is $x$. We first investigate the location of the vectors ${\bf q}(A)$ in the set of fixed points of the progeny generating vector and prove that $q_x(\{x\})$ is larger than or equal to the $x$th entry of any fixed point, whenever it is different from 1. Next, we present equivalent conditions for $q_x(A)< q_x (B)$ for any initial type $x$ and $A,B\subseteq \mathcal{X}$. Finally, we develop a general framework to characterise all \emph{distinct} extinction probability vectors, and thereby to determine whether there are finitely many, countably many, or uncountably many distinct vectors. We illustrate our results with examples, and conclude with open questions.
