We consider a class of multitype Galton–Watson branching processes with a countably infinite type set $\mathcal{X}_d$ whose mean progeny matrices have a block lower Hessenberg form. For these processes, we study the probabilities $\textbf{\textit{q}}(A)$ of extinction in sets of types $A\subseteq \mathcal{X}_d$ . We compare $\textbf{\textit{q}}(A)$ with the global extinction probability $\textbf{\textit{q}} = \textbf{\textit{q}}(\mathcal{X}_d)$ , that is, the probability that the population eventually becomes empty, and with the partial extinction probability $\tilde{\textbf{\textit{q}}}$ , that is, the probability that all types eventually disappear from the population. After deriving partial and global extinction criteria, we develop conditions for $\textbf{\textit{q}} < \textbf{\textit{q}}(A) < \tilde{\textbf{\textit{q}}}$ . We then present an iterative method to compute the vector $\textbf{\textit{q}}(A)$ for any set A. Finally, we investigate the location of the vectors $\textbf{\textit{q}}(A)$ in the set of fixed points of the progeny generating vector.

中文翻译：

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带上分支随机游走的灭绝概率

我们考虑一类具有可数无限类型集的多类型 Galton-Watson 分支过程$\mathcal{X}_d$其平均子代矩阵具有块下 Hessenberg 形式。对于这些过程，我们研究概率$\textbf{\textit{q}}(A)$类型集合中的灭绝$A\subseteq \mathcal{X}_d$. 我们比较$\textbf{\textit{q}}(A)$与全球的灭绝概率$\textbf{\textit{q}} = \textbf{\textit{q}}(\mathcal{X}_d)$，即总体最终变为空的概率，并且随着部分的灭绝概率$\波浪号{\textbf{\textit{q}}}$，即所有类型最终从总体中消失的概率。在推导出部分和全球灭绝标准后，我们​​为$\textbf{\textit{q}} < \textbf{\textit{q}}(A) < \代字号{\textbf{\textit{q}}}$. 然后我们提出了一种迭代方法来计算向量$\textbf{\textit{q}}(A)$对于任何集合一种. 最后，我们调查向量的位置$\textbf{\textit{q}}(A)$在后代生成向量的固定点集合中。