Perron–Frobenius theory developed for irreducible non-negative kernels deals with so-called R-positive recurrent kernels. If the kernel M is R-positive recurrent, then the main result determines the limit of the scaled kernel iterations $R^nM^n$ as $n\to\infty$ . In Nummelin (1984) this important result is proven using a regeneration method whose major focus is on M having an atom. In the special case when $M=P$ is a stochastic kernel with an atom, the regeneration method has an elegant explanation in terms of an associated split chain. In this paper we give a new probabilistic interpretation of the general regeneration method in terms of multi-type Galton–Watson processes producing clusters of particles. Treating clusters as macro-individuals, we arrive at a single-type Crump–Mode–Jagers process with a naturally embedded renewal structure.

中文翻译：

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内核的 Perron-Frobenius 理论和具有宏观个体的 Crump-Mode-Jagers 过程

为不可约非负核开发的 Perron-Frobenius 理论处理所谓的R-正循环内核。如果内核米是R- 正循环，则主要结果决定了缩放内核迭代的极限$R^nM^n$作为$n\to\infty$. 在 Nummelin (1984) 中，这一重要结果通过一种主要关注于米有一个原子。在特殊情况下$M=P$是一个带有原子的随机核，再生方法在关联的分裂链方面有一个优雅的解释。在本文中，我们根据产生粒子簇的多类型 Galton-Watson 过程对一般再生方法进行了新的概率解释。将集群视为宏观个体，我们得到了具有自然嵌入更新结构的单一类型 Crump-Mode-Jagers 过程。