Perron-Frobenius theory for kernels and Crump-Mode-Jagers processes with macro-individuals

Perron-Frobenius theory developed for irreducible non-negative kernels deals with so-called $R$-positive recurrent kernels. If 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 the Nummelin's monograph 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 explaination 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.