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

Journal article, 2020

Perron-Frobenius theory developed for irreducible non-negative kernels deals with socalled R-positive recurrent kernels. If the kernel M is R-positive recurrent, then the main result determines the limit of the scaled kernel iterations (RMn)-M-n as n -> infinity. 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.