PERRON-FROBENIUS THEORY FOR KERNELS AND CRUMP-MODE-JAGERS PROCESSES WITH MACRO-INDIVIDUALS
Artikel i vetenskaplig tidskrift, 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.

positive recurrent kernel

multi-type Galton-Watson process

Irreducible non-negative kernels

Författare

Serik Sagitov

Göteborgs universitet

Chalmers, Matematiska vetenskaper, Tillämpad matematik och statistik

Journal of Applied Probability

0021-9002 (ISSN)

Vol. 57 3 720-733

Ämneskategorier

Filosofi

Annan fysik

Sannolikhetsteori och statistik

DOI

10.1017/jpr.2020.23

Mer information

Senast uppdaterat

2020-10-07