Linear-fractional branching processes with countably many types

TL;DR

This paper analyzes multi-type linear-fractional branching processes with countably many types, providing a clear recurrence criterion based on the Malthusian parameter and mean age at childbearing, using advanced analytical tools.

Contribution

It introduces a transparent criterion for R-positive recurrence in countably many type branching processes, linking it to the Malthusian parameter and age at childbearing.

Findings

Established a criterion for R-positive recurrence.

Linked recurrence to the Malthusian parameter.

Utilized contour process, spinal representation, and Perron-Frobenius theorem.

Abstract

We study multi-type Bienaym\'e-Galton-Watson processes with linear-fractional reproduction laws using various analytical tools like contour process, spinal representation, Perron-Frobenius theorem for countable matrices, renewal theory. For this special class of branching processes with countably many types we present a transparent criterion for $R$-positive recurrence with respect to the type space. This criterion appeals to the Malthusian parameter and the mean age at childbearing of the associated linear-fractional Crump-Mode-Jagers process.