Linear-fractional branching processes with countably many types

We study multi-type Bienayme-Galton-Watson processes with linear-fractional reproduction laws using various analytical tools like the contour process, spinal representation, Perron-Frobenius theorem for countable matrices, and 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