Sharp $L \log L$ condition for supercritical Galton-Watson processes with countable types

We investigate Kesten-Stigum-like results for multi-type Galton-Watson processes with a countable number of types in a general setting, allowing us in particular to consider processes with an infinite total population at each generation.

Specifically, a sharp $L\log L$ condition is found under the only assumption that the mean reproduction matrix is positive recurrent in the sense of Vere-Jones (1967).

The type distribution is shown to always converge in probability in the recurrent case, and under conditions covering many cases it is shown to converge almost surely.

We finally apply these results to the study of a directed network built from countably many configuration models as a model of epidemics.