Mathilde André
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 \([1]\). 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.
This is a joint work with Jean-Jil Duchamps (Université de
Franche-Comté).
Bibliography
\([1]\) Vere-Jones, D. (1967) Ergodic properties of nonnegative matrices.
I. Pacific Journal of Mathematics, 22(2):361–386.