A Kesten Stigum theorem for Galton-Watson processes with infinitely many types in a random environment

In this paper, we study a Galton-Watson process $(Z_n)$ with infinitely many types in a random ergodic environment $\bar{\xi}=(\xi_n)_{n\geq 0}$. We focus on the supercritical regime of the process, where the quenched average of the size of the population grows exponentially fast to infinity. We work under Doeblin-type assumptions coming from a previous paper, which ensure that the quenched mean semi group of $(Z_n)$ satisfies some ergodicity property and admits a $\bar{\xi}$-measurable family of space-time harmonic functions. We use these properties to derive an associated nonnegative martingale $(W_n)$. Under a $L\log(L)^{1+\varepsilon}$-integrabilty assumption on the offspring distribution, we prove that the almost sure limit $W$ of the martingale $(W_n)$ is not degenerate. Assuming some uniform $L^2$-integrability of the offspring distribution, we prove that conditionally on $\{W>0\}$, at a large time $n$, both the size of the population and the distribution of types correspond to those of the quenched mean of the population $\mathbb{E}[Z_n|\bar{\xi}, Z_0]$. We finally introduce an example of a process modelling a population with a discrete age structure. In this context, we provide more tractable criterions which guarantee our various assumptions are met.