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

TL;DR

This paper extends the Kesten-Stigum theorem to Galton-Watson processes with infinitely many types in a random environment, establishing conditions for non-degenerate martingale limits and describing population behavior.

Contribution

It introduces a Kesten-Stigum type result for complex multi-type processes in random environments, with new criteria ensuring martingale convergence and population characterization.

Findings

Proves non-degenerate martingale limit under integrability conditions.

Shows population size and type distribution align with quenched mean at large times.

Provides tractable criteria for models with age structure.

Abstract

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.