Weak law of large numbers for some Markov chains along non homogeneous genealogies

We consider a population with non-overlapping generations, whose size goes to infinity. It is described by a discrete genealogy which may be time non-homogeneous and we pay special attention to branching trees in varying environments. A Markov chain models the dynamic of the trait of each individual along this genealogy and may also be time nonhomogeneous. Such models are motivated by transmission processes in the cell division, reproduction-dispersion dynamics or sampling problems in evolution. We want to determine the evolution of the distribution of the traits among the population, namely the asymptotic behavior of the proportion of individuals with a given trait. We prove some quenched laws of large numbers which rely on the ergodicity of an auxiliary process, in the same vein as [21, 15]. Applications to time inhomogeneous Markov chains lead us to derive a backward (with respect to the environment) law of large numbers and a law of large numbers on the whole population until generation n. A central limit is also established in the transient case.