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

¹ Ecole Polytechnique, CMAP, 91128 Palaiseau, France
bansaye@cmap.polytechnique.fr
² Harbin institute of technology at Weihai, Department of mathematics, Weihai 264209, P.R. China
cmhuang@hitwh.edu.cn.

Received: 7 September 2013
Revised: 11 June 2014

Abstract

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 non-homogeneous. 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. A central limit is also established in the transient case.