As soon as the Markov chain along a random lineage of the binary tree is ergodic, a law of large numbers holds. It yields the convergence for the proportions of individuals in generation n whose trait has some given value. In the present paper we are concerned with the problem of proving the law of large numbers (LLN) for random dynamical systems. The question of establishing the LLN for an additive functional of a Markov process is one of the most fundamental in probability theory and there exists a rich literature on the subject, see e.g. and the citations therein.
Download PDF Abstract: We consider a population with non-overlapping generations, whose size goes toinfinity. It is described by a discrete genealogy which may be timenon-homogeneous and we pay special attention to branching trees in varyingenvironments. A Markov chain models the dynamic of the trait of each individualalong this genealogy and may also be time non-homogeneous. Such models aremotivated by transmission processes in the cell division,reproduction-dispersion dynamics or sampling problems in evolution. We want todetermine the evolution of the distribution of the traits among the population,namely the asymptotic behavior of the proportion of individuals with a giventrait. We prove some quenched laws of large numbers which rely on theergodicity of an auxiliary process, in the same vein as cite{guy,delmar}.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 largenumbers on the whole population until generation $n$. A central limit is alsoestablished in the transient case. Submission history[v1]Tue, 21 May 2013 09:21:35 UTC (27 KB) Full-text links: Download:Current browse context:
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