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.
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机译:我们考虑具有不重叠世代的人口,其人口数量达到无穷大。它由一个离散的族谱描述,该族谱可能在时间上是不均匀的,因此我们特别注意变化环境中的分支树。马尔可夫链模拟了这个谱系中每个个体特征的动态,并且可能在时间上是非均匀的。此类模型是由细胞分裂中的传播过程,繁殖-分散动力学或进化中的采样问题所驱动。我们要确定人群中性状分布的演变,即具有给定性状的个体比例的渐近行为。我们证明了一些依赖于辅助过程的遍历性的淬灭定律,与 cite {guy,delmar}相同。对时间非齐次马尔可夫链的应用使我们得出了一个倒向(相对于环境)定律在整个人口中产生大数和大数定律,直到产生$ n $。在瞬态情况下也建立了中心极限。
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