Bifurcating Markov chains (BMC) are Markov chains indexed by a full binary tree representing the evolution of a trait along a population where each individual has two children. Motivated by the functional estimation of the density of the invariant probability measure which appears as the asymptotic distribution of the trait, we prove the consistence and the Gaussian fluctuations for a kernel estimator of this density based on late generations. In this setting, it is interesting to note that the distinction of the three regimes on the ergodic rate identified in a previous work (for fluctuations of average over large generations) disappears. This result is a first step to go beyond the threshold condition on the ergodic rate given in previous statistical papers on functional estimation.
翻译:马可夫链条(BMC)是由一整棵二进制树组成的马可夫链条(Markov),它代表着一个人有两个孩子的人口中一个特征的演化,其动机是用功能来估计不同概率度量的密度,这种概率度量是该特性的无症状分布,我们证明了这种密度基于后几代的内核测算器的一致性和高斯波动性。在这种背景下,值得注意的是,在以往工作中查明的(关于几代人的平均波动),三种制度对异性比率的区别已经消失。这是超越以往关于功能估计的统计文件中给出的ergodic比率门槛条件的第一步。