  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.
