Central Limit Theorem for Kernel Estimator of Invariant Density in Bifurcating Markov Chains Models

Bifurcating Markov chains are indexed by a full binary tree representing the evolution of trait along population where each individual has two children. Motivated functional estimation density invariant probability measure which appears as asymptotic distribution trait, we prove consistency and Gaussian fluctuations for kernel estimator this based on late generations. In setting, it is interesting to note that distinction three regimes ergodic rate identified in previous work (for average over large generations) disappears. This result first step go beyond threshold condition given statistical papers estimation.