Central limit theorem for kernel estimator of invariant density in bifurcating Markov chains models

TL;DR

This paper establishes a central limit theorem for a kernel estimator of the invariant density in bifurcating Markov chains, showing consistency and Gaussian fluctuations, and clarifies the ergodic rate regimes.

Contribution

It introduces a CLT for the kernel density estimator in BMC models, advancing understanding of density estimation without ergodic rate restrictions.

Findings

Proves consistency of the kernel estimator.

Establishes Gaussian fluctuations (CLT) for the estimator.

Shows the disappearance of ergodic rate regimes in this setting.

Abstract

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.