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

arXiv eprints
 Pub Date:
 June 2021
 arXiv:
 arXiv:2106.08626
 Bibcode:
 2021arXiv210608626V
 Keywords:

 Mathematics  Statistics Theory;
 Mathematics  Probability;
 62G05;
 62F12;
 60J05;
 60F05;
 60J80
 EPrint:
 40 pages, 8 figures. arXiv admin note: substantial text overlap with arXiv:2012.04741