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

S. Valère Bitseki Penda () and Jean-François Delmas ()
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S. Valère Bitseki Penda: Université de Bourgogne Franche- Comté
Jean-François Delmas: École des Ponts

Journal of Theoretical Probability, 2023, vol. 36, issue 3, 1591-1625

Abstract: Abstract Bifurcating Markov chains 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 consistency 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.

Keywords: Bifurcating Markov chains; Kernel estimator; Density estimation; Bifurcating autoregressive process; Binary trees; Fluctuations for trees indexed Markov chains; 62G05; 62G07; 62G20; 60J80; 60F05 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s10959-022-01205-w

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