Central limit theorem for bifurcating Markov chains: the mother-daughters triangles case

TL;DR

This paper establishes a central limit theorem for additive functionals of bifurcating Markov chains, extending previous results under various ergodic conditions and analyzing the asymptotic variance behavior.

Contribution

It extends the central limit theorem to bifurcating Markov chains with new ergodic conditions and analyzes the asymptotic variance for certain function classes.

Findings

Central limit theorem established for bifurcating Markov chains.

Asymptotic variance can be non-zero at faster convergence rates.

Results extend previous CLT frameworks under ergodic conditions.

Abstract

The main objective of this article is to establish a central limit theorem for additive three-variable functionals of bifurcating Markov chains. We thus extend the central limit theorem under point-wise ergodic conditions studied in Bitseki-Delmas (2022) and to a lesser extent, the results of Bitseki-Delmas (2022) on central limit theorem under $L^{2}$ ergodic conditions. Our results also extend and complement those of Guyon (2007) and Delmas and Marsalle (2010). In particular, when the ergodic rate of convergence is greater than $1/\sqrt{2}$, we have, for certain class of functions, that the asymptotic variance is non-zero at a speed faster than the usual central limit theorem studied by Guyon and Delmas-Marsalle.