Central limit theorem for bifurcating Markov chains under $L^{2}$-ergodic conditions

TL;DR

This paper establishes a central limit theorem for additive functionals of bifurcating Markov chains under $L^2$-ergodic conditions, revealing phase transitions in fluctuations and extending previous pointwise approaches.

Contribution

It provides the first CLT for BMCs under $L^2$-ergodic conditions with multiple regimes, completing prior pointwise methods and applying to symmetric bifurcating autoregressive processes.

Findings

Three regimes identified for the CLT of BMCs.

Phase transition observed in the fluctuation behavior.

Application to symmetric bifurcating autoregressive process.

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. We provide a central limit theorem for additive functionals of BMC under $L^2$-ergodic conditions with three different regimes. This completes the pointwise approach developed in a previous work. As application, we study the elementary case of symmetric bifurcating autoregressive process, which justify the non-trivial hypothesis considered on the kernel transition of the BMC. We illustrate in this example the phase transition observed in the fluctuations.