Central limit theorem for bifurcating Markov chains under point-wise ergodic conditions

TL;DR

This paper establishes a central limit theorem for bifurcating Markov chains, revealing three regimes based on the interplay between reproduction and trait evolution rates, extending previous work to more general trait dynamics.

Contribution

It introduces a CLT for additive functionals of BMC under point-wise ergodic conditions, identifying multiple regimes and generalizing prior results to broader trait evolution models.

Findings

Three regimes identified based on reproduction and ergodicity rates.

Extension of CLT to general trait evolution beyond martingale sums.

Connection to continuous-time branching processes with Ornstein-Uhlenbeck traits.

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 general additive functionals of BMC, and prove the existence of three regimes. This corresponds to a competition between the reproducing rate (each individual has two children) and the ergodicity rate for the evolution of the trait. This is in contrast with the work of Guyon (2007), where the considered additive functionals are sums of martingale increments, and only one regime appears. Our result can be seen as a discrete time version, but with general trait evolution, of results in the time continuous setting of branching particle system from Adamczak and Mi\l{}o\'{s} (2015), where the evolution of the trait is given by an Ornstein-Uhlenbeck process.