Central limit theorem for bifurcating markov chains under <i>L²</i>-ergodic conditions

Abstract Bifurcating Markov chains (BMCs) are indexed by a full binary tree representing the evolution of trait along population where each individual has two children. We provide central limit theorem for additive functionals BMCs under $L^2$ -ergodic conditions with three different regimes. This completes pointwise approach developed in previous work. As an application, we study elementary case symmetric bifurcating autoregressive process, which justifies nontrivial hypothesis considered on kernel transition BMCs. illustrate this example phase observed fluctuations.