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

S. Valère Bitseki Penda; Jean-François Delmas

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arXiv · 2106.07711

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

S. Valère Bitseki Penda·Jean-François Delmas

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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.

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BibTeX RIS

S. Valère Bitseki Penda, Jean-François Delmas. 2021-06-14. Central limit theorem for bifurcating Markov chains under $L^{2}$-ergodic conditions. https://arxiv.org/abs/2106.07711

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