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Central limit theorem for bifurcating markov chains under L2-ergodic conditions
- Part of
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- Journal:
- Advances in Applied Probability / Volume 54 / Issue 4 / December 2022
- Published online by Cambridge University Press:
- 15 June 2022, pp. 999-1031
- Print publication:
- December 2022
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- Article
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Bifurcating Markov chains (BMCs) 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 BMCs under
$L^2$
-ergodic conditions with three different regimes. This completes the pointwise approach developed in a previous work. As an application, we study the elementary case of a symmetric bifurcating autoregressive process, which justifies the nontrivial hypothesis considered on the kernel transition of the BMCs. We illustrate in this example the phase transition observed in the fluctuations.
