October 2022 Central limit theorem for bifurcating Markov chains under pointwise ergodic conditions
S. Valère Bitseki Penda, Jean-François Delmas
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Ann. Appl. Probab. 32(5): 3817-3849 (October 2022). DOI: 10.1214/21-AAP1774


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 (Ann. Appl. Probab. 17 (2007) 1538–1569), 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 Miłoś (Electron. J. Probab. 20 (2015) 42), where the evolution of the trait is given by an Ornstein–Uhlenbeck process.


The authors would like to thank the referees and the Associate Editor for their useful comments which helped to improve the presentation.


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S. Valère Bitseki Penda. Jean-François Delmas. "Central limit theorem for bifurcating Markov chains under pointwise ergodic conditions." Ann. Appl. Probab. 32 (5) 3817 - 3849, October 2022. https://doi.org/10.1214/21-AAP1774


Received: 1 February 2021; Revised: 1 August 2021; Published: October 2022
First available in Project Euclid: 18 October 2022

MathSciNet: MR4497859
zbMATH: 1500.60013
Digital Object Identifier: 10.1214/21-AAP1774

Primary: 60F05 , 60J05 , 60J80

Keywords: Bifurcating Markov chains , binary trees , central limit theorem , tree indexed Markov chain

Rights: Copyright © 2022 Institute of Mathematical Statistics


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Vol.32 • No. 5 • October 2022
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