CENTRAL LIMIT THEOREM FOR BIFURCATING MARKOV CHAINS UNDER POINTWISE ERGODIC CONDITIONS

• 成果类型: Article
• 署名作者: Penda, S. Valere Bitseki; Delmas, Jean-Francois
• 署名单位: Universite Bourgogne Europe; Institut Polytechnique de Paris; Ecole Nationale des Ponts et Chaussees
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/21-AAP1774
• 发表日期: 2022
• 页码: 3817-3849
• 关键词: deviation inequalities; Moderate Deviations; MODEL

摘要

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 Milo ' s (Electron. J. Probab. 20 (2015) 42), where the evolution of the trait is given by an Ornstein-Uhlenbeck process.