Moderate deviation principles for bifurcating Markov chains: case of functions dependent of one variable

TL;DR

This paper establishes moderate deviation principles for additive functionals of bifurcating Markov chains, focusing on functions depending on a single variable, under geometric ergodicity assumptions, extending previous CLT results.

Contribution

It introduces moderate deviation principles for additive functionals of bifurcating Markov chains with functions depending on one variable, under spectral gap conditions.

Findings

Moderate deviation principles are proven for specific bifurcating Markov chains.

Results extend previous CLT findings to moderate deviations.

The work relies on martingale decomposition and spectral gap assumptions.

Abstract

The main purpose of this article is to establish moderate deviation principles for additive functionals of bifurcating Markov chains. Bifurcating Markov chains are a class of processes which are indexed by a regular binary tree. They can be seen as the models which represent the evolution of a trait along a population where each individual has two offsprings. Unlike the previous results of Bitseki, Djellout \& Guillin (2014), we consider here the case of functions which depend only on one variable. So, mainly inspired by the recent works of Bitseki \& Delmas (2020) about the central limit theorem for general additive functionals of bifurcating Markov chains, we give here a moderate deviation principle for additive functionals of bifurcating Markov chains when the functions depend on one variable. This work is done under the uniform geometric ergodicity and the uniform ergodic property based on the second spectral gap assumptions. The proofs of our results are based on martingale decomposition recently developed by Bitseki \& Delmas (2020) and on results of Dembo (1996), Djellout (2001) and Puhalski (1997).