Deviation inequalities, Moderate deviations and some limit theorems for bifurcating Markov chains with application

TL;DR

This paper establishes limit theorems, deviation inequalities, and moderate deviation principles for bifurcating Markov chains, with applications to statistical estimation in cell lineage models, under geometric ergodicity assumptions.

Contribution

It extends asymptotic results for bifurcating Markov chains by deriving deviation inequalities and moderate deviation principles, including for unbounded functionals and statistical estimators.

Findings

Established limit theorems and deviation inequalities under geometric ergodicity.

Derived moderate deviation principles for functionals of BMCs with different conditions.

Proved superexponential convergence and deviation inequalities for parameter estimators.

Abstract

First, under a geometric ergodicity assumption, we provide some limit theorems and some probability inequalities for the bifurcating Markov chains (BMC). The BMC model was introduced by Guyon to detect cellular aging from cell lineage, and our aim is thus to complete his asymptotic results. The deviation inequalities are then applied to derive first result on the moderate deviation principle (MDP) for a functional of the BMC with a restricted range of speed, but with a function which can be unbounded. Next, under a uniform geometric ergodicity assumption, we provide deviation inequalities for the BMC and apply them to derive a second result on the MDP for a bounded functional of the BMC with a larger range of speed. As statistical applications, we provide superexponential convergence in probability and deviation inequalities (for either the Gaussian setting or the bounded setting), and the MDP for least square estimators of the parameters of a first-order bifurcating autoregressive process.