Deviation inequalities for bifurcating Markov chains on Galton-Watson tree

TL;DR

This paper establishes deviation inequalities for sums of bifurcating Markov chains on Galton-Watson trees, extending previous models to include cell death and missing data, with applications to autoregressive parameter estimation.

Contribution

It introduces deviation inequalities for bifurcating Markov chains on Galton-Watson trees, incorporating cell death and missing data, and applies these to parameter estimation.

Findings

Derived deviation inequalities under geometric ergodicity.

Applied inequalities to autoregressive parameter estimation with missing data.

Extended bifurcating Markov chain models to include cell death.

Abstract

We provide deviation inequalities for properly normalized sums of bifurcating Markov chains on Galton-Watson tree. These processes are extension of bifurcating Markov chains (which was introduced by Guyon to detect cellular aging from cell lineage) in case the index set is a binary Galton-Watson process. As application, we derive deviation inequalities for the least-squares estimator of autoregressive parameters of bifurcating autoregressive processes with missing data. These processes allow, in case of cell division, to take into account the cell's death. The results are obtained under an uniform geometric ergodicity assumption of an embedded Markov chain.