Limit theorems for bifurcating Markov chains. Application to the detection of cellular aging

TL;DR

This paper develops limit theorems for bifurcating Markov chains, a model for dependent data on binary trees, and applies these results to detect cellular aging in bacteria using a bifurcating autoregressive model.

Contribution

It introduces a general framework for analyzing bifurcating Markov chains and derives laws of large numbers and central limit theorems for these processes.

Findings

Established laws of large numbers for bifurcating Markov chains.

Proved central limit theorems for dependent data on binary trees.

Applied the theoretical results to detect cellular aging in Escherichia coli.

Abstract

We propose a general method to study dependent data in a binary tree, where an individual in one generation gives rise to two different offspring, one of type 0 and one of type 1, in the next generation. For any specific characteristic of these individuals, we assume that the characteristic is stochastic and depends on its ancestors' only through the mother's characteristic. The dependency structure may be described by a transition probability $P(x,dy dz)$ which gives the probability that the pair of daughters' characteristics is around $(y,z)$, given that the mother's characteristic is $x$. Note that $y$, the characteristic of the daughter of type 0, and $z$, that of the daughter of type 1, may be conditionally dependent given $x$, and their respective conditional distributions may differ. We then speak of bifurcating Markov chains. We derive laws of large numbers and central limit theorems for such stochastic processes. We then apply these results to detect cellular aging in Escherichia Coli, using the data of Stewart et al. and a bifurcating autoregressive model.