The Annals of Applied Probability

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

Julien Guyon

Full-text: Open access

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, dydz) 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.

Article information

Source
Ann. Appl. Probab., Volume 17, Number 5/6 (2007), 1538-1569.

Dates
First available in Project Euclid: 3 October 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1191419175

Digital Object Identifier
doi:10.1214/105051607000000195

Mathematical Reviews number (MathSciNet)
MR2358633

Zentralblatt MATH identifier
1143.62049

Subjects
Primary: 60F05: Central limit and other weak theorems 60F15: Strong theorems
Secondary: 60F25: $L^p$-limit theorems 60J27: Continuous-time Markov processes on discrete state spaces 62M02: Markov processes: hypothesis testing 62M05: Markov processes: estimation 62P10: Applications to biology and medical sciences

Keywords
Bifurcating Markov chains limit theorems ergodicity bifurcating autoregression AR(1) cellular aging

Citation

Guyon, Julien. Limit theorems for bifurcating Markov chains. Application to the detection of cellular aging. Ann. Appl. Probab. 17 (2007), no. 5/6, 1538--1569. doi:10.1214/105051607000000195. https://projecteuclid.org/euclid.aoap/1191419175


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