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October 2007 Limit theorems for bifurcating Markov chains. Application to the detection of cellular aging
Julien Guyon
Ann. Appl. Probab. 17(5-6): 1538-1569 (October 2007). DOI: 10.1214/105051607000000195


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.


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Julien Guyon. "Limit theorems for bifurcating Markov chains. Application to the detection of cellular aging." Ann. Appl. Probab. 17 (5-6) 1538 - 1569, October 2007.


Published: October 2007
First available in Project Euclid: 3 October 2007

zbMATH: 1143.62049
MathSciNet: MR2358633
Digital Object Identifier: 10.1214/105051607000000195

Primary: 60F05, 60F15
Secondary: 60F25, 60J27, 62M02, 62M05, 62P10

Rights: Copyright © 2007 Institute of Mathematical Statistics


Vol.17 • No. 5-6 • October 2007
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