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

By Julien Guyon. 2007

98 citations recorded by the source corpus.

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.

Subfields: Mathematical Physics, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/247e06b8b7609142a8ca74f16e84366271ef101e

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In Mathematical Physics

1. Vincent Bansaye
2. Viet Chi Tran
3. Jean-François Delmas
4. Laurence Marsalle
5. Bertrand Cloez
6. S. M'el'eard

In Molecular Biology

1. Jean-François Delmas
2. Laurence Marsalle
3. Viet Chi Tran
4. Vincent Bansaye

Citations in the corpus

• Limit theorems for Markov processes indexed by continuous time Galton--Watson trees (2009, methodology): “2 Cellular-aging process We now present a generalization to the continuous time of Guyon [32] and Delmas and Marsalle [17] about cellular aging.”
• Detection of cellular aging in a Galton-Watson process (2008, background): “In particular, given |G∗q | = k, (Π∗q(1), . . . ,Π∗q(k)) can be viewed as a random drawing of all the elements of G∗q , without replacement.”