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

• 成果类型: Article
• 署名作者: Guyon, Julien
• 署名单位: Institut Polytechnique de Paris; Ecole Nationale des Ponts et Chaussees
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/105051607000000195
• 发表日期: 2007
• 页码: 1538-1569
• 关键词: Autoregressive Model; FIELDS; inference; trees

摘要

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.