  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.
