  First, under a geometric ergodicity assumption, we provide some limit
theorems and some probability inequalities for the bifurcating Markov chains
(BMC). The BMC model was introduced by Guyon to detect cellular aging from cell
lineage, and our aim is thus to complete his asymptotic results. The deviation
inequalities are then applied to derive first result on the moderate deviation
principle (MDP) for a functional of the BMC with a restricted range of speed,
but with a function which can be unbounded. Next, under a uniform geometric
ergodicity assumption, we provide deviation inequalities for the BMC and apply
them to derive a second result on the MDP for a bounded functional of the BMC
with a larger range of speed. As statistical applications, we provide
superexponential convergence in probability and deviation inequalities (for
either the Gaussian setting or the bounded setting), and the MDP for least
square estimators of the parameters of a first-order bifurcating autoregressive
process.
