Deviation inequalities, Moderate deviations ans some limit theorems for bifurcating Markov chains with application.

Valère Bitseki Penda (1) , Hacène Djellout (1) , Arnaud Guillin (1)

1 LMBP - Laboratoire de Mathématiques Blaise Pascal

Firstly, under geometric ergodicity assumption, we provide some limit theorems and some probability inequalities for bifurcating Markov chains introduced by Guyon to detect cellular aging from cell lineage, thus completing the work of Guyon. This probability inequalities are then applied to derive a first result on moderate deviation principle for a functional of bifurcating Markov chains with a restricted range of speed, but with a function which can be unbounded. Next, under uniform geometric ergodicity assumption, we provide deviation inequalities for bifurcating Markov chains and apply them to derive a second result on moderate deviation principle for bounded functional of bifurcating Markov chains with a more larger range of speed. As statistical applications, we provide superexponential convergence in probability and deviation inequalities (under the gaussian setting or the bounded setting), and moderate deviation principle for least square estimators of the parameters of a first order bifurcating autoregressive process.

Mots clés: Bifurcating Markov chains; limit theorems; ergodicity; deviation inequalities; moderate deviation; martingale; first order bifurcating autoregressive process; cellular aging

Domaines: Probabilités [math.PR]

Valère Bitseki Penda, Hacène Djellout, Arnaud Guillin. Deviation inequalities, Moderate deviations ans some limit theorems for bifurcating Markov chains with application.. The Annals of Applied Probability, 2014, 24 (1), pp.235-291. ⟨hal-00646588v1⟩