Volume 19, 2015
|689 - 724
|11 December 2015
Deviation inequalities for bifurcating Markov chains on Galton−Watson tree
CMAP, UMR 7641, École polytechnique CNRS,
Route de Saclay, 91128
Revised: 1 December 2014
We are interested in bifurcating Markov chains on Galton−Watson tree. These processes are an extension of bifurcating Markov chains, which was introduced by Guyon to detect cellular aging from cell lineage, in case the index set is a binary Galton−Watson process. First, under geometric ergodicity assumption of an embedded Markov chain, we provide polynomial deviation inequalities for properly normalized sums of bifurcating Markov chains on Galton−Watson tree. Next, under some uniformity, we derive exponential inequalities. These results allow to exhibit different regimes of convergence which correspond to a competition between the geometric ergodic speed of the underlying Markov chain and the exponential growth of the Galton−Watson tree. As application, we derive deviation inequalities (for either the Gaussian setting or the bounded setting) for the least-squares estimator of autoregressive parameters of bifurcating autoregressive processes with missing data which allow, in the case of cell division, to take into account the cell’s death.
Mathematics Subject Classification: 60E15 / 60J80 / 60J10
Key words: Bifurcating Markov chains / Galton−Watson processes / ergodicity / deviation inequalities / first order bifurcating autoregressive process with missing data / cellular aging
© EDP Sciences, SMAI, 2015
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