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.
DOI : 10.1051/ps/2015007
Mots clés : Bifurcating Markov chains, Galton−Watson processes, ergodicity, deviation inequalities, first order bifurcating autoregressive process with missing data, cellular aging
@article{PS_2015__19__689_0,
author = {Bitseki Penda, S. Val\`ere},
title = {Deviation inequalities for bifurcating {Markov} chains on {Galton\ensuremath{-}Watson} tree},
journal = {ESAIM: Probability and Statistics},
pages = {689--724},
publisher = {EDP-Sciences},
volume = {19},
year = {2015},
doi = {10.1051/ps/2015007},
zbl = {1335.60136},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps/2015007/}
}
TY - JOUR AU - Bitseki Penda, S. Valère TI - Deviation inequalities for bifurcating Markov chains on Galton−Watson tree JO - ESAIM: Probability and Statistics PY - 2015 SP - 689 EP - 724 VL - 19 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps/2015007/ DO - 10.1051/ps/2015007 LA - en ID - PS_2015__19__689_0 ER -
%0 Journal Article %A Bitseki Penda, S. Valère %T Deviation inequalities for bifurcating Markov chains on Galton−Watson tree %J ESAIM: Probability and Statistics %D 2015 %P 689-724 %V 19 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps/2015007/ %R 10.1051/ps/2015007 %G en %F PS_2015__19__689_0
Bitseki Penda, S. Valère. Deviation inequalities for bifurcating Markov chains on Galton−Watson tree. ESAIM: Probability and Statistics, Tome 19 (2015), pp. 689-724. doi : 10.1051/ps/2015007. http://www.numdam.org/articles/10.1051/ps/2015007/
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