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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.
Bitseki Penda, S. Valère 1
@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://geodesic.mathdoc.fr/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://geodesic.mathdoc.fr/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://geodesic.mathdoc.fr/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://geodesic.mathdoc.fr/articles/10.1051/ps/2015007/
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