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/
, Large deviations for branching processes-I. Single type case. Ann. Appl. Probab. 5 (1994) 779–790 | Zbl
K.B. Athreya and P.E Ney, Branching Process. Springer, Berlin (1972). | Zbl
, Weighted sums of certain dependent random variables. Tôhoku Math. J. 19 (1967) 357–367. | Zbl
, , and , Limit theorems for Markov processes indexed by continuous time Galton−Watson trees. Ann. Appl. Probab. 21 (2011) 2263–2314 | Zbl
G. Bennett, Probability inequalities for sum of independant random variables. J. Am. Stat. Assoc.57 (1962) 33–45. | Zbl
, and , A Rademacher−Menchov approach for randon coefficient bifurcating autoregressive processes. Stochastic Processes Appl. 125 (2015) 1218–1243. | Zbl
, and , Asymptotic analysis for bifurcating autoregressive processes via a martingale approach. Electronic. J. Probab. 14 (2009) 2492–2526. | Zbl
and , Deviation inequalities and moderate deviations for estimators of parameters in bifurcating autoregressive models. Ann. Inst. Henri Poincaré 50 (2014) 806–844. | Zbl
, and , Deviation inequalities, Moderate deviations and some limit theorems for bifurcating Markov chains with application. Ann. Appl. Probab. 24 (2014) 235–291. | Zbl
, Asymptotic results for random coefficient bifurcating autoregressive processes. Statistics 48 (2013) 1202–1232. | Zbl
and , The bifurcating autoregressive model in cell lineage studies. Biometrics 42 (1986) 769–783. | Zbl
, and , Parameters estimation for asymmetric bifurcating autoregressive processes with missing data. Electron. J. Stat. 5 (2011) 1313–1353. | Zbl
, and , Asymmetry tests for Bifurcating Auto-Regressive Processes with missing data. Stat. Probab. Lett. 82 (2012) 1439–1444. | Zbl
, and , Random coefficients bifurcating autoregressive processes ESAIM: PS 18 (2014) 365–399. | Zbl
and , Detection of cellular aging in Galton−Watson process. Stochastic Process. Appl. 120 (2010) 2495-2519 | Zbl
and , Limit theorems and estimation theory for branching processes with increasing random number of ancestors. J. Appl. Probab. 34 (1997) 309–327. | Zbl
, Limit theorems for bifurcating markov chains. Application to the detection of cellular aging. Ann. Appl. Probab. 17 (2007) 1538–1569. | Zbl
, , , , and , Statistical study of cellular aging. Proc. of CEMRACS 2004. ESAIM: Proc. 14 (2005) 100–114. | Zbl
, Probability inequalities for sums of bounded random variables. J. Am. Stat. Assoc. 58 (1963) 13–30. | Zbl
R.B. Karp and Y. Zhang, Finite branching processes and AND/OR tree evaluation TR. International Computer Science Institute; 93-043, ICSI. Berkeley, Calif. (1994).
, On the method of bounded differences. Electron. Comm. Probab. 11 (2006) 64–77.
, , and , Aging and death in an organism that reproduces by morphologically symmetric division. PLoS Biol. 3 (2005) e45.
Cité par Sources :
