The Annals of Applied Probability

Deviation inequalities, moderate deviations and some limit theorems for bifurcating Markov chains with application

S. Valère Bitseki Penda, Hacène Djellout, and Arnaud Guillin

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Abstract

First, under a geometric ergodicity assumption, we provide some limit theorems and some probability inequalities for the bifurcating Markov chains (BMC). The BMC model was introduced by Guyon to detect cellular aging from cell lineage, and our aim is thus to complete his asymptotic results. The deviation inequalities are then applied to derive first result on the moderate deviation principle (MDP) for a functional of the BMC with a restricted range of speed, but with a function which can be unbounded. Next, under a uniform geometric ergodicity assumption, we provide deviation inequalities for the BMC and apply them to derive a second result on the MDP for a bounded functional of the BMC with a larger range of speed. As statistical applications, we provide superexponential convergence in probability and deviation inequalities (for either the Gaussian setting or the bounded setting), and the MDP for least square estimators of the parameters of a first-order bifurcating autoregressive process.

Article information

Source
Ann. Appl. Probab., Volume 24, Number 1 (2014), 235-291.

Dates
First available in Project Euclid: 9 January 2014

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1389278725

Digital Object Identifier
doi:10.1214/13-AAP921

Mathematical Reviews number (MathSciNet)
MR3161647

Zentralblatt MATH identifier
1293.60036

Subjects
Primary: 60F05: Central limit and other weak theorems 60F10: Large deviations 60F15: Strong theorems 60E15: Inequalities; stochastic orderings
Secondary: 60G42: Martingales with discrete parameter 60J05: Discrete-time Markov processes on general state spaces 62M02: Markov processes: hypothesis testing 62M05: Markov processes: estimation 62P10: Applications to biology and medical sciences

Keywords
Bifurcating Markov chains limit theorems ergodicity deviation inequalities moderate deviation martingale first-order bifurcating autoregressive process cellular aging

Citation

Bitseki Penda, S. Valère; Djellout, Hacène; Guillin, Arnaud. Deviation inequalities, moderate deviations and some limit theorems for bifurcating Markov chains with application. Ann. Appl. Probab. 24 (2014), no. 1, 235--291. doi:10.1214/13-AAP921. https://projecteuclid.org/euclid.aoap/1389278725


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