The Annals of Applied Probability

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

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

https://projecteuclid.org/euclid.aoap/1389278725

Digital Object Identifier
doi:10.1214/13-AAP921

Mathematical Reviews number (MathSciNet)
MR3161647

Zentralblatt MATH identifier
1293.60036

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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