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February 2014 Deviation inequalities, moderate deviations and some limit theorems for bifurcating Markov chains with application
S. Valère Bitseki Penda, Hacène Djellout, Arnaud Guillin
Ann. Appl. Probab. 24(1): 235-291 (February 2014). DOI: 10.1214/13-AAP921

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.

Citation

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S. Valère Bitseki Penda. Hacène Djellout. Arnaud Guillin. "Deviation inequalities, moderate deviations and some limit theorems for bifurcating Markov chains with application." Ann. Appl. Probab. 24 (1) 235 - 291, February 2014. https://doi.org/10.1214/13-AAP921

Information

Published: February 2014
First available in Project Euclid: 9 January 2014

zbMATH: 1293.60036
MathSciNet: MR3161647
Digital Object Identifier: 10.1214/13-AAP921

Subjects:
Primary: 60E15, 60F05, 60F10, 60F15
Secondary: 60G42, 60J05, 62M02, 62M05, 62P10

Rights: Copyright © 2014 Institute of Mathematical Statistics

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Vol.24 • No. 1 • February 2014
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