General Bernstein-like inequality for additive functionals of Markov chains
Abstract
Using the renewal approach we prove Bernstein-like inequalities for additive functionals of geometrically ergodic Markov chains, thus obtaining counterparts of inequalities for sums of independent random variables. The coefficient in the sub-Gaussian part of our estimate is the asymptotic variance of the additive functional, i.e. the variance of the limiting Gaussian variable in the Central Limit Theorem for Markov chains. This refines earlier results by R. Adamczak and W. Bednorz, which were obtained under the additional assumption of strong aperiodicity of the chain.
Cite
@article{arxiv.1811.09870,
title = {General Bernstein-like inequality for additive functionals of Markov chains},
author = {Michał Lemańczyk},
journal= {arXiv preprint arXiv:1811.09870},
year = {2020}
}
Comments
In previous version of the paper, Lemma C.4 contained minor error in the proof. The correct statement of Lemma C.4 (in current version of the paper) can be found between equations (3.8) and (3.9) on page 6