Probability and moment inequalities for additive functionals of geometrically ergodic Markov chains

TL;DR

This paper derives moment and Bernstein-type inequalities for additive functionals of geometrically ergodic Markov chains, extending classical results for independent variables to dependent Markov processes with explicit conditions.

Contribution

It introduces new inequalities for unbounded functions of Markov chains that converge geometrically, covering $V$-norms and weighted Wasserstein distances, with explicit constants.

Findings

Established moment inequalities for additive functionals

Extended Bernstein inequalities to Markov chains

Applicable to unbounded functions with explicit constants

Abstract

In this paper, we establish moment and Bernstein-type inequalities for additive functionals of geometrically ergodic Markov chains. These inequalities extend the corresponding inequalities for independent random variables. Our conditions cover Markov chains converging geometrically to the stationary distribution either in $V$-norms or in weighted Wasserstein distances. Our inequalities apply to unbounded functions and depend explicitly on constants appearing in the conditions that we consider.