Uniform Chernoff and Dvoretzky-Kiefer-Wolfowitz-type inequalities for Markov chains and related processes

TL;DR

This paper introduces new measure concentration inequalities for Markov chains and related processes using Markov contraction, providing simple, dimension-free bounds that extend to infinite state spaces.

Contribution

It presents a unified, elementary approach to derive Chernoff and Dvoretzky-Kiefer-Wolfowitz-type inequalities for Markov chains and hidden Markov models, including infinite state spaces.

Findings

Derived a general concentration inequality for Markov and hidden Markov chains

Established dimension-free bounds applicable to countably infinite state spaces

Extended results to processes like Markov trees

Abstract

We observe that the technique of Markov contraction can be used to establish measure concentration for a broad class of non-contracting chains. In particular, geometric ergodicity provides a simple and versatile framework. This leads to a short, elementary proof of a general concentration inequality for Markov and hidden Markov chains (HMM), which supercedes some of the known results and easily extends to other processes such as Markov trees. As applications, we give a Dvoretzky-Kiefer-Wolfowitz-type inequality and a uniform Chernoff bound. All of our bounds are dimension-free and hold for countably infinite state spaces.