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

Abstract

We observe that the technique of Markov contraction can be used to establish measure concentration for a broad class of noncontracting 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, which supersedes some of the known results and easily extends to other processes such as Markov trees. As applications, we provide 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.