Bernstein-type Inequalities for Markov Chains and Markov Processes: A Simple and Robust Proof

TL;DR

This paper introduces a new Bernstein-type deviation inequality for Markov chains and processes, using a simple approach that requires only an iterated Poincaré inequality, making it more robust and broadly applicable.

Contribution

The authors present a novel Bernstein-type inequality for non-reversible Markov chains that relies on minimal assumptions and extends to continuous-time processes.

Findings

Provides a simple proof technique for Bernstein inequalities

Applicable to non-reversible Markov chains and continuous-time processes

Enhances robustness over existing deviation inequalities

Abstract

We establish a new Bernstein-type deviation inequality for general (non-reversible) discrete-time Markov chains via an elementary approach. More robust than existing works in the literature, our result only requires the Markov chain to satisfy an iterated Poincar\'e inequality. Moreover, our method can be readily generalized to continuous-time Markov processes.