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

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 Poincaré inequality. Moreover, our method can be readily generalized to continuous-time Markov processes.