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

TL;DR

This work proves a Bernstein-type deviation inequality for general (non-reversible) Markov chains by exploiting an elementary iterated Poincaré inequality (IP gap) $\eta_p$, which provides a robust measure of ergodicity beyond traditional spectral gaps. The authors develop explicit exponential-moment bounds for sums of bounded, mean-zero functions and translate these into Bernstein-type tail bounds that depend only on $\eta_p$, $M$, and $\sigma$, with explicit trade-offs for the initial distribution. Their framework unifies discrete- and continuous-time settings by defining the IP gap for the Markov operator or generator and showing analogous bounds in both cases, emphasizing robustness to non-reversibility and independence from state-space size. The results show $\eta_p \ge \eta_s \ge \eta_a$ and $\eta_p \ge \eta_{ps}/2$, making the IP gap particularly effective for broad classes of chains. Overall, the paper delivers simple, robust concentration tools for Markov processes, with potential broader applicability to non-reversible dynamics and continuous-time systems.

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

Theorems & Definitions (24)

• Lemma 2.1
• proof
• Proposition 2.2
• proof
• Corollary 2.3
• proof
• Theorem 2.4
• proof
• Lemma 3.1
• proof