Bernsteins inequality for general Markov chains

We establish Bernstein inequalities for functions of general (general-state-space, not necessarily reversible) Markov chains. These inequalities achieve sharp variance proxies and recover the classical Bernsteins inequality under independence. The key analysis lies in upper bounding the operator norm of a perturbed Markov transition kernel by the limiting operator norm of a sequence of finite-rank and perturbed Markov transition kernels. For each finite-rank and perturbed Markov kernel, we bound its norm by the sum of two convex functions. One coincides with what delivers the classical Bernsteins inequality, and the other reflects the influence of the Markov dependence. A convex analysis on conjugates of these two functions then derives our Bernstein inequalities.