Bernstein's inequalities for general Markov chains

We establish Bernstein's inequalities for functions of general (general-state-space and possibly non-reversible) Markov chains. These inequalities achieve sharp variance proxies and encompass the classical Bernstein inequality for independent random variables as special cases. The key analysis lies in bounding the operator norm of a perturbed Markov transition kernel by the exponential of sum of two convex functions. One coincides with what delivers the classical Bernstein inequality, and the other reflects the influence of the Markov dependence. A convex analysis on these two functions then derives our Bernstein inequalities. As applications, we apply our Bernstein inequalities to the Markov chain Monte Carlo integral estimation problem and the robust mean estimation problem with Markov-dependent samples, and achieve tight deviation bounds that previous inequalities can not.