  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.
