Bernstein type’s concentration inequalities for symmetric Markov processes

Using the method of transportation-information inequality introduced in [28], we establish Bernstein type’s concentration inequalities for empirical means 1 t ∫ t 0 g(Xs)ds where g is a unbounded observable of the symmetric Markov process (Xt). Three approaches are proposed : functional inequalities approach ; Lyapunov function method ; and an approach through the Lipschitzian norm of the solution to the Poisson equation. Several applications and examples are studied.