BERNSTEIN-TYPE CONCENTRATION INEQUALITIES FOR SYMMETRIC MARKOV PROCESSES

Using the method of transportation- information inequality introduced in [A. Guillin et al., Probab. Theory Related Fields, 144 (2009), pp. 669- 695], we establish Bernstein- type concentration inequalities for empirical means of functions of the Markov process (integral(t)(0) g(X-s) ds)/t where g is an unbounded observable of the symmetric Markov process (X-t). Three approaches are proposed: a functional inequalities approach, a Lyapunov function method, and an approach through the Lipschitzian norm of the solution to the Poisson equation. Several applications and examples are studied.