Bernstein type's concentration inequalities for symmetric Markov processes

Fuqing Gao (1) , Arnaud Guillin (2) , Liming Wu (2, 3)

1 Department of Mathematics Wuhan University
2 LMBP - Laboratoire de Mathématiques Blaise Pascal
3 Institute of Applied Mathematics

Using the method of transportation-information inequality introduced in \cite{GLWY}, we establish Bernstein type's concentration inequalities for empirical means $\frac 1t \int_0^t g(X_s)ds$ where $g$ is a unbounded observable of the symmetric Markov process $(X_t)$. 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.

Mots clés: transportation information inequality; Markov process; Bernstein's inequality; logarithmic Sobolev inequality; Lyapunov conditions

Domaines: Probabilités [math.PR]

Fuqing Gao, Arnaud Guillin, Liming Wu. Bernstein type's concentration inequalities for symmetric Markov processes. 2010. ⟨hal-00455598v1⟩