Bernstein type's concentration inequalities for symmetric Markov processes

TL;DR

This paper develops Bernstein-type concentration inequalities for empirical means of symmetric Markov processes using transportation-information inequalities, with three different methodological approaches and various applications.

Contribution

It introduces three novel approaches to establish Bernstein-type inequalities for symmetric Markov processes, expanding the theoretical toolkit in this area.

Findings

Established Bernstein inequalities for unbounded observables

Proposed three different methodological approaches

Applied results to various examples and applications

Abstract

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.