Concentration Inequalities for Markov Jump Processes

We derive concentration inequalities for empirical means $\frac{1}{t}{\int }_0^{t}f(X_s)ds$ where $X_s$ is an irreducible Markov jump process on a finite state space and $f$ some observable. Using a Feynman-Kac semigroup we first derive a general concentration inequality. Then, based on this inequality we derive further concentration inequalities. Hereby we use three different methods; perturbation theory, Poincaré inequalities and information inequalities. We also obtain a Bernstein type concentration inequality.