New Bernstein and Hoeffding type inequalities for regenerative Markov chains

The purpose of this paper is to present Bernstein and Hoeffding type inequalities for regenerative Markov chains. Furthermore, we generalize these results and establish exponential bounds for suprema of empirical processes over a class of functions which size is controlled by its uniform entropy number. All constants involved in the bounds of the considered inequalities are given in an explicit form which can be advantageous for practical considerations. We present the theory for regenerative Markov chains, however the inequalities are also valid in the Harris recurrent case.