Appears in collection : Les probabilités de demain 2017

The purpose of this talk is to present Bernstein and Hoeffding type functional inequalities for regenerative Markov chains. Furthermore, we generalize these results and show exponential bounds for suprema of empirical processes over a class of functions F 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 in practical considerations. We present the theory for regenerative Markov chains, however the inequalities are also valid in the Harris recurrent case.