We prove concentration inequalities for some classes of Markov
chains and $\Phi$-mixing processes, with constants independent of the size of
the sample, that extend the inequalities for product measures of Talagrand. The
method is based on information inequalities put forwardby Marton in case of
contracting Markov chains. Using a simple duality argument on entropy, our
results also include the family of logarithmic Sobolev inequalities for convex
functions. Applications to bounds on supremum of dependent empirical processes
complete this work.