We present a tail inequality for suprema of empirical processes generated by variables with finite $\psi_\alpha$ norms and apply it to some geometrically ergodic Markov chains to derive similar estimates for empirical processes of such chains, generated by bounded functions. We also obtain a bounded difference inequality for symmetric statistics of such Markov chains.
Radoslaw Adamczak. "A tail inequality for suprema of unbounded empirical processes with applications to Markov chains." Electron. J. Probab. 13 1000 - 1034, 2008. https://doi.org/10.1214/EJP.v13-521