  The aim of this note is to investigate the concentration properties of
unbounded functions of geometrically ergodic Markov chains. We derive
concentration properties of centered functions with respect to the square of
the Lyapunov's function in the drift condition satisfied by the Markov chain.
We apply the new exponential inequalities to derive confidence intervals for
MCMC algorithms. Quantitative error bounds are providing for the regenerative
Metropolis algorithm of [5].
