  The Nummellin's split chain construction allows to decompose a Markov chain
Monte Carlo (MCMC) trajectory into i.i.d. "excursions". RegenerativeMCMC
algorithms based on this technique use a random number of samples. They have
been proposed as a promising alternative to usual fixed length simulation [25,
33, 14]. In this note we derive nonasymptotic bounds on the mean square error
(MSE) of regenerative MCMC estimates via techniques of renewal theory and
sequential statistics. These results are applied to costruct confidence
intervals. We then focus on two cases of particular interest: chains satisfying
the Doeblin condition and a geometric drift condition. Available explicit
nonasymptotic results are compared for different schemes of MCMC simulation.
