A note on one of the Markov chain Monte Carlo novice's questions

We introduce a novel time-homogeneous Markov embedding of a class of time inhomogeneous Markov chains widely used in the context of Monte Carlo sampling algorithms which allows us to answer one of the most basic, yet hard, question about the practical implementation of these techniques. We also show that this embedding sheds some light on the recent result of [#maire-douc-olsson2013]. We discuss further applications of the technique.