  Given access to a single long trajectory generated by an unknown irreducible
Markov chain $M$, we simulate an $\alpha$-lazy version of $M$ which is ergodic.
This enables us to generalize recent results on estimation and identity testing
that were stated for ergodic Markov chains in a way that allows fully empirical
inference. In particular, our approach shows that the pseudo spectral gap
introduced by Paulin [2015] and defined for ergodic Markov chains may be given
a meaning already in the case of irreducible but possibly periodic Markov
chains.
