On the α-lazy version of Markov chains in estimation and testing problems

Given access to a single long trajectory generated by an unknown irreducible Markov chain M, we simulate an α-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.