On the $$\alpha $$-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 $$\alpha $$ -lazy version of M which is ergodic. This enables us generalize recent results on estimation and identity testing, that were stated for ergodic chains, in way allows fully empirical inference. In particular, our approach shows the pseudo spectral gap introduced Paulin (Electron J Probab 20:32, 2015) defined chains may be given meaning already case but possibly periodic chains.