On the $\alpha$-lazy version of Markov chains in estimation and testing problems

Perhaps surprisingly, it appears that despite its extensive usage, seemingly no guidelines exist regarding when moving to the lazy version is appropriate. In this paper we make a first step towards a characterization of such scenarios, beginning with Markov chains statistical estimation and testing problems. Parallel to this work, Chan et al. [2021] gave a unified treatment of Markov chains estimation and testing problems in the single trajectory setting, based on the works of Wolfer and Kontorovich mentioned above. Their results hold for irreducible Markov chains and this was achieved by replacing the pseudo spectral gap, which is defined only for ergodic Markov chains, with the cover time, which is defined for every irreducible Markov chains. They then used deep results that connect the cover time with the blanket time.