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

We formulate extendibility of the minimax one-trajectory length of several statistical Markov chains inference problems and give sufficient conditions for both the possibility and impossibility of such extensions. We follow up and apply this framework to recently published results on learning and identity testing of ergodic Markov chains. In particular, we show that for some of the aforementioned results, we can omit the aperiodicity requirement by simulating an $\alpha$-lazy version of the original process, and quantify the incurred cost of removing this assumption.