  A discrete-time Markov chain can be transformed into a new Markov chain by
looking at its states along iterations of an almost surely finite stopping
time. By the optional stopping theorem, any bounded harmonic function with
respect to the transition function of the original chain is harmonic with
respect to the transition function of the transformed chain. The reverse
inclusion is in general not true. Our main result provides a sufficient
condition on the stopping time which guarantees that the space of bounded
harmonic functions for the transformed chain embeds in the space of bounded
harmonic sequences for the original chain. We also obtain a similar result on
positive unbounded harmonic functions, under some additional conditions. Our
work was motivated by and is analogous to Forghani-Kaimanovich, the
well-studied case when the Markov chain is a random walk on a discrete group.
