Abstract
We consider the question of whether a function of a finite-state Markov chain is also Markovian, that is whether the chain is lumpable with respect to the partition determined by the function. We explore how an initial distribution with respect to which the chain is lumpable may differ from a pseudo-stationary initial distribution. Our results give insight into Peng's condition under which the Chapman-Kolmogorov equation implies that the lumped chain is Markovian. We illustrate these ideas by treating the question of whether the absorption time of a finite-state absorbing Markov chain is geometric
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