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On the equivalence of certain Markov chains

It is shown that an irreducible and aperiodic Markov chain can be altered preserving irreducibility without altering the nature of the chain in the sense that, the modified chain is transient (recurrent) if and only if the original chain is transient (recurrent). Furthermore, it is shown by means of a counterexample that ergodicity (null-recurrence) is not preserved. An interesting application of this result is a simple proof of Pakes's generalization of Foster's criterion for a chain to be recurrent.