When is a Markov chain regenerative?

A sequence of random variables {Xn}n≥0{Xn}n≥0 is called regenerative if it can be broken up into iid components. The problem addressed in this paper is that of determining under what conditions a Markov chain is regenerative. It is shown that an irreducible Markov chain with a countable state space is regenerative for any initial distribution if and only if it is recurrent (null or positive). An extension of this to the general state space case is also discussed.