= Irreducibility and recurrence pass between a CTMC and its jump chain
For a <continuous-time Markov chain> whose jump times tend to infinity, the chain and its <jump chain> have the same directed paths of positive-probability jumps. They are therefore <irreducible Markov chain>[irreducible] simultaneously, and a state is <recurrent state>[recurrent] for one exactly when it is recurrent for the other: the event of returning after departure depends only on the sequence of visited states.
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