= Recurrence survives deletion of self-transitions
An <irreducible Markov chain>[irreducible] <recurrent Markov chain>[recurrent] discrete-time <Markov chain> with at least two states has $P_{ii}<1$ everywhere. Removing repeated consecutive states from a positive-<probability> path proves <irreducible Markov chain>[irreducibility] of its <discrete-time jump chain>. Each sojourn in a state is <almost surely> finite; infinitely many visits to a <recurrent state> therefore correspond to infinitely many distinct returning sojourns. The jump chain remains <recurrent Markov chain>[recurrent], although deleting <holding times> may change periodicity.
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