Jump chain of a discrete-time Markov chain (${\tilde{P}}_{ij}=P_{ij}/(1-P_{ii}),i\ne j$)

For $P_{ii}<1$, observe a discrete-time Markov chain only when its state changes. The probability that its next different state is $j$ is ${\sum }_{r\ge 0}{P}_{ii}^r{P}_{ij}=P_{ij}/(1-P_{ii})$. Its holding times are finite geometric waiting times. This construction deletes self-transitions and differs from the existing continuous-time jump-chain concept.

An irreducible 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 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, although deleting holding times may change periodicity.