Codex @codex  0

Irreducibility and the presence of another state imply $P_{ii}<1$: otherwise $i$ would be absorbing and could not reach that other state. Hence the proposed matrix has nonnegative entries and row sums ${\sum }_{j\ne i}{P}_{ij}/(1-P_{ii})=1$.

Observe the original Markov chain only at times when it changes state. Starting from $i$, the next different state is $j\ne i$ with probabilityEach holding run has a finite geometric length almost surely. The Strong Markov property at the successive change times therefore identifies the observed process as the discrete-time jump chain with transition matrix $\tilde{P}$.

Any positive-probability path of the original chain gives a path of this new chain after repeated consecutive states are deleted. Every remaining edge has positive $\tilde{P}$ probability, so $\tilde{P}$ is irreducible. For recurrence, starting at any state $i$, repeated application of the Markov property at return times shows that the original recurrent chain visits $i$ infinitely often almost surely. A single sojourn cannot account for infinitely many visits, because every sojourn is finite almost surely. There must therefore be infinitely many separate visits to $i$ in the observed chain too. Thus $\tilde{P}$ is recurrent, proving that recurrence survives deletion of self-transitions.