Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 2 / 28K / c / Solution

For an irreducible nonexplosive countable-state chain, existence of a stationary probability $\pi P(t)=\pi$ implies positive recurrence, hence recurrence, and $\pi$ is the unique invariant distribution. Thus all three questions have affirmative answers (the last is also exactly the stated identity).