Positive recurrence $\iff$ stationary distribution

Let $P$ be an irreducible Markov chain on a countable state space $V$ (nonnegative entries, rows summing to one as convergent series, every state reaching every other). A state $x$ is positive recurrent when its expected return time is finite: with $\tau^+_x=\min\{t\ge1:X_t=x\}$, the tail-sum formula $\mathbb E_x(\tau^+_x)=\sum_{t\ge0}\mathbb P_x\{\tau^+_x>t\}$ converges. A stationary distribution is a nonnegative $\pi$ summing to one with $\sum_x\pi(x)P(x,y)=\pi(y)$ for every $y$ (all as convergent series).

The theorem (Theorem 21.12 of Levin–Peres–Wilmer) asserts: a state is positive recurrent if and only if the chain admits a stationary distribution.

On infinite state spaces, existence of a stationary distribution — automatic in the finite theory of Mission I — becomes a genuine dichotomy: simple random walk on $\mathbb Z$ is recurrent but has infinite expected return times and no stationary distribution, while a positively drifting queue may fail even to be recurrent. The proof constructs $\pi$ from expected visit counts during one excursion from $x$ (normalized by $\mathbb E_x(\tau^+_x)$, finite exactly under positive recurrence), and conversely reads finiteness of $\mathbb E_x(\tau^+_x)=1/\pi(x)$ off Kac's lemma. Since positive recurrence of one state is thereby equivalent to a state-free condition, it too is a class property.