Suppose $X_n$, $n \in \mathbb{Z}$ is an irreducible Markov chain with state space $S$, which has stationary probabilities $\pi_i = P(X_n = i)$ for all $i \in S$. My textbook seems to avoid talking about the stationary probabilities in the case that $X_n$ is transient. Why is that? My question is:
- Are $\pi_i$ infinite, or zero? Or something else?
- How can I prove either?
Existence of a stationary distribution is in fact equivalent to positive recurrence (not just recurrence) for an irreducible Markov chain. If the chain is only assumed recurrent and still irreducible, then we can find a unique stationary measure $\pi$ where $0 < \pi_i < \infty$ for all $i \in S$, but we potentially have $\sum_{i \in S} \pi_i = \infty$.
As for proof of these facts, they can be found in most textbooks on discrete time Markov chains, for example Grimmett and Welsh's "Probability: An Introduction".