Mean recurrence time and stationary distribution of a Markov chain?

Question

In a Markov chain is there a theorem relating the existence of the stationary distribution and the mean recurrence time?

E.g. impossible for stationary distribution to exist therefore mean recurrence time is infinity?

Answer 1

If the stationary distribution $\pi$ exists then this means that the "long-term" probability of being in state $i$ is given by $\pi(i)$. You can think of these probabilities as the proportion of time on average that the system spends in states. So the mean recurrence times are given by $1/\pi(i)$.

However this does not mean that the mean recurrence times associated with a stationary distribution are finite, in particular if $\pi(i)=0$ for some $i$, then there is no chance of being in state $i$ long-term and it never recurs. On the other hand if there is no stationary distribution then it doesn't make sense to consider stationary probabilities. For example if the chain is reducible into two sub-chains, then in the long term the system is is one sub-chain or the other so then it might make more sense to consider the stationary probabilities and mean recurrence times conditional on the sub-chain.

Answer 2

Only for irreducible, aperiodic chains. For these chains either a stationary distribution exists or the mean recurrence time is infinite but not both. See Theorem 8.8 of Billingsley's Probability & Measure, 3e.