Discussion
If all states of a Markov chain are persistent non-null :
Answer: D
The chain has at least one stationary distribution.
Each closed class of positive recurrent (persistent non-null) states carries a unique stationary distribution, \(\pi_{j} = 1/\mu_{j}\) with \(\mu_{j}\) the mean recurrence time. So a chain whose states are all positive recurrent has at least one stationary distribution.
It need not be unique (A): with two closed classes there are infinitely many, as in Q176. B is false by the argument above. And \(\lim p_{ij}^{(n)} \gt 0\) for all \(i, j\) (C) needs irreducibility and aperiodicity; a periodic chain has no limit at all.
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