Discussion
The following are four statements about a Markov chain. A unique stationary distribution exists if the Markov chain is irreducible and (I) ergodic (II) non-null persistent, (III) transient and (IV) null persistent.
Which of the following is true ?
Answer: C
Both (I) and (II) are true.
An irreducible Markov chain has a stationary distribution if and only if it is positive (non-null) recurrent, and then the stationary distribution is unique, with \(\pi_{j} = 1/\mu_{j}\), the reciprocal of the mean recurrence time. So (II) is sufficient. An ergodic chain is aperiodic and non-null persistent, so (I) is also sufficient, and in that case \(p_{ij}^{(n)} \to \pi_{j}\) as well.
For transient or null persistent irreducible chains, \(p_{ij}^{(n)} \to 0\) and no stationary distribution exists (a stationary measure, if any, cannot be normalised). So (III) and (IV) fail, and the option 'only (I)' or 'only (II)' leaves out the other correct condition.
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