Discussion
Following are three statements. In a Markov chain, a persistent state i is a null persistent if :
(I) \(\limsup p_{ii}^{(n)} = 0\)
(II) mean recurrence time is \(\infty\)
(III) \(\limsup p_{ii}^{(n)} \gt 0\)
Which of the following is true ?
Answer: B
Both (I) and (II) are true.
A persistent (recurrent) state is null persistent when its mean recurrence time \(\mu_{i}\) is infinite; that is the definition, so (II) is true. For such a state \(p_{ii}^{(n)} \to 0\), so \(\limsup p_{ii}^{(n)} = 0\) and (I) is true.
A positive persistent state has finite mean recurrence time and, if aperiodic, \(p_{ii}^{(n)} \to 1/\mu_{i} \gt 0\). So \(\limsup p_{ii}^{(n)} \gt 0\) characterises positive persistence, and (III) is false for a null persistent state.
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