Discussion
The following are three statements related to a Markov Chain :
(I) For any state \(i\), \(\mu i \ge fii\)
(II) For a persistent state \(i\), \(\mu i \ge 1\)
(III) For a transient state \(i\), \(\mu i \lt 1\)
where \(\mu i\) denote the mean recurrence time for state \(i\) and \(fii\) is the probability of first return to state \(i\).
Which of the following is true ?
Answer: B
Both (I) and (II) are true.
Write \(f_{ii}^{(n)}\) for the probability of first return at step \(n\), so \(f_{ii} = \sum_{n} f_{ii}^{(n)}\) and the mean recurrence time is \(\mu_{i} = \sum_{n} n f_{ii}^{(n)}\) (taken as \(\infty\) for a transient state).
(I) Since \(n \ge 1\) in every term, \(\mu_{i} = \sum n f_{ii}^{(n)} \ge \sum f_{ii}^{(n)} = f_{ii}\). (II) For a persistent state \(f_{ii} = 1\), so (I) gives \(\mu_{i} \ge 1\); a return takes at least one step.
(III) is false: for a transient state the chain may never return, so the mean recurrence time is infinite, not less than 1. That rules out the options containing (III) and the option 'only (I)'.
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