Discussion
Suppose \(f_{ii}^{(n)}\) denotes the probability of the first return to \(i\) in \(n\) steps in a Markov chain. The following are three statements :
(I) For any state \(i\), \(f_{ii}^{(n)} \le p_{ii}^{(n)}\), \(n \ge 1\).
(II) For any state \(i\), \(\sum_{n \ge 1} p_{ii}^{(n)} \ge 1\).
(III) For a persistent state \(i\), \(\Sigma p_{ii}^{(n)} \ge 1\).
Which of the following is true ?
Answer: C
Both (I) and (III) are true.
(I) The event 'first return to \(i\) at step \(n\)' is contained in the event 'at \(i\) at step \(n\)', so \(f_{ii}^{(n)} \le p_{ii}^{(n)}\) for every \(n \ge 1\).
(III) A state is persistent exactly when \(\sum_{n} p_{ii}^{(n)} = \infty\), so the sum is certainly at least 1.
(II) fails for transient states. If \(i\) can be left and never revisited, all \(p_{ii}^{(n)}\) are 0 and the sum is 0. More generally \(\sum p_{ii}^{(n)} = f_{ii}/(1 - f_{ii})\), which is below 1 whenever \(f_{ii} \lt 1/2\). So 'all three' and 'both (I) and (II)' are wrong, and 'only (I)' misses (III).
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