Discussion
Suppose in a Markov chain, \(f_{22}\) denotes the probability of the first return to state 2 and \(\mu_{2}\) denotes the mean recurrence time to state 2. Suppose \(\sum_{n \ge 1} p_{22}^{(n)} = 2.45\). Then which of the following options is true ?
Answer: A
The true option is \(f_{22} \lt 1\).
A state is recurrent exactly when \(\sum_{n} p_{22}^{(n)} = \infty\). Here the sum is the finite number 2.45, so state 2 is transient and \(f_{22} \lt 1\). In fact the sum equals \(\frac{f_{22}}{1 - f_{22}}\), so
\[ f_{22} = \frac{2.45}{3.45} \approx 0.71 \]
A transient state has \(f_{22} \lt 1\) (so option B fails) and an infinite mean recurrence time (option C fails), and since the series of \(p_{22}^{(n)}\) converges, its terms tend to 0 (option D fails).
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