Discussion
Suppose \(\{X_{n}, n \ge 0\}\) is a Markov chain with state space \(S = \{1, 2, 3, 4\}\) and the transition probability matrix P given by :
\[ P = \begin{bmatrix} \frac{1}{2} & 0 & \frac{1}{2} & 0 \\ 0 & \frac{1}{3} & 0 & \frac{2}{3} \\ \frac{2}{3} & 0 & \frac{1}{3} & 0 \\ 0 & \frac{1}{4} & 0 & \frac{3}{4} \end{bmatrix} \]
(rows and columns are indexed by the states 1, 2, 3, 4)
Suppose \(f_{ij}\) denotes the probability of first visit to j from i. The following are four statements :
(I) \(f_{12} = 0\), (II) \(f_{24} = 1\), (III) \(f_{13} = 1\), (IV) \(f_{33} \lt 1\).
Which of the following is true ?
Answer: D
Only (I), (II) and (III) are true.
From the matrix, state 1 moves only to 1 or 3, and state 3 only to 1 or 3; state 2 moves only to 2 or 4, and state 4 only to 2 or 4. So the chain splits into two closed classes, {1, 3} and {2, 4}.
(I) Starting in {1, 3}, the chain can never reach 2, so \(f_{12} = 0\). (II) and (III) A finite closed irreducible class is recurrent, so from any state of the class every other state is reached with probability 1: \(f_{24} = 1\) and \(f_{13} = 1\).
(IV) State 3 is recurrent, so \(f_{33} = 1\), not less than 1. That leaves exactly (I), (II) and (III).
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