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 (rows and columns indexed by the states 1, 2, 3, 4) :
\[ P = \begin{bmatrix} \frac{1}{2} & 0 & \frac{1}{2} & 0 \\ 0 & \frac{1}{3} & 0 & \frac{2}{3} \\ \frac{2}{3} & 0 & \frac{1}{3} & 0 \\ 0 & \frac{1}{4} & 0 & \frac{3}{4} \end{bmatrix} \]
Which of the following options is true ?
Answer: A
States 1 and 3 are essential states.
From state 1 the chain moves only to 1 or 3, and from 3 only to 1 or 3. So {1, 3} is a closed class: once entered it is never left, and 1 and 3 communicate. A state is essential if every state it leads to leads back to it, which holds for 1 and 3.
A finite closed communicating class is positive persistent, so 1 and 3 are not transient or inessential, and not null persistent (a finite chain has no null persistent states). The same holds for the separate closed class {2, 4}.
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