A Markov chain on the state-space $I=\{1,2,3,4,5,6,7\}$ has transition matrix

$$P=\left(\begin{array}{ccccccc} 0 & 1 / 2 & 1 / 4 & 0 & 1 / 4 & 0 & 0 \\ 1 / 3 & 0 & 1 / 2 & 0 & 0 & 1 / 6 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 / 2 & 0 & 1 / 2 \end{array}\right)$$

Classify the chain into its communicating classes, deciding for each what the period is, and whether the class is recurrent.

For each $i, j \in I$ say whether the $\operatorname{limit}^{-1} \lim _{n \rightarrow \infty} p_{i j}^{(n)}$ exists, and evaluate the limit when it does exist.