Question

Specify the classes of the following Markov chains, and determine whether they are transient or recurrent: $$ \begin{aligned} &\mathbf{P}_{1}=\left|\begin{array}{ccc} 0 & \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & 0 & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} & 0 \end{array} \| \quad \mathbf{P}_{2}=\right| \begin{array}{llll} 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 \\ \frac{1}{2} & \frac{1}{2} & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array} \mid\\ &\mathbf{P}_{3}=\left|\begin{array}{ccccc} \frac{1}{2} & 0 & \frac{1}{2} & 0 & 0 \\ \frac{1}{4} & \frac{1}{2} & \frac{1}{4} & 0 & 0 \\ \frac{1}{2} & 0 & \frac{1}{2} & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{2} & \frac{1}{2} \\ 0 & 0 & 0 & \frac{1}{2} & \frac{1}{2} \end{array}\left\|\quad \mathbf{P}_{4}=\mid \begin{array}{lllll} \frac{1}{2} & 4 & 0 & 0 & 0 \\ \frac{1}{2} & \frac{1}{2} & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & + & - & 0 \\ 1 & 0 & 0 & 0 & 0 \end{array}\right\|\right. \end{aligned} $$

          Specify the classes of the following Markov chains, and determine whether they are transient or recurrent:
$$
\begin{aligned}
&\mathbf{P}_{1}=\left|\begin{array}{ccc}
0 & \frac{1}{2} & \frac{1}{2} \\
\frac{1}{2} & 0 & \frac{1}{2} \\
\frac{1}{2} & \frac{1}{2} & 0
\end{array} \| \quad \mathbf{P}_{2}=\right| \begin{array}{llll}
0 & 0 & 0 & 1 \\
0 & 0 & 0 & 1 \\
\frac{1}{2} & \frac{1}{2} & 0 & 0 \\
0 & 0 & 1 & 0
\end{array} \mid\\
&\mathbf{P}_{3}=\left|\begin{array}{ccccc}
\frac{1}{2} & 0 & \frac{1}{2} & 0 & 0 \\
\frac{1}{4} & \frac{1}{2} & \frac{1}{4} & 0 & 0 \\
\frac{1}{2} & 0 & \frac{1}{2} & 0 & 0 \\
0 & 0 & 0 & \frac{1}{2} & \frac{1}{2} \\
0 & 0 & 0 & \frac{1}{2} & \frac{1}{2}
\end{array}\left\|\quad \mathbf{P}_{4}=\mid \begin{array}{lllll}
\frac{1}{2} & 4 & 0 & 0 & 0 \\
\frac{1}{2} & \frac{1}{2} & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 \\
0 & 0 & + & - & 0 \\
1 & 0 & 0 & 0 & 0
\end{array}\right\|\right.
\end{aligned}
$$

Added by Manuel D.

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