Consider the Markov chain whose states are 1,2,3,4,5 and...

Consider the Markov chain whose states are {1,2,3,4,5} and whose transition probability matrix is: P = egin{pmatrix} 0.5 & 0 & 0.5 & 0 & 0 \ 0.25 & 0.5 & 0.25 & 0 & 0 \ 0.5 & 0 & 0.5 & 0 & 0 \ 0 & 0 & 0 & 0.5 & 0.5 \ 0 & 0 & 0 & 0.5 & 0.5 end{pmatrix} Specify the classes of the following Markov chains, and determine whether they are transient or recurrent

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

Aarti K.

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