I am having a hard time with this exercise question.
Let $X_n$ be an irreducible, positive recurrent DTMC with period $d \ge 2$ and transition matrix $P$. Let $Y_n$ be a DTMC with transition matrix $P_d$.
a. Is $Y_n$ irreducible? Justify your answer
b. Find the period of each state in $Y_n$.
My approach to part a first involves creating examples. I have found an example in which $Y_n$ would be irreducible and one where $Y_n$ would be reducible. Consider $$P = \begin{bmatrix}0&1&0\\0&0&1\\1&0&0\end{bmatrix}$$
$X_n$ has $d = 3$, and is irreducible and positive recurrent. However, $$P_d = \begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}$$
which is reducible. On the other hand, if $$P = \begin{bmatrix}0&.5&0.5\\0.3&0&0.7\\0.4&0.6&0\end{bmatrix}$$
then $d = 2$ and $$P_d = \begin{bmatrix}0.35&0.3&0.35\\0.28&0.57&0.15\\0.18&0.2&0.62\end{bmatrix}$$
which is irreducible. Therefore, I'm not sure how to approach this question.
For part b, any help is appreciated!