Irreducible Periodic Markov Chains

Question

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!

Answer 1

Your second transition matrix doesn’t have period $2$; it’s aperiodic. The fact that the states never transition to themselves doesn’t lead to a period $2$; the chain can still return to a state in an odd number of steps by visiting both other states in between.

If you choose your examples correctly, part b) will probably become apparent to you. (If not, consider what it would mean for the period of $X_n$ if $Y_n$ were to have a non-trivial period.)