Question: Fix $\alpha \in (0,1)$. Let $(Y_n)_{n=0,1,2,...}$ be a Markov Chain with state space $S=\{1,2,3,4,5\}$ and transition matrix $$P= \begin{bmatrix} 0 & \alpha & 1 - \alpha & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ \end{bmatrix}. $$ Define a new matrix $Q$ from $P$ such that $q_{ii} = 1- \delta$, where $\delta \in (0,1)$, and $q_{ij} = \delta p_{ij}$ for $i \neq j$. The following fact is given: for a transition matrix $P$ with $p_{ii} = 0$, if $\pi$ is stationary for the chain with matrix $P$, then $\pi$ is stationary for the chain with matrix $Q$.
The question is to find the limiting distributions of $P$ and $Q$ if they exist.
Attempt: I started by finding the stationary distribution of $P$, which I got as $$\pi = \frac{1}{3} \begin{bmatrix} 1 & \alpha & 1-\alpha & \alpha & 1-\alpha \end{bmatrix}.$$
I have managed to verify this calculation using Python code.
I also noted that the chain with matrix $P$ is irreducible and has period $3$. It is also positive recurrent as the chain is closed and finite. Since it is not aperiodic, it does not have a limiting distribution.
For $Q$, I note that it is aperiodic, irreducible and positive recurrent. It therefore has a unique limiting distribution. I write $\pi Q = Q$, I deduced the same equations as I had obtained for $P$. So $\pi$ is stationary for $Q$ as well, and it is the unique limiting distribution.
Can someone please let me know if this is correct? Thank you for your time!
Your conclusions are correct, but the limiting distribution of $Q$ can be deduced without need for further computation.
Since $P_{ii}=0$ for all $i$, we can write $Q=(1-\delta)I+\delta P$, hence $$ \pi Q = \pi(1-\delta)I + \delta\pi P = \pi-\delta\pi + \delta \pi = \pi, $$ so that $\pi$ is a stationary distribution for $Q$ (as was given in the question). Moreover, $Q$ is irreducible and recurrent since $P_{ij}>0$ implies $Q_{ij}>0$, and $Q$ is necessarily aperiodic as $Q_{ii}=\delta>0$ for all $i$. It follows that $\pi$ is the unique limiting distribution of $Q$.