Consider a Markov Chain of state space $S=\{1,2,3,4\}$ with transition matrix $P$ as follows \begin{equation*}P=\left( \begin{array}{rrrr} 0 & 1/2 & 0 & 1/2 \\ 1/4 & 0 & 3/4 & 0 \\ 0 & 1/3 & 0 & 2/3 \\ 1/2 & 0 & 1/2 & 0 \\ \end{array}\right) \end{equation*}
If it exists, find $\lim_{n\to \infty} P^{n}$.
I've set up the balance equations and solved them with the mass equation to get the stationary distribution as $(0.2,\;\; 0.2,\;\; 0.3,\;\; 0.3)$ and am confused with how this helps me calculate the limit. I checked, for high powers of n what the matrix approaches and it seems to reach $$\left( \begin{array}{rrrr} 0.4 & 0 & 0.6 & 0 \\ 0 & 0.4 & 0 & 0.6 \\ 0.4 & 0 & 0.6 & 0 \\ 0 & 0.4 & 0 & 0.6 \\ \end{array}\right) $$
So how am I supposed to use the limit/stationary law and $P$ to get to that matrix? I ask, even though I can evidently numerically solve for the limit but there are some other questions such as $P^{2n}$ and $P^{2n-1}$ that I also need to know.
Thanks