Given a Markov Chain with three states 0,1,2 with the following State Transition Probabilites:
$$M = \left( \begin{array}{ccc} 0.3 & 0.3 & 0.4 \\ 0.2 & 0.7 & 0.1 \\ 0.2 & 0.3 & 0.5 \\ \end{array} \right)$$
Calculate $P(X_{16}=2|X_0=0)$ and $P(X_{12}=2,X_{16}=2|X_0=0)$, without using brute force calculation.
$$P(X_{16}|X_0=0) = (M^{16})_{0,2}$$$M^n$ seems to converge to the following matrix starting from $n_0=16$:
$$\left( \begin{array}{ccc} 0.222222 & 0.5 & 0.277778 \\ 0.222222 & 0.5 & 0.277778 \\ 0.222222 & 0.5 & 0.277778 \\ \end{array} \right)$$
How can I show this without using brute force calculation. Is there any trick to show that this converges to the given matrix from $n=16$ onwards?
For the second part of the question, I think $P(X_{12}=2,X_{16}=2|X_0=0)=P(X_{16}|X_{12})P(X_{12}|X_0=0)=(M^4)_{2,2}(M^{12}_{0,2}$, again is there a computational trick to obtain this elegantly?