I am to find $\lim_{n \to \infty} P^n$(if it exsits) for the following stochastic matrices:
$$
\begin{pmatrix}
0.2 & 0 & 0.8 \\
0.7 & 0.3 & 0 \\
0 & 0.4 & 0.6
\end{pmatrix}
$$
and
$$
\begin{pmatrix}
0 & \frac{1}{2} & \frac{1}{2} & 0 \\
0 & 1 & 0 & 0 \\
\frac{1}{2} & 0 & 0 & \frac{1}{2} \\
\frac{1}{3} & 0 & \frac{1}{3} & \frac{1}{3}
\end{pmatrix}
$$
Now, the first one is easy – I transpose the matrix, then solve:
$$0.2 t_1 + 0.7 t_2 = t_1$$
$$0.3 t_2 + 0.4 t_3 = t_2$$
$$0.8 t_1 + 0.6 t_3 = t_3$$
$$t_1 + t_2 + t_3 = 1$$
to obtain $(t_1, t_2, t_3) = (\frac{7}{29}, \frac{8}{29}, \frac{14}{29})$. Each row of $P^n$ will be just $(t_1, t_2, t_3)$ and Wolfram Alpha agrees here.
However, the same method does not work with second Matrix – the similar equations here seem not to lead anywhere. This makes me think that such limit does not exists – how do I prove it though?
Is there an easy method to first check if the limit even exists, before trying to solve equations?
Best Answer
A stationary distribution for the second matrix is $(0,1,0,0)$. Notice that the Markov chain will end up getting stuck in state 2 eventually.