I'm new to Markov Chains. I have this probability matrix:
$P = \begin{bmatrix}0&1/2&1/2\\1/3&0&2/3\\1/2&1/2&0\end{bmatrix}$
I see it's a 3 state(0,1 and 2) recurrent markov chain that looks like this:
I'm interested in finding this limit:
$ \lim_{n \to \infty}P(X_n = 2 | X_0 = 1)$
I understand we start in state $X_0 = 1$ and we would like to find the probability of going to state 2 in the limit. So the probability matrix should be stable after running it many times. I know I can find a numerically value by taking the matrix power many times: $P^{1000}$
But I would love to know if this can be solved analytically? I have tried doing some searching and I think it's possible to start with the $\pi$ vector: $\pi = [1/3, 0, 2/3]$
Not sure... I would love your input on this
As pointed out in the comment by joriki there is no MC with your matrix as transition matrix. (Perhaps you have copied the second row third column entry wrongly).
In an irreducible aperiodic MC with finite state space there is a unique probability vector $\pi$ with $\pi P=\pi$ and $\lim P(X_n=j|X_0=i)=\pi_j$. I will let you solve the equation $\pi P=\pi$ to find $\pi_2$.