I have two Markov chains $X_n$, $Y_n$ with the same transision matrix P, which is non-periodic and non separable. The initial distribution is $\pi_x = \frac{1}{3}[1,1,1]$ and $\pi_y$ is unknown.
Define the stopping time: $T = \inf\{n\geq 0 : X_n=Y_n\}$ , I need to find $P(T>n)$
Now, I know that the first distribution makes $X_n$ invariant, and that from some $n$ they will be of the same distribution... But I don't really know how to approach the problem.
Appreciate and help.