# Invariant markov chains and stopping time question

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.

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are $X_n$ and $Y_n$ independent, and what do you know about $P$ that tells you that $\pi$ is its stationary measure ? – mike Jun 7 '12 at 14:59
Any luck with my answer below? – Did Jul 12 '12 at 10:41
Bis repetita... – Did Jul 23 '12 at 10:29

## 1 Answer

For every $a\geqslant0$ and $b\geqslant0$ such that $0\lt a+b\leqslant1$, the transition matrix $$P=\begin{pmatrix}1-a-b&a&b\\ b&1-a-b&a\\ a& b&1-a-b\end{pmatrix}$$ is irreducible and aperiodic with uniform stationary distribution, but, even assuming that $(X_n)$ and $(Y_n)$ are independent, the distribution of $T$ depends very much on $(a,b)$. You can try to compute the cases (i) $a=b=\frac12$, (ii) $a=b=\frac13$ and (iii) $b=0$.

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