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Suppose we have a Markov chain on a finite state space $X=\{1, \ldots, r\}$ with transition matrix $Q = (q_{ij})$ and initial distribution $\mu$. I believe that we can write

$P(\omega_{k} \neq 1: k = 0, 1, \ldots, n-1))$

as

$$\mu'Q'\textbf{1}$$

where $\mu'=[\mu_{2}, \ldots, \mu_{r}]$, $Q'$ is the minor of $q_{11}$ (the $(r-1) \times (r-1)$ matrix with the first row and column deleted, and $\textbf{1}$ is the vector of all ones. I'm wondering is there a way to write

$P(\omega_{n}=1, \omega_{k} \neq 1: k=0,1, \ldots, n-1)$

in connection to what's written above, I have written out a few examples and haven't been able to see a pattern other then you take each of the terms above and multiply by $q_{k1}$.

I suppose one way to think of this is to think of $1$ as an absorbing state....

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