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I need to find an expression for the first state in an MC with transition matrix $P$ with tridiagonal entries. The state space is $U={1,2,..n}$ with the last state being absorbing. Expressions for limiting distribution for the other states are given by $ \pi_1 = r_{11} \pi_1 + q_{21} \pi_2 \\ \pi_j = r_{jj} \pi_j + q_{j+1, j} \pi_{j+1} + p_{j-1,j} \pi_{j-1} $

I transform this matrix by adding probability $p_{n,1}=1$ to remove the absorbing state. By solving the recurent equation I get the expression for $\pi_j, j>1$:

$ \pi_j = \frac{p_{12} p_{23} ... p_{j-1,j}}{q_{21}q_{23}...q_{j,j-1}} \pi_1 $

So my question is, how to find $\pi_1$? It's not a constant value, my impression it is also expressed in the form $\frac{p}{q}$, but I'm not too sure.

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    @Sigma Ne za 4to2012-02-15

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