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Given a markov chain where the next state is related to the previous state by the following matrix: $\begin{array}{c|ccc} & A & B & C\\ \hline A & p_1 & q_1 & r_1\\ B & p_2 & q_2 & r_2\\ C & p_3 & q_3 & r_3\\ \end{array}$ Where the system of equations is given by

A$_{n+1} = p_1A_n + p_2B_n+p_3C_n$

B$_{n+1} = q_1A_n + q_2B_n+q_3C_n$

C$_{n+1} = r_1A_n + r_2B_n+r_3C_n$

How can the final relationships be found? (i.e. the limit of each function as n $\rightarrow \infty$, assuming A$_0 = B_0 = C_0$).

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If there is a limit, it will satisfy $\eqalign{A&=p_1A+p_2B+p_3C\cr B&=q_1A+q_2B+q_3C\cr C&=r_1A+r_2B+r_3C\cr}$ so it's just a matter of solving a system of three linear equations in three unknowns.

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    I understand what you're saying now - as n approaches infinity, S$_n$ = S$_{n+1}$ and so on, so that's how you don't have different unknowns. Thanks Ilmari for the tip on including the normalization condition. I couldn't solve my problem until I used that.2012-03-07