<s>[INST] Limit distribution of a reducible Markov chain I had a simple question yesterday when I was trying to solve an exercise on a reducible,aperiodic Markov Chain. The state spase S was $$S=\{1,...,7\}$$ and we could partition it into two closed classes $\{5,6,7\} \bigcup \{3,4\}$ but the class $\{1,2\}$ was open (thus transient). The exercise was giving the transition matrix and was asking to find the stationary distributions. At the end it was also asking to find the limit distribution, given that the INITIAL distribution was $$P[X_0=1]=P[X_0=2]=\frac{1}{2}$$ I only know what to do in case the initial probability gives mass 1 to some state (I find the absorption probabilities in each closed class, etc..) but when I have initial distribution which gives mass to both of these transient states I do not know what to do. Any helpful ideas/theorems about this?? 
Thanks a lot ! 
$$ \begin{pmatrix}
1/3 & 1/6 & 1/3 & 0 & 0 & 1/6 \\ 
3/5 & 0 & 0 & 1/5 & 1/5 & 0 & 0 \\
0 & 0 & 1/2 & 1/2 & 0 & 0 & 0 \\
0 & 0 & 1/4 & 3/4 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 1/4 & 3/4 \\
0 & 0 & 0 & 0 & 1/2 & 0 & 1/2 \end{pmatrix} $$ 
 [/INST] As dan_fulea noticed above, it suffices to calculate the case when $ P[X_0=1]=1 $ and $ P[X_0=2]=1 $ . The first is already treated in the notes: it gives the limit vector $$ \begin{pmatrix} 0, & 0, & \frac{11}{51}, & \frac{22}{51}, & \frac{18}{221}, & \frac{24}{221}, & \frac{36}{221} \end{pmatrix} $$ 
For the case $ P[X_0=2]=1 $ , since $q_{2,E_1}=\frac{10}{17}$ and $ q_{2,E_2}=\frac{7}{17}$, we get the limit vector : $$\begin{pmatrix} 0, & 0, & \frac{10}{51}, & \frac{20}{51}, & \frac{21}{221}, & \frac{28}{221}, & \frac{42}{221} \end{pmatrix} $$ Hence, the final answer to our question is : $$ \frac{1}{2}\begin{pmatrix} 0, & 0, & \frac{11}{51}, & \frac{22}{51}, & \frac{18}{221}, & \frac{24}{221}, & \frac{36}{221} \end{pmatrix} + \frac{1}{2}  \begin{pmatrix} 0, & 0, & \frac{10}{51}, & \frac{20}{51}, & \frac{21}{221}, & \frac{28}{221}, & \frac{42}{221} \end{pmatrix} $$
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