 Solution

Here the valid parameters lie in the irreducible aperiodic range. The finite-state theorem on convergence to an invariant distribution therefore implies convergence to the unique stationary distribution, independently of the initial state.

Solve $\pi P=\pi$ and ${\sum }_i{\pi }_i=1$. The equations for states $1,2,3$ giveHence ${\pi }_3={\pi }_1$ and ${\pi }_4=5{\pi }_1/3$. Normalization gives ${\pi }_1(1+1/2+1+5/3)=1$, soStationarity alone would not guarantee convergence of the one-time distributions; irreducibility and aperiodicity justify that final limit.