= 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 $1,2,3$ give
$$
\pi_2=\frac12\pi_1,\qquad
\pi_1=\frac13\pi_2+\frac12\pi_4,\qquad
\pi_3=\frac13\pi_2+\frac12\pi_4.
$$
Hence $\pi_3=\pi_1$ and $\pi_4=5\pi_1/3$. Normalization gives $\pi_1(1+1/2+1+5/3)=1$, so
$$
\pi=\frac1{25}(6,3,6,10),\qquad
\boxed{\lim_{n\to\infty}\Pr(X_n=1\mid X_0=2)=\frac6{25}.}
$$
Stationarity alone would not guarantee convergence of the one-time distributions; irreducibility and aperiodicity justify that final limit.
Back to article page
