23 - Convergence to Equilibrium

Convergence to Equilibrium

Let $P$ be a transition matrix that is irreducible and aperiodic with stationary distribution $\pi$
Let $X_n$ be a Markov Chain with transition matrix $P$ and any distribution then

$$P(X_n=j)\to {\pi }_j\text{ as }n\to \infty \text{ for all }j\in I$$

In particular then

$$p_{ij}^{(n)}\to {\pi }_j\text{ as }n\to \infty \text{ for all }i,j\in I$$