For an Ergodic Markov chain with unique stationary distribution $\pi$ we have

$$\lim_{k \rightarrow \infty} P^k = \left ( \begin{array}{ccc} \cdots & \pi & \cdots\\ \cdots & \pi & \cdots\\ \vdots & \vdots & \vdots\\ \cdots & \pi & \cdots\\ \end{array}\right ).$$