For a irreducible finite Markov chain, I know that the definition of a stationary distribution is as follows:
$\pi P = \pi$, where $P$ is a transition matrix and $\pi$ is a stationary distribution (as a vector).
Based on the definition above, I was wondering if the following definition can be valid for a stationary distribution.
$\pi_j = \lim_{n \rightarrow \infty} \frac{1}{n} \sum^n_{m=1} Pr\{X_m = j\}$
That is, are the two definitions equivalent? If so, could you please give me some links (or books) in order to check out their relationship?