In the study of discrete-time Markov models, it is said that if a limiting distribution $\pi$ exists, then it satisfies $\pi_j = \sum_{i = 1}^N \pi_i p_{i, j}, \ j \in S$ and $\sum_{j = 1}^N \pi_j = 1$. Is the converse also true? That is, if $\pi_j = \sum_{i = 1}^N \pi_i p_{i, j}, \ j \in S$ and $\sum_{j = 1}^N \pi_j = 1$, then is $\pi$ a limiting distribution?
I would greatly appreciate it if people would please take the time to clarify this.