# Existence of limiting distribution in the context of discrete-time Markov models

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.

• No. Even if we clarify that the pi are non-negative, there’s no reason for a limit to exist... the chain could be periodic. – user8675309 Mar 27 at 9:17
• @user8675309 Thanks for the clarification. – The Pointer Mar 27 at 18:19