# Long run proportion of transitions in a Markov chain

Let $$S$$ be a set of states for a Markov chain and let $$S^C$$ be the remaining states. Explain the identity $$\sum_{i\in S}\sum_{j\in S^C}\pi_iP_{ij}=\sum_{i\in S^C}\sum_{j\in S}\pi_iP_{ij}$$

I know that LHS refers the long run proportion of transitions going from one state in $$S$$ to another in $$S^C$$ and vice versa for RHS, but how do I explain that they are equal? A qualitative answer would be sufficient.

This is a form of reversibility, namely that under stationarity, the probability to be in $$S$$ at time $$t$$ and move to $$S^c$$ at time $$t+1$$ is the same as the probability to be in $$S^c$$ at time $$t$$ and move to $$S$$ at time $$t+1$$. It obviously holds if detailed balance$$\pi_i P_{ij} = \pi_j P_{ji}$$applies.