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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.

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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.

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