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Invariant distribution

The vector π is the invariant distribution of a time-homogeneous Markov chain? with state space S and transition matrix (p ij) if the components sum to one,

iSπ i=1\sum_{i \in S} \pi_i = 1

and if it satisfies

π i= jSπ jp ji\pi_i = \sum_{j \in S} \pi_j p_{ji}

for all iS.