Convergence of stationary MC

For a Markov chain to converge to have a Stationary MC distribution then in the transition matrix $K$:

$$\underset{t\to \infty }{\lim }{p}_{ij}(t)=p_j$$

or for the whole matrix

$$\underset{t\to \infty }{\lim }{K}^t-K^{t+1}=0$$

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References

@lancaster2004