I don't know how to prove it. Stationary distribution is a stochastic process {Xt: t>_0} is stationary if for any k>_ 0 and any m>_t, the joint distribution of (Xt,....Xm) is the same as the joint distribution of (Xt+k,...Xm+k), right?
I don't know how to apply it to this problem
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Stationary states can be derived with an expression for the values of the matrix.
The idea is that pi_(j) = Sum [i=0 to M] pi(i) * p(i,j) where pi(x) is the stationary probability for state x and p(i,j) is the probability value in the Markov chain matrix.
They are also probabilities meaning they are between 0 and 1 inclusive and all add up to unity.