This is an example demonstrating my answer to this math.stackexchange post: Let be a Markov chain on with transition matrix

where . Then is irreducible and aperiodic, so there exists a unique stationary distribution satisfying

where denotes the row of . From the global balance equation we see that , and normalization yields

To actually compute , we could write where the columns of are linearly independent eigenvectors of and is a diagonal matrix whose entries are the corresponding eigenvalues. This is possible because the Perron-Frobenius theorem implies that the eigenspace corresponding to the eigenvalue is one-dimensional, and the rank of is two. Instead, we will compute the generating function of . Let

then

where

Partial fraction decomposition yields

and so we have

Similar computations yield

and it follows that

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Great stuff, thanks for sharing!

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