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Assuming the state space = {0,1,2,3}, I know that this Markov Chain is reducible and I believe that there are 4 communicating classes here: Class 1: {State 0} , Class 2: {State 1}, Class 3: {State 2} and Class 4: {State 3}. Is this true? Also, how can we find a stationary distribution for this Markov chain (if it exists) and how do we prove that the stationary distribution is unique?

popcornchicken
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    {0,1} make a class. If you start in one of those states, you stay there. Then {2,3} are a class. – jdods Apr 19 '21 at 15:58
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    Put the transition matrix in software and raise it to a very large power, like $P^{10000}$. Then every row will be a limiting distribution. In this case, there will be 2 distinct rows, each copied a second time. Those two distinct rows are your two extremal stationary distributions, and any convex combination of those is also a stationary distribution. Otherwise solve for the right eigenvalues and eigenvectors. Any eigenvector associated with an eigenvalue of one will be a stationary distribution (and any convex combo of those too). – jdods Apr 19 '21 at 16:58
  • @jdods I see, thanks a lot ! – popcornchicken Apr 21 '21 at 13:26

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