For the question, if we go a bit into the details of the problem, here we do not mean the classification of states. First of all, we take into account a transition probability matrix of an irreducible Markov chain. In fact, we take into account a sequence of transition probability matrices of irreducible Markov chains which have the same limiting distribution. From the spectral representation of stochastic matrices, we know the limiting distribution of a transition probability matrix is the same as the rows of the “base” matrix corresponding to the eigenvalue 1. And the other “base” matrices of transition matrices may be different than each others because they may have different eigenvalues other than 1. Now, a second question comes to our mind: Is there an approach such that the transition probability matrices having the same limiting distribution are in the same convex cone in the sense of convex analysis?

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