Lumpability for non-irreducible finite markov chains
Atef M. Abdel-moneim, Frederick W. Leysieffer · Journal of Applied Probability · 1984
Conditions under which a function of a finite, discrete-time Markov chain, X ( t ), is again Markov are given, when X ( t ) is not irreducible. These conditions are given in terms of an interrelationship between two partitions of the state space of X ( t ), the partition induced by the minimal essential classes of X ( t ) and the partition with respect to which lumping is to be considered.