Some results pertaining to lumpings of a Markov chain

When the state space of a discrete parameter finite Markov chain is partitioned, and the chain defined in a natural way over the member sets of the partition retains the Markov property, then the partition is called a lumping. When the state space is large, the number of possible partitions is very large, and it will usually be very difficult to determine those few which are lumpings. The usefulness of three necessary conditions is discussed, and it is shown, that in principle at least, the problem can be solved by formulating and solving a mixed integer program. A method of speeding up the solution of mixed integer programs in which the number of rows greatly exceeds the number of columns is suggested. A generalization of a theorem pertaining to lumpings of a promotion matrix is given, and the extension of the notion of lumping to that of ‘approximate lumping’ is noted.