An Eigenvector Condition for Markov Chain Lumpability

Abstract

Under certain conditions the state space of a discrete parameter Markov chain may be partitioned to form a smaller lumped chain that retains the Markov property. Existing statements of conditions on the transition matrix P of the original chain characterizing such lumpability are difficult to verify in practice, especially when the dimension of P is large. An alternate approach, based on the eigenvectors of P, is presented and illustrated with examples.