A geometric invariant in weak lumpability of finite Markov chains

Abstract

We consider weak lumpability of finite homogeneous Markov chains, that is when a lumped Markov chain with respect to a partition of the initial state space is also a homogeneous Markov chain. We show that weak lumpability is equivalent to the existence of a direct sum of polyhedral cones which is is positively invariant by the transition probability matrix of the original chain. It allows us, in a unified way, to derive new results on lumpability of reducible Markov chains and to obtain spectral properties associated with lumpability.

Comparison and aggregation of max-plus linear systems

“…These lumpability conditions are the counterparts of those existing for Markov chains [6]. For our activity network, considering the lumping map…”