A finite characterization of weak lumpable Markov processes. Part I: The discrete time case

A necessary condition for weak lumpability in finite Markov processes

“…This being stated, we can use this necessary condition to obtain "candidate" lumpings of E leading to an aggregated markovian process, as in [1]. This condition allows the preselection of "candidate" partitions before applying the characterization of weak lumpability in terms of the solution to some linear systems [6].…”

“…These conditions are very expensive to verify: in the worst case, the cost grows exponentially with the number of states [6].…”

“…Necessary and sufficient conditions have been studied in [3], Rubino and Sericola [6], [7] for irreducible processes. These conditions are very expensive to verify: in the worst case, the cost grows exponentially with the number of states [6].…”

Weak lumpability and pseudo-stationarity of finite markov chains

“…The approach developed in [6], [12] consists of identifying the conditional probability IP α (X n+1 ∈ C(m)|X n ∈ C(l), . .…”

“…The aim of our paper is to explore how the set A M may differ from A π . We adopt in Section 3 a somewhat different approach from Rubino and Sericola's [12] to characterize the set A M . Indeed we directly use the necessary stochastic equivalence between the aggregated process and a homogeneous Markov chain with state spaceŜ and t.p.m.…”