A geometric invariant in weak lumpability of finite Markov chains

James Ledoux (1, 2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 INSA Rennes - Institut National des Sciences Appliquées - Rennes

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.

Mots clés: States aggregation; Positive invariance of cones

Domaines: Probabilités [math.PR]

James Ledoux. A geometric invariant in weak lumpability of finite Markov chains. Journal of Applied Probability, 1997, 34 (4), pp.847-858. ⟨hal-00852311⟩