On Weak Lumpability of Denumerable Markov Chains

We consider weak lumpability of denumerable discrete or continuous time Markov chains. Firstly, we are concerned with irreducible recurrent positive and R-positive Markov chains evolving in discrete time. We study the properties of the set of all initial distributions of the starting chain leading to an aggregated homogeneous Markov chain with respect to a partition of the state space. In particular, the asymptotic interpretation of the quasi-stationary distribution is addressed and it is fruitfully used for weak lumpability of R-positive Markov chains. Furthermore, we present a simple example which shows that a denumerable Markov chain can be (weakly) lumped into a finite Markov chain. Finally, it is stated that weak lumpability for any continuous time Markov chain with an uniform transition semi-group can be handled in discrete time context. The sequel of this result are also discussed for irreducible positive-recurrent or -positive continuous time Markov chains. Key-words: Weak Lumpability, Positive recurrence, R-positivity, Quasi-stationary distribution, Uniform transition semi-group.