In this paper, we obtain Markovian bounds on a function of a homogeneous discrete time Markov chain. For deriving such bounds, we use well-known results on stochastic majorization of Markov chains and the Rogers–Pitman lumpability criterion. The proposed method of comparison between functions of Markov chains is not equivalent to generalized coupling method of Markov chains, although we obtain same kind of majorization. We derive necessary and sufficient conditions for existence of our Markovian bounds. We also discuss the choice of the geometric invariant related to the lumpability condition that we use.

We characterize the conditions under which an absorbing Markovian finite process (in discrete or continuous time) can be transformed into a new aggregated process conserving the Markovian property, whose states are elements of a given partition of the original state space. To obtain this characterization, a key tool is the quasi-stationary distribution associated with absorbing processes. It allows the absorbing case to be related to the irreducible one. We are able to calculate the set of all initial distributions of the starting process leading to an aggregated homogeneous Markov process by means of a finite algorithm. Finally, it is shown that the continuous-time case can always be reduced to the discrete one using the uniformization technique.

Criteria are given to determine whether a given finite Markov chain can be lumped weakly with respect to a given partition of its state space. These conditions are given in terms of solution classes of systems of linear equations associated with the transition probability matrix of the Markov chain and the given partition.