Markovian bounds on functions of finite Markov chains

Abstract

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's 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.