Linear dynamics for the state vector of Markov chain functions

Abstract

Let (φ(X n )) n be a function of a finite-state Markov chain (X n ) n . In this article, we investigate the conditions under which the random variables φ( n ) have the same distribution as Y n (for every n), where (Y n ) n is a Markov chain with fixed transition probability matrix. In other words, for a deterministic function φ, we investigate the conditions under which (X n ) n is weakly lumpable for the state vector. We show that the set of all probability distributions of X 0, such that (X n ) n is weakly lumpable for the state vector, can be finitely generated. The connections between our definition of lumpability and the usual one (i.e. as the proportional dynamics property) are discussed.