Algebraic characterization of infinite Markov chains where movement to the right is limited to one step

Abstract

We consider an infinite Markov chain with states E0, E1, …, such that E1, E2, … is not closed, and for i ≧ 1 movement to the right is limited by one step. Simple algebraic characterizations are given for persistency of all states, and, if E0 is absorbing, simple expressions are given for the probabilities of staying forever among the transient states. Examples are furnished, and simple necessary conditions and sufficient conditions for the above characterizations are given.