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.