We consider an infinite Markov chain with states E
0, E
1, …, such that E
1, E
2, … 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 E
0 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.