Necessary and sufficient conditions for recurrence and transience of Markov chains, in terms of inequalities

Abstract

For an aperiodic, irreducible Markov chain with the non-negative integers as state space it is shown that the existence of a solution to in which y i → ∞is necessary and sufficient for recurrence, and the existence of a bounded solution to the same inequalities, with y k < y o , · · ·, y N –1 for some k ≧ N , is necessary and sufficient for transience.