Necessary and sufficient conditions for recurrence and transience of Markov chains, in terms of inequalities
Jean‐François Mertens, Ester Samuel‐Cahn, Shmuel Zamir · 1978
For an aperiodic, irreducible Markov chain with the non-negative integers as state space it is shown that the existence of a solution to E-,op,ijy,- y,; i N> 0 in which y,- * 0 is necessary and sufficient for recurrence, and the existence of a bounded solution to the same inequalities, with yk O, such that lim y, = c. j =( Sufficiency of the condition of Theorem 1 is well known. For N = 1 this is Theorem 5 in Foster (1953). Foster states that 'the condition would appear necessary only under certain additional assumptions'. For N- 1 the sufficiency is proved in Pakes (1969), Theorem 3. For the sake of completeness, and in its own interest, we include a different proof of the sufficiency. Proof of sufficiency. Let y be a solution of (1) such that limy, = c. By possibly adding a constant to all components of y we may without loss of generality assume y, i 0. Define y * by y * = y, for i- N, y * for i N, and p *i = 1 for i < N. Then P * and y * satisfy =(,p*y * y y * i 0. Let Zo = N and let Z,; n 1, be the state at time n of the Markov chain governed by P*. Note that the original chain