Let be a stochastic matrix indexed by the set of positive integers, assumed irreducible and positive recurrent, and let be the unique P-invariant probability distribution. For each n, let be the restriction of P to 1,...,n, let be any stochastic matrix such that (elementwise), and let be any invariant distribution for . It was known previously [the second author, Linear Algebra Appl. 34, 259-267 (1980; Zbl 0484.65086)] that if and only if is tight.
In this paper, the authors show that tightness holds provided P is stochastically monotone; that is, if whenever , the probability distribution P(i, is stochastically less than P(k,, in the sense that for every .