Monotone infinite stochastic matrices and their augmented truncations (Q1091679)

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