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Monotone infinite stochastic matrices and their augmented truncations. (English) Zbl 0623.60089

Let $P=\left[P\left(i,j\right)\right]$ 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 $\left\{$ 1,...,n$\right\}×\left\{1,···,n\right\}$, let ${\stackrel{˜}{P}}_{n}$ be any $n×n$ stochastic matrix such that ${\stackrel{˜}{P}}_{n}\ge {P}_{n}$ (elementwise), and let ${\pi }_{n}$ be any invariant distribution for ${\stackrel{˜}{P}}_{n}$. 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 $i, the probability distribution P(i,$·\right)$ is stochastically less than P(k,$·\right)$, in the sense that ${\sum }_{j=1}^{\ell }P\left(i,j\right)\ge {\sum }_{j=1}^{\ell }P\left(k,j\right)$ for every $\ell$.

Reviewer: A.F.Karr
##### MSC:
 60J10 Markov chains (discrete-time Markov processes on discrete state spaces) 15A51 Stochastic matrices (MSC2000)