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Monotone infinite stochastic matrices and their augmented truncations. (English) Zbl 0623.60089
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 $P\sb n$ be the restriction of P to $\{$ 1,...,n$\}\times \{1,...,n\}$, let $\tilde P\sb n$ be any $n\times n$ stochastic matrix such that $\tilde P\sb n\ge P\sb n$ (elementwise), and let $\pi\sb n$ be any invariant distribution for $\tilde P\sb n$. It was known previously [the second author, Linear Algebra Appl. 34, 259-267 (1980; Zbl 0484.65086)] that $\pi\sb n\to \pi$ if and only if $\pi\sb n$ 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 $\sum\sp{\ell}\sb{j=1}P(i,j)\ge \sum\sp{\ell}\sb{j=1}P(k,j)$ for every $\ell$.
Reviewer: A.F.Karr

60J10Markov chains (discrete-time Markov processes on discrete state spaces)
15B51Stochastic matrices
Full Text: DOI
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