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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 π be the unique P-invariant probability distribution. For each n, let P n be the restriction of P to { 1,...,n}×{1,···,n}, let P ˜ n be any n×n stochastic matrix such that P ˜ n P n (elementwise), and let π n be any invariant distribution for P ˜ n . It was known previously [the second author, Linear Algebra Appl. 34, 259-267 (1980; Zbl 0484.65086)] that π n π if and only if π 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,·) is stochastically less than P(k,·), in the sense that j=1 P(i,j) j=1 P(k,j) for every .

Reviewer: A.F.Karr
MSC:
60J10Markov chains (discrete-time Markov processes on discrete state spaces)
15A51Stochastic matrices (MSC2000)