Monotone infinite stochastic matrices and their augmented truncations
type: Item
instance of: scholarly article
title: Monotone infinite stochastic matrices and their augmented truncations
zbMATH Open document ID: 0623.60089
DOI: 10.1016/0304-4149(87)90019-6
author: Diana Gibson
author: Eugene Seneta
published in: Stochastic Processes and their Applications
publication date: 1987-01-01T00:00:00Z
review text: 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 \(\ell\).
Mathematics Subject Classification ID: 60J10
Mathematics Subject Classification ID: 15B51
zbMATH DE Number: 4011599
zbMATH Keywords: stochastic matrix
zbMATH Keywords: irreducible and positive recurrent
zbMATH Keywords: tightness
zbMATH Keywords: stochastically monotone
reviewed by: Alan F. Karr
MaRDI profile type: Publication
full work available at URL: 0304-4149(87)90019-6
OpenAlex ID: W2076457397
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description: scientific article; zbMATH DE number 4011599