On the imbedding problem for stochastic and doubly stochastic matrices
type: Item
instance of: scholarly article
title: On the imbedding problem for stochastic and doubly stochastic matrices
zbMATH Open document ID: 0639.15007
DOI: 10.1007/BF00356104
publication date: 1988-01-01T00:00:00Z
review text: It is shown that, for any \(n\times n\) stochastic matrix \(A=(a_{ij})\neq (\delta _{ij})\) which can be imbedded in a continuous time Markov chain, there exist distinct indices i,j such that for all k \(a_{ik}=0\) implies \(a_{jk}=0\), and likewise distinct indices i',j' such that, for all k, \(a_{ki'}=0\) implies \(a_{jk'}=0\). The present proof of this does not use Kingman and Williams' characterization of the patterns of zero entries which can occur in imbeddable stochastic matrices. In the analogous doubly stochastic situation the same result holds, even with ``implies'' replaced by ``if and only if''. The main result is that the set \({\mathcal F}\) of imbeddable stochastic (or doubly stochastic) matrices is a Lipschitz manifold with boundary. For any Markov chain leading to a matrix on the boundary of \({\mathcal F}\) the associated intensity matrix in the Kolmogorov differential equation has, at almost every time, at least one zero entry.
Mathematics Subject Classification ID: 15B51
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 60J27
zbMATH DE Number: 4042090
zbMATH Keywords: stochastic matrix
zbMATH Keywords: continuous time Markov chain
zbMATH Keywords: imbeddable stochastic matrices
zbMATH Keywords: stochastic matrices
zbMATH Keywords: Lipschitz manifold
zbMATH Keywords: intensity matrix
zbMATH Keywords: Kolmogorov differential equation
MaRDI profile type: Publication
cites work: Q3755956
cites work: The embedding problem for Markov chains with three states
cites work: A structure of the Bang-Bang representation for 3�3 embeddable matrices
cites work: A structure of doubly stochastic Markov chains
cites work: Continuous selection in a convexity theorem of Minkowski
cites work: The characterization of mixtures of non-stationary Markov transition matrices
cites work: A Bang-Bang representation for 3�3 embeddable stochastic matrices
cites work: Continuity and convexity of projections and barycentric coordinates in convex polyhedra
cites work: Q5650526
cites work: Q5542360
author: Bent Fuglede
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 4042090