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Inhomogeneous products of cyclic irreducible nonnegative matrices

Published online by Cambridge University Press:  09 April 2009

G. C. Taylor
G. C. Taylor
Affiliation:
School of Economic and Financial Studies, Macquarie University, North Ryde 2113 N.S.W., Australia.
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Abstract

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The now well-known Coale–Lopez theorem, and variants of it, state that, under certain conditions, the product of the first r members of an infinite inhomogeneous sequence of nonnegative matrices approaches, as r → ∞, the class of matrices which, apart from scalar multiples, have only one distinct column. The aim of the present paper is to lay down conditions under which such products approach the class of matrices which, apart from scalar multiples, have no more than d distinct columns. A stronger result is then obtained by considering stochastic matrices instead of just nonnegative ones.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 24 , Issue 4 , December 1977 , pp. 409 - 416
Copyright
Copyright © Australian Mathematical Society 1977

References

Hajnal, J. (1956). ‘The ergodic properties of non-homogeneous finite Markov chains’, Proceedings of the Cambridge Philosophical Society, 52, 67–77.Google Scholar
Hajnal, J. (1958), ‘Weak ergodicity in non-homogeneous finite Markov chains’, Proceedings of the Cambridge Philosophical Society, 54, 233–46.Google Scholar
Lopez, A. (1961), Problems in stable population theory, Office of Population Research, Princeton University.Google Scholar
Parlett, B. K. (1970), ‘Ergodic properties of populations I: the one sex model’, Theoretical Population Biology, 1, 191–207.CrossRefGoogle ScholarPubMed
Seneta, E. (1973), Non-negative matrices. An introduction to theory and applications, George Allen and Unwin Limited.Google Scholar
Wolfwitz, J. (1963), ‘Products of indecomposable, aperiodic, stochastic matrices’. Proceedings of the Ameriian Mathematical Society, 733–7.CrossRefGoogle Scholar