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Total positivity and the embedding problem for Markov chains

Published online by Cambridge University Press:  24 October 2008

Halina Frydman  and
Burton Singer
Halina Frydman
Affiliation:
New York University
Burton Singer
Affiliation:
Columbia University and Rockefeller University
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Abstract

We prove that the class of transition matrices for the finite state time-inhomogeneous birth and death processes coincides with the class of non-singular totally positive stochastic matrices. Thus we obtain a complete solution to the embedding problem for this class of Markov chains.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 86 , Issue 2 , September 1979 , pp. 339 - 344
Copyright
Copyright © Cambridge Philosophical Society 1979

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References

REFERENCES

(1)Gantmacher, F. R.The Theory of Matrices, vol. 2 (New York; Chelsea, 1960).Google Scholar
(2)Johansen, S.A central limit theorem for finite semigroups and its application to the imbedding problem for finite state Markov chains, Z. Wahr. 26 (1973), 171–190.CrossRefGoogle Scholar
(3)Johansen, S.The bang-bang problem for stochastic matrices. Z. Wahr. 26 (1973), 191–195.CrossRefGoogle Scholar
(4)Johansen, S.Some results on the imbedding problem for finite Markov chains. J. London Math. Soc. 8 (1974), 345–351.CrossRefGoogle Scholar
(5)Karlin, S.Total Positivity, vol. 1 (Stanford, California, Stanford University Press, 1968).Google Scholar
(6)Karlin, S. and McGregor, J. L.A characterization of birth and death processes. Proc. Nat. Acad. Sci. U.S.A. 45 (1959), 375–379.CrossRefGoogle ScholarPubMed
(7)Kingman, J. F. C.Geometrical aspects of the theory of non-homogeneous Markov chains. Math. Proc. Camb. Philos. Soc. 77 (1975), 171–183.CrossRefGoogle Scholar
(8)Schlesinger, L.Neue Grundlagen für einem Infinitesimalkalkül der Matrisen. Math. Zeitschr 33 (1931), 33–61.CrossRefGoogle Scholar
(9)Whitney, A.A reduction theorem for totally positive matrices. J. d'Analyse Math. Jerusalem, 2 (1952), 88–92.CrossRefGoogle Scholar