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The embedding problem for Markov chains with three states

Published online by Cambridge University Press:  24 October 2008

Halina Frydman
Halina Frydman
Affiliation:
New York University
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Extract

In this paper we consider the embedding problem for Markov chains with three states. A non-singular stochastic matrix P is called embeddable if there exists a two-parameter family of stochastic matrices

satisfying

and such that

Though extensive characterizations of embeddable n × n stochastic matrices have been given in (l), (2), (3), (6), and further characterizations of embeddable 3 × 3 stochastic matrices in (4), they do not provide, except in the case of 2 × 2 stochastic matrices, easily applicable necessary and sufficient conditions for embeddability.

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Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 87 , Issue 2 , March 1980 , pp. 285 - 294
Copyright
Copyright © Cambridge Philosophical Society 1980

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References

REFERENCES

(1)Goodman, G. S.An intrinsic time for nonstationary finite Markov chains. Z. Wahrschein-lichkeitstheorie 16 (1970), 165–180.CrossRefGoogle Scholar
(2)Johansen, S.A central limit theorem for finite semigroups and its applications to the imbedding problem for finite state Markov chains. Z. Wahrscheinlichkeitstheorie 26 (1973), 171–190.CrossRefGoogle Scholar
(3)Johansen, S.The Bang–Bang problem for stochastic matrices. Z. Wahrscheinlichkeitstheorie 26 (1973), 191–195.CrossRefGoogle Scholar
(4)Johansen, S. and Ramsey, F. L. A representation theorem for imbeddable 3 × 3 stochastic matrices (1973). Preprint no. 5, Institute of Mathematical Statistics, University of Copenhagen.Google Scholar
(5)Johansen, S. and Ramsey, F. L.A Bang–Bang representation for 3 × 3 imbeddable stochastic matrices. Z. Wahrscheinlichkeitstheorie 47 (1979), 107–118.CrossRefGoogle Scholar
(6)Kingman, J. F. C. and Williams, David. A combinatorial structure of nonhomogeneous Markov chains. Z. Wahrscheinlichkeitstheorie 26 (1973), 77–86.CrossRefGoogle Scholar
(7)Frydman, H. and Singer, B.Total positivity and the embedding problem for Markov chains. Math. Proc. Cambridge Philos. Soc. 86 (1979), 399–44.CrossRefGoogle Scholar