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Geometric Methods in System Theory pp 227–236Cite as

The Imbedding Problem for Finite Markov Chains

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Part of the book series: NATO Advanced Study Institutes Series ((ASIC,volume 3))

Abstract

The problem of characterizing the stochastic matrices which can occur in a continuous time Markov chain was first formulated by Elfving in 1937, see [7 ] and [8 ]. The problem was mentioned by Chung in 1960, [ 1 ] p 203, and in the last 10 years a number of papers have appeared.

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References

  1. Chung, K.L.: Markov Chains with Stationary Transition Probabilities. Berlin-Göttingen-Heidelberg: Springer 1960.

    MATH  Google Scholar 

  2. Cohen, J.W.: A Note on Skeleton Chains.Nieuw Arch. Wisk. 10, 180–186 (1962).

    MathSciNet  MATH  Google Scholar 

  3. Cuthbert, J.R.: On Uniqueness of the Logarithm for Markov Semigroups. J. London Math. Soc., 4, 623–630 (1972).

    Article  MathSciNet  MATH  Google Scholar 

  4. Cuthbert, J.R.: The Logarithm Function for Finite-state Markov Semi-groups. J. London Math. Soc. 6, 524–532 (1973).

    Article  MathSciNet  MATH  Google Scholar 

  5. Davidson, R.: Arithmetic and Other Properties of Certain Delphic Semi-groups l.Z. Wahrscheinlichkeitstheorie verw. Gebiete 10, 120–145 (1968).

    Article  Google Scholar 

  6. Dobrusin, R.L.: Generalization of Kolmogorov’s Equations for Markov Processes with a Finite Number of Possible States. Mat. Sb. N.S. 33, (75), 567–596 (1953).

    MathSciNet  Google Scholar 

  7. Elfving, G.: Zur Theorie der Markoffsehen Ketten. Acta Soc. Sci. Fennicae, n.Ser. A 2, No. 8, 1–17 (1937).

    Google Scholar 

  8. Elfving, G.: Über die Interpolation von Markoffschen Ketten. Soc. Sci. Fenn. Comment. Phys.-Math. 10, No.3, 1–8 (1939).

    Google Scholar 

  9. Gnedenko,B.W., Kolmogorov, A.N.: Grenzverteilungen von Summen unabhängiger Zufallsgrössen. Berlin:Akademie-Verlag 1959.

    MATH  Google Scholar 

  10. Goodman, G.: An Intrinsic Time for Non-Stationary Finite Markov Chains. Z. Wahrscheinlichkeitstheorie verw. Gebiete 16, 165–180 (1970).

    Article  MATH  Google Scholar 

  11. Grenander, U.: Probabilities on Algebraic Structures. New York: Wiley 1963.

    MATH  Google Scholar 

  12. Johansen, S.: An Application of Extreme Point Methods to the Representation of Infinitely Divisible Distributions. Z Wahrscheinlichkeitstheorie verw. Geb. 5, 304–316 (1966).

    Article  MathSciNet  MATH  Google Scholar 

  13. Johansen, S. and Goodman, G.S.: Kolmogorov’s Differential Equations for Non-stationary, Countable State Markov Processes with Uniformly Continuous Transition Probabilities. Proc. Camb. Phil. Soc. 73, 119–138 (1973).

    Article  MathSciNet  MATH  Google Scholar 

  14. Johansen, S.: Some Results on the Imbedding Problem for Finite Markov Chains. J. London Math. Soc.(1973).

    Google Scholar 

  15. Johansen, S.: The Bang-Bang Problem for Stochastic Matrices. Z. Wahrscheinlichkeitstheorie verw. Gebiete 26, 191–195 (1973).

    Article  MathSciNet  MATH  Google Scholar 

  16. Johansen, S.: A Central Limit Theorem for Finite Semigroups and Its Application to the Imbedding Problem for Finite State Markov Chains. Z. Wahrscheinlichkeitstheorie verw. Gebiete 26, 171–190 (1973).

    Article  MathSciNet  MATH  Google Scholar 

  17. Johansen, S.: The Stochastic Matrices Generated by Symmetric Intensities. Imperial College and University of Copenhagen 30 pp (1972).

    Google Scholar 

  18. Johansen, S. and Ramsey, F.: A Representation Theorem for 3x3 Stochastic Matrices. University of Copenhagen (1973).

    Google Scholar 

  19. Karpelewitch, F.I.: On the Characteristic Roots of a Matrix with Non Negative Elements. Isvestija Ser. Mat. 15, 361–383 (1951).

    Google Scholar 

  20. Kendall, D.G.: Delphic Semigroups, Infinitely Divisible Regenerative Phenomena, and the Arithmetic of p-functions. Z. Wahrscheinlichkeitstheorie verw. Gebiete 9, 163–195 (1968).

    Article  MathSciNet  MATH  Google Scholar 

  21. Kingman, J.F.C.: The Imbedding Problem for Finite Markov Chains. Z. Wahrscheinlichkeitstheorie verw. Gebiete 1, 14–24 (1962).

    Article  MathSciNet  MATH  Google Scholar 

  22. Kingman, J.F.C. and Williams, D.: The Combinatorial Structure of Non-homogeneous Markov Chains. Z. Wahrscheinlichkeitstheorie verw. Gebiete 26, 77–86 (1973).

    Article  MathSciNet  MATH  Google Scholar 

  23. Lee, E.B., Markus,L.: Foundations of Optimal Control Theory. New York: John Wiley(1968).

    Google Scholar 

  24. Loewner, C.: A Theorem on the Partial Order Derived from a Certain Transformation Semigroup. Math. Z. 72, 53–60 (1959).

    Article  MathSciNet  MATH  Google Scholar 

  25. Loewner, K.: Untersuchungen über schlichte konforme Abbildungen des Einheitskreises. I. Math. Ann. 89, 103–121 (1923).

    Article  MathSciNet  MATH  Google Scholar 

  26. Maksimov, B.M.: On the Relation Between Limit Theorems of Finite Groups and an Ergodic Theorem for Markov Chains with Doubly Stochastic Transition Matrices. Theor. Probability Appl. 10, 493–496 (1965).

    Article  MathSciNet  MATH  Google Scholar 

  27. Maksimov, B.M.: On the Convergence of Products of Independent Random Variables Taking Values from an Arbitrary Finite Group. Theor. Probability Appl. 12, 619–637 (1967).

    Article  MATH  Google Scholar 

  28. Maksimov, B.M.: Random Processes with Independent Increments with Values in an Arbitrary Finite Group. Theor. Probability Appl. 15, 215–228 (1970).

    Article  MATH  Google Scholar 

  29. Maksimov, B.M.: Divisible Distributions on Finite Groups. Theor. Probability Appl. 16, 306–318 (1971).

    MathSciNet  Google Scholar 

  30. Martin-Löf, P.: Probability Theory on Discrete Semigroups. Z. Wahrscheinlichkeitstheorie verw. Gebiete 4, 78–101 (1965).

    Article  Google Scholar 

  31. Ott, J.T.: Infinitely Divisible Stochastic Matrices. University of Rochester 33 pp (1972).

    Google Scholar 

  32. Runnenburg, J.Th.: On Elfving’s Problem of Imbedding a Time-discrete Markov Chain in a Time-continuous One for Finitely many States I. Nederl. Akad. Wetensch. Proc. Ser. A. 65, 536–541 (1962).

    MathSciNet  MATH  Google Scholar 

  33. Scheffer, C.L.: On Elfving’s Problem of Imbedding a Time-discrete Markov Chein in a Time-continuous One for Finitely many States II. Nederl. Akad. Wetensch. Proc. Ser.A. 65, 542–548 (1962).

    MathSciNet  MATH  Google Scholar 

  34. Schlesinger, L.: Neue Grundlagen für einen Infinitesimalkalkül der Matrisen. Math. Z. 33, 33–61 (1931).

    Article  MathSciNet  Google Scholar 

  35. Speakman, J.M.O.: Two Markov Chains with a Common Skeleton. Z. Wahrscheinlichkeitstheorie verw. Geb. 7, 224 (1967).

    Article  MathSciNet  Google Scholar 

  36. Vere-Jones,D.: Finite Bivariate Distributions and Semigroups of Non-negative Matrices. Quart. J. Math. Oxford Ser. 22, 247–270 (1971).

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. University of Copenhagen, Denmark

    Søren Johansen

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  1. Søren Johansen
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D. Q. Mayne R. W. Brockett

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© 1973 D. Reidel Publishing Company, Dordrecht

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Johansen, S. (1973). The Imbedding Problem for Finite Markov Chains. In: Mayne, D.Q., Brockett, R.W. (eds) Geometric Methods in System Theory. NATO Advanced Study Institutes Series, vol 3. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-2675-8_13

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  • DOI: https://doi.org/10.1007/978-94-010-2675-8_13

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-2677-2

  • Online ISBN: 978-94-010-2675-8

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