Electronic Journal of Probability

Embeddable Markov Matrices

E. Davies

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We give an account of some results, both old and new, about any $n\times n$ Markov matrix that is embeddable in a one-parameter Markov semigroup. These include the fact that its eigenvalues must lie in a certain region in the unit ball. We prove that a well-known procedure for approximating a non-embeddable Markov matrix by an embeddable one is optimal in a certain sense.

Article information

Electron. J. Probab., Volume 15 (2010), paper no. 47, 1474-1486.

Accepted: 28 September 2010
First available in Project Euclid: 1 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60J22: Computational methods in Markov chains [See also 65C40] 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 65C40: Computational Markov chains

Markov matrix embeddability Markov generator eigenvalues

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Davies, E. Embeddable Markov Matrices. Electron. J. Probab. 15 (2010), paper no. 47, 1474--1486. doi:10.1214/EJP.v15-733. https://projecteuclid.org/euclid.ejp/1464819832

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  • Arendt, W.; Grabosch, A.; Greiner, G.; Groh, U.; Lotz, H. P.; Moustakas, U.; Nagel, R.; Neubrander, F.; Schlotterbeck, U. One-parameter semigroups of positive operators.Lecture Notes in Mathematics, 1184. Springer-Verlag, Berlin, 1986. x+460 pp. ISBN: 3-540-16454-5
  • Chung, Kai Lai. Markov chains with stationary transition probabilities.Die Grundlehren der mathematischen Wissenschaften, Bd. 104 Springer-Verlag, Berlin-Göttingen-Heidelberg 1960 x+278 pp.
  • Cuthbert, James R. On uniqueness of the logarithm for Markov semi-groups. J. London Math. Soc. (2) 4 (1972), 623–630.
  • Cuthbert, James R. The logarithm function of finite-state Markov semi-groups. J. London Math. Soc. (2) 6 (1973), 524–532.
  • Davies, E. B. Triviality of the peripheral point spectrum. J. Evol. Equ. 5 (2005), no. 3, 407–415.
  • Davies, E. Brian. Linear operators and their spectra.Cambridge Studies in Advanced Mathematics, 106. Cambridge University Press, Cambridge, 2007. xii+451 pp. ISBN: 978-0-521-86629-3; 0-521-86629-4
  • Dunford, N.; Schwartz, J. T. Linear Operators, Part 1, Interscience Publ., New York, 1966.
  • Eisner, Tanja. Embedding operators into strongly continuous semigroups. Arch. Math. (Basel) 92 (2009), no. 5, 451–460.
  • Elfving, G. Zur Theorie der Markoffschen Ketten, Acta Soc. Sci. Fennicae, n. Ser. A2 no. 8, (1937) 1-17.
  • Gantmacher, F. R. The theory of matrices. Vol. 1.Translated from the Russian by K. A. Hirsch.Reprint of the 1959 translation.AMS Chelsea Publishing, Providence, RI, 1998. x+374 pp. ISBN: 0-8218-1376-5
  • Haase, Markus. Functional calculus for groups and applications to evolution equations. J. Evol. Equ. 7 (2007), no. 3, 529–554.
  • Higham, Nicholas J. Functions of matrices.Theory and computation.Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008. xx+425 pp. ISBN: 978-0-89871-646-7
  • Higham N J and Lin L., On $p$th roots of stochastic matrices. MIMS preprint, 2009.
  • Israel, Robert B.; Rosenthal, Jeffrey S.; Wei, Jason Z. Finding generators for Markov chains via empirical transition matrices, with applications to credit ratings. Math. Finance 11 (2001), no. 2, 245–265.
  • Johansen, S. A central limit theorem for finite semigroups and its application to the imbedding problem for finite state Markov chains. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 26 (1973), 171–190.
  • Karpelevič, F. I. On the characteristic roots of matrices with nonnegative elements.(Russian) Izvestiya Akad. Nauk SSSR. Ser. Mat. 15, (1951). 361–383. (13,201a)
  • Kingman, J. F. C. The imbedding problem for finite Markov chains. Z. Wahrscheinlichkeitstheorie Verw. Gebiete 1 1962 14–24.
  • Kreinin, A.; Sidelnikova, M. Regularization algorithms for transition matrices, Algo Research Quarterly, 4 (2001), 23–40.
  • Minc, Henryk. Nonnegative matrices.Wiley-Interscience Series in Discrete Mathematics and Optimization. A Wiley-Interscience Publication.John Wiley & Sons, Inc., New York, 1988. xiv+206 pp. ISBN: 0-471-83966-3
  • Runnenberg, 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 = Indag. Math. 24 1962 536–541.
  • Singer, B.; Spilerman, S. The representation of social processes by Markov models, Amer. J. Sociology, 82 (1976), 1–54.
  • Speakman, J. M. O. Two Markov chains with a common skeleton. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 7 1967 224.
  • Zahl, S. A Markov process model for follow-up studies, Human Biology, 27 (1955), 90–120.