Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-05T06:41:00.375Z Has data issue: false hasContentIssue false
  • English
  • Français

On equal-input and monotone Markov matrices

Part of: Markov processes

Published online by Cambridge University Press:  06 June 2022

Michael Baake  and
Jeremy Sumner
Michael Baake*
Affiliation:
Bielefeld University
Jeremy Sumner*
Affiliation:
University of Tasmania
*
*Postal address: Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany. Email: [email protected]
**Postal address: School of Natural Sciences, Discipline of Mathematics, University of Tasmania, Private Bag 37, Hobart, TAS 7001, Australia. Email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The practically important classes of equal-input and of monotone Markov matrices are revisited, with special focus on embeddability, infinite divisibility, and mutual relations. Several uniqueness results for the classic Markov embedding problem are obtained in the process. To achieve our results, we need to employ various algebraic and geometric tools, including commutativity, permutation invariance, and convexity. Of particular relevance in several demarcation results are Markov matrices that are idempotents.

Keywords

Markov matrixembedding problemmonotone partial orderconvexity

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Original Article
Information
Advances in Applied Probability , Volume 54 , Issue 2 , June 2022 , pp. 460 - 492
Check for updates
Copyright
© The Author(s) 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Baake, M. and Sumner, J. (2020). Notes on Markov embedding. Linear Algebra Appl. 594, 262–299.10.1016/j.laa.2020.02.016CrossRefGoogle Scholar
Casanellas, M., Fernández-Sánchez, J. and Roca-Lacostena, J. (2020). The embedding problem for Markov matrices. Preprint. Available at https://arxiv.org/abs/2005.00818.Google Scholar
Culver, W. J. (1966). On the existence and uniqueness of the real logarithm of a matrix. Proc. Amer. Math. Soc. 17, 1146–1151.10.1090/S0002-9939-1966-0202740-6CrossRefGoogle Scholar
Daley, D. J. (1968). Stochastically monotone Markov chains. Z. Wahrscheinlichkeitsth. 10, 305–317.10.1007/BF00531852CrossRefGoogle Scholar
Davies, E. B. (2010). Embeddable Markov matrices. Electron. J. Prob. 15, 1474–1486.10.1214/EJP.v15-733CrossRefGoogle Scholar
Eisner, T. and Radl, A. (2021). Embeddability of real and positive operators. To appear in Linear Multilinear Algebra. Available at https://arxiv.org/abs/2003.08186.Google Scholar
Elfving, G. (1937). Zur Theorie der Markoffschen Ketten. Acta Soc. Sci. Fennicae A2, 1–17.Google Scholar
Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edn. John Wiley, New York.Google Scholar
Fernández-Sánchez, J., Sumner, J. G., Jarvis, P. D. and Woodhams, M. D. (2015). Lie Markov models with purine/pyrimidine symmetry. J. Math. Biol. 70, 855–891.10.1007/s00285-014-0773-zCrossRefGoogle ScholarPubMed
Gantmacher, F. R. (1986). Matrizentheorie. Springer, Berlin.CrossRefGoogle Scholar
Göndöcs, F., Michaletzky, G., Móri, T. F. and Székely, G. J. (1985). A characterization of infinitely divisible Markov chains with finite state space. Ann. Univ. Sci. Budapest R. Eötvös Nom. 27, 137–141.Google Scholar
Guerry, M.-A. (2019). Sufficient embedding conditions for three-state discrete-time Markov chains with real eigenvalues. Linear Multilinear Algebra 67, 106–120.CrossRefGoogle Scholar
Higham, N. J. (2002). Accuracy and Stability of Numerical Algorithms, 2nd edn. Society for Industrial and Applied Mathematics, Philadelphia.10.1137/1.9780898718027CrossRefGoogle Scholar
Higham, N. J. (2008). Functions of Matrices: Theory and Computation. Society for Industrial and Applied Mathematics, Philadelphia.10.1137/1.9780898717778CrossRefGoogle Scholar
Higham, N. J. and Lin, L. (2011). On pth roots of stochastic matrices. Linear Algebra Appl. 435, 448–463.10.1016/j.laa.2010.04.007CrossRefGoogle Scholar
Högnäs, G. and Mukherjea, A. (2011). Probability Measures on Semigroups: Convolution Products, Random Walks and Random Matrices, 2nd edn. Springer, New York.10.1007/978-0-387-77548-7CrossRefGoogle Scholar
Horn, R. A. and Johnson, C. R. (2013). Matrix Analysis, 2nd edn. Cambridge University Press.Google Scholar
Keilson, J. and Kester, A. (1977). Monotone Markov matrices and monotone Markov processes. Stoch. Process. Appl. 5, 231–241.10.1016/0304-4149(77)90033-3CrossRefGoogle Scholar
Kijima, M. (1997). Markov Processes for Stochastic Modeling. Springer, Boston.CrossRefGoogle Scholar
Kingman, J. F. C. (1962). The imbedding problem for finite Markov chains. Z. Wahrscheinlichkeitsth. 1, 14–24.10.1007/BF00531768CrossRefGoogle Scholar
Lang, S. (1993). Algebra, 3rd edn. Addison-Wesley, Reading, MA.Google Scholar
Lindqvist, B. H. (1987). Monotone and associated Markov chains, with applications to reliability theory. J. Appl. Prob. 24, 679–695.CrossRefGoogle Scholar
Luther, U. and Rost, K. (2004). Matrix exponentials and inversion of confluent Vandermonde matrices. Electron. Trans. Numer. Anal. 18, 91–100.Google Scholar
Marcus, M. and Minc, H. (1992). A Survey of Matrix Theory and Matrix Inequalities. Dover, New York.Google Scholar
Norris, J. R. (2005). Markov Chains. Cambridge University Press.Google Scholar
Sloane, N. J. A. (1964). The On-Line Encyclopedia of Integer Sequences. Available at https://oeis.org.Google Scholar
Steel, M. (2016). Phylogeny—Discrete and Random Processes in Evolution. Society for Industrial and Applied Mathematics, Philadelphia.10.1137/1.9781611974485CrossRefGoogle Scholar
Sumner, J. (2017). Multiplicatively closed Markov models must form Lie algebras. ANZIAM J. 59, 240–246.10.1017/S1446181117000359CrossRefGoogle Scholar
Sumner, J. G., Fernández-Sánchez, J. and Jarvis, P. D. (2012). Lie Markov models. J. Theoret. Biol. 298, 16–31.CrossRefGoogle ScholarPubMed
Webster, R. (1994). Convexity. Oxford University Press.Google Scholar
Yang, Z. and Ranala, B. (2012). Molecular phylogenetics: principles and practice. Nature Rev. Genet. 13, 303–314.CrossRefGoogle ScholarPubMed