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On the Imbedding Problem for Three-State Time Homogeneous Markov Chains with Coinciding Negative Eigenvalues

Author

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  • Yong Chen

    (Hunan University of Science and Technology)

  • Jianmin Chen

    (Hunan University of Science and Technology)

Abstract

For an indecomposable 3×3 stochastic matrix (i.e., 1-step transition probability matrix) with coinciding negative eigenvalues, a new necessary and sufficient condition of the imbedding problem for time homogeneous Markov chains is shown by means of an alternate parameterization of the transition rate matrix (i.e., intensity matrix, infinitesimal generator), which avoids calculating matrix logarithm or matrix square root. In addition, an implicit description of the imbedding problem for the 3×3 stochastic matrix in Johansen (J. Lond. Math. Soc. 8:345–351, 1974) is pointed out.

Suggested Citation

  • Yong Chen & Jianmin Chen, 2011. "On the Imbedding Problem for Three-State Time Homogeneous Markov Chains with Coinciding Negative Eigenvalues," Journal of Theoretical Probability, Springer, vol. 24(4), pages 928-938, December.
    • Handle: RePEc:spr:jotpro:v:24:y:2011:i:4:d:10.1007_s10959-010-0316-5
    DOI: 10.1007/s10959-010-0316-5
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    References listed on IDEAS

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    1. McCausland, William J., 2007. "Time reversibility of stationary regular finite-state Markov chains," Journal of Econometrics, Elsevier, vol. 136(1), pages 303-318, January.
    2. Robert B. Israel & Jeffrey S. Rosenthal & Jason Z. Wei, 2001. "Finding Generators for Markov Chains via Empirical Transition Matrices, with Applications to Credit Ratings," Mathematical Finance, Wiley Blackwell, vol. 11(2), pages 245-265, April.
    3. Mogens Bladt & Michael Sørensen, 2005. "Statistical inference for discretely observed Markov jump processes," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(3), pages 395-410, June.
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