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A solution to the reversible embedding problem for finite Markov chains

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  • Jia, Chen

Abstract

The embedding problem for Markov chains is a famous problem in probability theory and only partial results are available up till now. In this paper, we propose a variant of the embedding problem called the reversible embedding problem which has a deep physical and biochemical background and provide a complete solution to this new problem. We prove that the reversible embedding of a stochastic matrix, if it exists, must be unique. Moreover, we obtain the sufficient and necessary conditions for the existence of the reversible embedding and provide an effective method to compute the reversible embedding. Some examples are also given to illustrate the main results of this paper.

Suggested Citation

  • Jia, Chen, 2016. "A solution to the reversible embedding problem for finite Markov chains," Statistics & Probability Letters, Elsevier, vol. 116(C), pages 122-130.
    • Handle: RePEc:eee:stapro:v:116:y:2016:i:c:p:122-130
    DOI: 10.1016/j.spl.2016.04.020
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    1. Frydman, Halina, 1983. "On a number of poisson matrices in Bang-Bang representations for 3 - 3 embeddable matrices," Journal of Multivariate Analysis, Elsevier, vol. 13(3), pages 464-472, September.
    2. 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.
    3. 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.
    4. Klara L Verbyla & Von Bing Yap & Anuj Pahwa & Yunli Shao & Gavin A Huttley, 2013. "The Embedding Problem for Markov Models of Nucleotide Substitution," PLOS ONE, Public Library of Science, vol. 8(7), pages 1-12, July.
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    Cited by:

    1. Xian Chen & Yong Chen & Yumin Cheng & Chen Jia, 2024. "Moderate and $$L^p$$ L p Maximal Inequalities for Diffusion Processes and Conformal Martingales," Journal of Theoretical Probability, Springer, vol. 37(4), pages 2990-3014, November.
    2. Manuel L. Esquível & Nadezhda P. Krasii & Gracinda R. Guerreiro, 2021. "Open Markov Type Population Models: From Discrete to Continuous Time," Mathematics, MDPI, vol. 9(13), pages 1-29, June.

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