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Second Order Markov Chains with Finite Vectors Known Stationary

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Abstract

We consider the stationary probability vectors of a second order Markov chain, and discuss the geometrical structure of the set of these vectors with finite ones known stationary. By elementary methods, we deduce that if three vectors on a line segment in the standard simplex are stationary probability vectors of a second order Markov chain, then the set of stationary probability vectors of the chain contains all vectors on the segment. Based on properties of quadratic stochastic operators and affine subspaces, we give two generalizations of the above result for any plane sections of the standard simplex. The generalizations claim that the stationarity of all vectors of any given plane section of the simplex is equivalent to the stationarity of finite vectors in some appropriate positions of the section. Furthermore, the obtained results are applied to discuss the minimum number of such known stationary vectors as interior points for faces.

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References

  1. Badocha, M., Bartoszek, W.: Quadratic stochastic operators on Banach lattices. Positivity 22(2), 477–492 (2018). https://doi.org/10.1007/s11117-017-0522-9

    Article  MathSciNet  MATH  Google Scholar 

  2. Bozorgmanesh, H., Hajarian, M.: Convergence of a transition probability tensor of a higher-order Markov chain to the stationary probability vector. Numer. Linear Algebra Appl. 23(6), 972–988 (2016). https://doi.org/10.1002/nla.2063

    Article  MathSciNet  MATH  Google Scholar 

  3. Chang, K.C., Pearson, K., Zhang, T.: Perron–Frobenius theorem for nonnegative tensors. Commun. Math. Sci. 6(2), 507–520 (2008). https://doi.org/10.4310/cms.2008.v6.n2.a12

    Article  MathSciNet  MATH  Google Scholar 

  4. Chang, K.C., Zhang, T.: On the uniqueness and non-uniqueness of the positive Z-eigenvector for transition probability tensors. J. Math. Anal. Appl. 408(2), 525–540 (2013). https://doi.org/10.1016/j.jmaa.2013.04.019

    Article  MathSciNet  MATH  Google Scholar 

  5. Ching, W., Ng, M.: Markov Chains: Models Algorithms and Applications. Springer, New York (2006)

    MATH  Google Scholar 

  6. Demmel, J.W.: Applied Numerical Linear Algebra. SIAM, Philadelphia (1997)

    Book  MATH  Google Scholar 

  7. Deng, A., Qu, Z., Ma, H., et al.: Stationary probability vectors of the second order Markov chains. J. Donghua Univ. (Engl. Ed.) 36(6), 580–585 (2019)

    Google Scholar 

  8. Fasino, D., Tudisco, F.: Ergodicity coefficients for higher-order stochastic processes. SIAM J. Math. Data Sci. 2(3), 740–769 (2020). https://doi.org/10.1137/19M1285214

    Article  MathSciNet  MATH  Google Scholar 

  9. Friedland, S., Gaubert, S., Han, L.: Perron–Frobenius theorem for nonnegative multilinear forms and extensions. Linear Algebra Appl. 438(2), 738–749 (2013). https://doi.org/10.1016/j.laa.2011.02.042

    Article  MathSciNet  MATH  Google Scholar 

  10. Ganikhodzhaev, R., Mukhamedov, F., Rozikov, U.: Quadratic stochastic operators and processes: results and open problems. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 14(02), 279–335 (2011). https://doi.org/10.1142/S0219025711004365

    Article  MathSciNet  MATH  Google Scholar 

  11. Góra, P., Boyarsky, A.: A matrix solution to the inverse Perron–Frobenius problem. Proc. Am. Math. Soc. 118(2), 409–414 (1993). https://doi.org/10.1090/S0002-9939-1993-1129877-8

    Article  MathSciNet  MATH  Google Scholar 

  12. Heiner, M., Kottas, A.: Estimation and selection for high-order Markov chains with Bayesian mixture transition distribution models. J. Comput. Graph. Stat. 31(1), 100–112 (2022). https://doi.org/10.1080/10618600.2021.1979565

    Article  MathSciNet  MATH  Google Scholar 

  13. Hu, S., Qi, L.: Convergence of a second order Markov chain. Appl. Math. Comput. 241, 183–192 (2014). https://doi.org/10.1016/j.amc.2014.05.011

    Article  MathSciNet  MATH  Google Scholar 

  14. Li, C.K., Zhang, S.: Stationary probability vectors of higher-order Markov chains. Linear Algebra Appl. 473, 114–125 (2015). https://doi.org/10.1016/j.laa.2014.03.043

    Article  MathSciNet  MATH  Google Scholar 

  15. Li, W., Ng, M.K.: On the limiting probability distribution of a transition probability tensor. Linear Multilinear Algebra 62(3), 362–385 (2014). https://doi.org/10.1080/03081087.2013.777436

    Article  MathSciNet  MATH  Google Scholar 

  16. Mukhamedov, F., Embong, A.F.: On stable b-bistochastic quadratic stochastic operators and associated non-homogenous Markov chains. Linear Multilinear Algebra 66(1), 1–21 (2018). https://doi.org/10.1080/03081087.2017.1281215

    Article  MathSciNet  MATH  Google Scholar 

  17. Mukhamedov, F., Ganikhodjaev, N.: Quantum Quadratic Operators and Processes. Springer, Switzerland (2015)

    Book  MATH  Google Scholar 

  18. Ng, M., Qi, L., Zhou, G.: Finding the largest eigenvalue of a non-negative tensor. SIAM J. Matrix Anal. Appl. 31, 1090–1099 (2009). https://doi.org/10.1137/09074838X

    Article  MathSciNet  MATH  Google Scholar 

  19. Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, New York (1999)

    Book  MATH  Google Scholar 

  20. Qaralleh, I., Mukhamedov, F.: Volterra evolution algebras and their graphs. Linear Multilinear Algebra 69(12), 2228–2244 (2021). https://doi.org/10.1080/03081087.2019.1664387

    Article  MathSciNet  MATH  Google Scholar 

  21. Qi, L., Luo, Z.: Tensor Analysis: Spectral Theory and Special Tensors. SIAM, Philadelphia (2017)

    Book  MATH  Google Scholar 

  22. Raftery, A.E.: A model for high-order Markov chains. J. R. Stat. Soc. Ser. B. 47(3), 528–539 (1985). https://doi.org/10.1111/j.2517-6161.1985.tb01383.x

    Article  MathSciNet  MATH  Google Scholar 

  23. Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)

    Book  MATH  Google Scholar 

  24. Saburov, M.: On regularity of diagonally positive quadratic doubly stochastic operators. Results Math. 72(4), 1907–1918 (2017). https://doi.org/10.1007/s00025-017-0723-3

    Article  MathSciNet  MATH  Google Scholar 

  25. Saburov, M., Yusof, N.A.: Counterexamples to the conjecture on stationary probability vectors of the second order Markov chains. Linear Algebra Appl. 507, 153–157 (2016). https://doi.org/10.1016/j.laa.2016.06.012

    Article  MathSciNet  MATH  Google Scholar 

  26. Saburov, M., Yusof, N.A.: On quadratic stochastic operators having three fixed points. J. Phys. Conf. Ser. 697, 012012 (2016). https://doi.org/10.1088/1742-6596/697/1/012012

    Article  Google Scholar 

  27. Saburov, M., Yusof, N.A.: On uniqueness of fixed points of quadratic stochastic operators on a 2D simplex. Methods Funct. Anal. Topol. 24(03), 255–264 (2018). https://doi.org/10.24193/fpt-ro.2018.1.30

    Article  MathSciNet  MATH  Google Scholar 

  28. Saburov, M., Yusof, N.: The structure of the fixed point set of quadratic operators on the simplex. Fixed Point Theory 19(1), 383–396 (2018). https://doi.org/10.24193/fpt-ro.2018.1.30

    Article  MathSciNet  MATH  Google Scholar 

  29. Seneta, E.: Non-negative Matrices and Markov Chains. Springer, New York (2006)

    MATH  Google Scholar 

  30. Snapper, E., Troyer, R.J.: Metric Affine Geometry. Academic Press, New York (1971)

    MATH  Google Scholar 

  31. Yakovyna, V., Symets, I.: Reliability assessment of CubeSat nanosatellites flight software by high-order Markov chains. Procedia Comput. Sci. 192, 447–456 (2021). https://doi.org/10.1016/j.procs.2021.08.046

    Article  Google Scholar 

  32. Zhou, D., Zhang, S., Yildirim, M.Y., et al.: High-order structure exploration on massive graphs: a local graph clustering perspective. ACM Trans. Knowl. Discov. Data 15(2), 1–26 (2021). https://doi.org/10.1145/3425637

    Article  Google Scholar 

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Acknowledgements

The authors are thankful to anonymous referees for their useful comments. This work was supported by National Natural Science Foundation of China (No.11671258).

Funding

This work was supported by National Natural Science Foundation of China (No. 11671258).

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  1. College of Science, Donghua University, Shanghai, China

    Aiping Deng & Yuting Hu

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  1. Aiping Deng
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All authors contributed to the study conception and design. The subject is proposed, guided and improved by Aiping Deng. The first draft of the manuscript was written by Yuting Hu. All authors commented on each versions of the manuscript, and approved the final manuscript.

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Correspondence to Aiping Deng.

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Deng, A., Hu, Y. Second Order Markov Chains with Finite Vectors Known Stationary. Results Math 78, 46 (2023). https://doi.org/10.1007/s00025-022-01823-0

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