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Convergence Rates of the Spectral Distributions of Large Random Quaternion Self-Dual Hermitian Matrices

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Abstract

In this paper, convergence rates of the spectral distributions of quaternion self-dual Hermitian matrices are investigated. We show that under conditions of finite 6th moments, the expected spectral distribution of a large quaternion self-dual Hermitian matrix converges to the semicircular law in a rate of \(O(n^{-1/2})\) and the spectral distribution itself converges to the semicircular law in rates \(O_p(n^{-2/5})\) and \(O_{a.s.}(n^{-2/5+\eta })\). Those results include GSE as a special case.

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References

  1. Bai, Z.D.: Convergence rate of expected spectral distributions of large random matrices. part i. wigner matrices. Ann. Probab. 21(2), 625–648 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bai, Z.D., Yin, Y.Q.: Limit of the smallest eigenvalue of a large dimensional sample covariance matrix. Ann. Probab. 21(3), 1275–1294 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bai, Z.D., Miao, B.Q., Tsay, J.: Remarks on the convergence rate of the spectral distributions of wigner matrices. J. Theor. Probab. 12(2), 301–311 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bai, Z.D., Miao, B.Q., Tsay, J.: A note on the convergence rate of the spectral distributions of large random matrices. Stat. Probab. Lett. 34(1), 95–101 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bai, Z.D., Miao, B.Q., Tsay, J.: Convergence rates of the spectral distributions of large wigner matrices. Int. Math. J. 1(1), 65–90 (2002)

    MATH  MathSciNet  Google Scholar 

  6. Bai, Z.D., Silverstein, J.W.: Spectral Analysis of Large Dimensional Random Matrices. Springer, Berlin (2010)

    Book  MATH  Google Scholar 

  7. Chevalley, C.: Lie Groups. Princeton UP, Princeton (1946)

    MATH  Google Scholar 

  8. Craig, A.: Tracy and Harold Widom. Level-spacing distributions and the airy kernel. Commun. Math. Phys. 159(1), 151–174 (1994)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  9. Dean, David S., Majumdar, Satya N.: Extreme value statistics of eigenvalues of gaussian random matrices. Phys. Rev. E 77, 041108 (Apr 2008)

  10. Deavours, C.A.: The quaternion calculus. Am. Math. Mon. 80(9), 995–1008 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  11. Dilworth, S.J.: Some probabilistic inequalities with applications to functional analysis. In: Banach Spaces, Contemporary Mathematics 144, AMS, Providence. Citeseer, 1993.

  12. Dumitriu, I., Koev, P.: Distributions of the extreme eigenvaluesof beta-jacobi random matrices. SIAM J. Matrix Anal. Appl. 30(1), 1–6 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  13. Füredi, Z., Komlós, J.: The eigenvalues of random symmetric matrices. Combinatorica 1(3), 233–241 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  14. Gtze, F., Tikhomirov, A.: Rate of convergence to the semi-circular law. Probab. Theory Rel. Fields 127(2), 228–276 (2003)

    Article  Google Scholar 

  15. Gtze, F., Tikhomirov, A.: Rate of convergence to the semicircular law for the gaussian unitary ensemble. Theory Probab. Appl. 47(2), 323–330 (2003)

    Article  MathSciNet  Google Scholar 

  16. Hamilton, W.R., Hamilton, W.E.: Elements of Quaternions. Longmans, Green, & Company, London (1866)

    Google Scholar 

  17. Juhász, Ferenc: On the spectrum of a random graph. Colloq. Math. Soc. Janos Bolyai 25, 313–316 (1978)

    Google Scholar 

  18. O’Rourke, S., Vu, V.: Universality of local eigenvalue statistics in random matrices with external source. arXiv preprint arXiv:1308.1057, 2013.

  19. Tao, Terence, Van, Vu: The wigner-dyson-mehta bulk universality conjecture for wigner matrices. Electron. J. Probab. 16, 2104–2121 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  20. Wang, Dong: The largest eigenvalue of real symmetric, hermitian and hermitian self-dual random matrix models with rank one external source, part i. J. Stat. Phys. 146(4), 719–761 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  21. Wigner, Eugene P.: Characteristic vectors of bordered matrices with infinite dimensions. Ann. Math. 62(3), 548–564 (1955)

    Article  MATH  MathSciNet  Google Scholar 

  22. Wigner, Eugene P.: On the distribution of the roots of certain symmetric matrices. Ann. Math. 67(2), 325–327 (1958)

    Article  MATH  MathSciNet  Google Scholar 

  23. Yin, Y.Q., Bai, Z.D., Hu, J.: On the semicircular law of large dimensional random quaternion matrices. arXiv preprint arXiv:1309.6937, 2013.

  24. Yin, Y.Q., Bai, Z.D., Hu, J.: On the limit of extreme eigenvalues of large dimensional random quaternion matrices. Phys. Lett. A 378(16–17), 1049–1058 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  25. Zhang, Fuzhen: Quaternions and matrices of quaternions. Linear Algebra Appl. 251, 21–57 (1997)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

Y. Q. Yin was partially supported by a Grant CNSF 11301063; Z. D. Bai was partially supported by CNSF 11171057 and PCSIRT

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  1. KLASMOE and School of Mathematics & Statistics, Northeast Normal University, Changchun, 130024, People’s Republic of China

    Yanqing Yin & Zhidong Bai

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  1. Yanqing Yin
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  2. Zhidong Bai
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Correspondence to Zhidong Bai.

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Yin, Y., Bai, Z. Convergence Rates of the Spectral Distributions of Large Random Quaternion Self-Dual Hermitian Matrices. J Stat Phys 157, 1207–1224 (2014). https://doi.org/10.1007/s10955-014-1096-6

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