A Note on Rate of Convergence in Probability to Semicircular Law

Zhidong Bai (Northeast Normal University)
Jiang Hu (Northeast Normal University)
Guangming Pan (Nanyang Technological University)
Wang Zhou (National University of Singapore)

Abstract


In the present paper, we prove that under the assumption of the finite sixth moment for elements of a Wigner matrix, the convergence rate of its empirical spectral distribution to the Wigner semicircular law in probability is $O(n^{-1/2})$ when the dimension n tends to infinity.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 2439-2451

Publication Date: November 23, 2011

DOI: 10.1214/EJP.v16-963

References

  1. G. W. Anderson, A. Guionnet, and O. Zeitouni. An introduction to random matrices. Cambridge University Press, 2010. Math. Review 2760897
  2. Z. D. Bai. Convergence rate of expected spectral distributions of large random matrices. Part I. Wigner matrices. The Annals of Probability, 21(2):625--648, 1993. Math. Review 1217559
  3. Z. D. Bai. Convergence rate of expected spectral distributions of large random matrices. Part II. Sample covariance matrices. The Annals of Probability, 21(2):649--672, 1993. Math. Review 1217560
  4. Z. D. Bai, B. Q. Miao, and J. Tsay. A note on the convergence rate of the spectral distributions of large random matrices. Statistics & Probability Letters, 34:95--101, 1997. Math. Review 1457501
  5. Z. D. Bai, B. Q. Miao, and J. Tsay. Convergence rates of the spectral distributions of large Wigner matrices. International Mathematical Journal, 1:65--90, 2002. Math. Review 1825933
  6. Z. D. Bai and J. W. Silverstein. Spectral analysis of large dimensional random matrices. Second Edition. Springer Verlag, 2010. Math. Review 2567175
  7. Z. D. Bai, X. Y. Wang, and W. Zhou. CLT for linear spectral statistics of Wigner matrices. Electronic Journal of Probability, 14:2391--2417, 2009. Math. Review 2556016
  8. L. Erdos, H-T. Yau, and J. Yin. Rigidity of eigenvalues of generalized Wigner matrices. Preprint. arXiv:1007.4652v4
  9. S. G. Bobkov, F. Gotze, and A. N. Tikhomirov. On concentration of empirical measures and convergence to the semi-circle law. Journal of Theoretical Probability, Apr. 2010.
  10. F. Gotze and A. Tikhomirov. Rate of convergence to the semi-circular law. Probability Theory and Related Fields, 127(2):228--276, 2003. Math. Review 2013983
  11. F. Gotze and A. Tikhomirov. The rate of convergence for spectra of GUE and LUE matrix ensembles. Central European Journal of Mathematics, 3(4):666--704, Dec. 2005. Math. Review 2171668
  12. F. Gotze, A. N. Tikhomirov, and D. a. Timushev. Rate of convergence to the semi-circle law for the Deformed Gaussian Unitary Ensemble. Central European Journal of Mathematics, 5(2):305--334, June 2007. Math. Review 2300275
  13. M. L. Mehta. Random matrices, Third Edition. Academic Press, 2004. Math. Review 2129906
  14. J. W. Silverstein and Z. D. Bai. On the empirical distribution of eigenvalues of a class of large dimensional random matrices. Journal of Multivariate Analysis, 54(2):175--192, Aug. 1995. Math. Review 1345534
  15. A. Tikhomirov. On the rate of convergence of the expected spectral distribution function of a Wigner matrix to the semi-circular law. Siberian Advances in Mathematics, 19(3):211--223, 2009. Math. Review 2655022
  16. E. P. Wigner. Characteristic vectors of bordered matrices with infinite dimensions. Annals of Mathematics, 62(3):548--564, 1955. Math. Review 0077805
Bookmark and Share


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.