Electronic Journal of Probability

CLT for Linear Spectral Statistics of Wigner matrices

Zhidong Bai, Xiaoying Wang, and Wang Zhou

Full-text: Open access


In this paper, we prove that the spectral empirical process of Wigner matrices under sixth-moment conditions, which is indexed by a set of functions with continuous fourth-order derivatives on an open interval including the support of the semicircle law, converges weakly in finite dimensions to a Gaussian process.

Article information

Electron. J. Probab., Volume 14 (2009), paper no. 83, 2391-2417.

Accepted: 1 November 2009
First available in Project Euclid: 1 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 15B52: Random matrices
Secondary: 60F15: Strong theorems 62H99: None of the above, but in this section

Bernstein polynomial central limit theorem Stieltjes transform Wigner matrices

This work is licensed under aCreative Commons Attribution 3.0 License.


Bai, Zhidong; Wang, Xiaoying; Zhou, Wang. CLT for Linear Spectral Statistics of Wigner matrices. Electron. J. Probab. 14 (2009), paper no. 83, 2391--2417. doi:10.1214/EJP.v14-705. https://projecteuclid.org/euclid.ejp/1464819544

Export citation


  • Arnold, Ludwig. On the asymptotic distribution of the eigenvalues of random matrices. J. Math. Anal. Appl. 20 1967 262–268.
  • Arnold, L. On Wigner's semicircle law for the eigenvalues of random matrices. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 19 (1971), 191–198.
  • Bai, Z. D. Convergence rate of expected spectral distributions of large random matrices. I. Wigner matrices. Ann. Probab. 21 (1993), no. 2, 625–648.
  • Bai, Z. D. Methodologies in spectral analysis of large-dimensional random matrices, a review.With comments by G. J. Rodgers and Jack W. Silverstein; and a rejoinder by the author. Statist. Sinica 9 (1999), no. 3, 611–677.
  • Bai, Z. D.; Miao, Baiqi; Tsay, Jhishen. Convergence rates of the spectral distributions of large Wigner matrices. Int. Math. J. 1 (2002), no. 1, 65–90.
  • Bai, Z. D.; Yao, J. On the convergence of the spectral empirical process of Wigner matrices. Bernoulli 11 (2005), no. 6, 1059–1092.
  • Bai, Z. D.; Silverstein, Jack W. No eigenvalues outside the support of the limiting spectral distribution of large-dimensional sample covariance matrices. Ann. Probab. 26 (1998), no. 1, 316–345.
  • Bai, Z. D. and Silverstein, J. W. (2006). Spectral analysis of large dimensional random matrices. Mathematics Monograph Series 2, Science Press, Beijing.
  • Bai, Z. D. and Yin, Y. Q. (1988). Necessary and sufficient conditions for the almost sure convergence of the largest eigenvalue of Wigner matrices. Ann. Probab. 16 1729–1741.
  • Costin, O. and Lebowitz, J. (1995). Gaussian fluctuations in random matrices. Physical Review Letters 75 69–72.
  • Horn, Roger A.; Johnson, Charles R. Matrix analysis.Corrected reprint of the 1985 original.Cambridge University Press, Cambridge, 1990. xiv+561 pp. ISBN: 0-521-38632-2
  • Johansson, Kurt. On fluctuations of eigenvalues of random Hermitian matrices. Duke Math. J. 91 (1998), no. 1, 151–204.
  • Khorunzhy, Alexei M.; Khoruzhenko, Boris A.; Pastur, Leonid A. Asymptotic properties of large random matrices with independent entries. J. Math. Phys. 37 (1996), no. 10, 5033–5060.
  • Collins, Benoˆıt; Mingo, James A.; Åšniady, Piotr; Speicher, Roland. Second order freeness and fluctuations of random matrices. III. Higher order freeness and free cumulants. Doc. Math. 12 (2007), 1–70 (electronic).
  • Mingo, James A.; Åšniady, Piotr; Speicher, Roland. Second order freeness and fluctuations of random matrices. II. Unitary random matrices. Adv. Math. 209 (2007), no. 1, 212–240.
  • Mingo, James A.; Speicher, Roland. Second order freeness and fluctuations of random matrices. I. Gaussian and Wishart matrices and cyclic Fock spaces. J. Funct. Anal. 235 (2006), no. 1, 226–270.
  • Pastur, L. A. and Lytova A. (2009) Central Limit Theorem for linear eigenvalue statistics of random matrices with independent entries. Ann. Prob. 37, 1778-1840.
  • Sinai, Ya.; Soshnikov, A. Central limit theorem for traces of large random symmetric matrices with independent matrix elements. Bol. Soc. Brasil. Mat. (N.S.) 29 (1998), no. 1, 1–24.
  • Wigner, Eugene P. Characteristic vectors of bordered matrices with infinite dimensions. Ann. of Math. (2) 62 (1955), 548–564.