CLT for Linear Spectral Statistics of Wigner matrices

Zhidong Bai (Northeast Normal University Changchun)
Xiaoying Wang (Northeast Normal University Changchun)
Wang Zhou (National University of Singapore)

Abstract


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.

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

Pages: 2391-2417

Publication Date: November 1, 2009

DOI: 10.1214/EJP.v14-705

References

  1. Arnold, Ludwig. On the asymptotic distribution of the eigenvalues of random matrices. J. Math. Anal. Appl. 20 1967 262--268. MR0217833 (36 #922)
  2. Arnold, L. On Wigner's semicircle law for the eigenvalues of random matrices. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 19 (1971), 191--198. MR0348820 (50 #1315)
  3. Bai, Z. D. Convergence rate of expected spectral distributions of large random matrices. I. Wigner matrices. Ann. Probab. 21 (1993), no. 2, 625--648. MR1217559 (95a:60039)
  4. 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. MR1711663 (2000e:60044)
  5. 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. MR1825933 (2003d:60071)
  6. Bai, Z. D.; Yao, J. On the convergence of the spectral empirical process of Wigner matrices. Bernoulli 11 (2005), no. 6, 1059--1092. MR2189081 (2006g:60034)
  7. 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. MR1617051 (99b:60041)
  8. Bai, Z. D. and Silverstein, J. W. (2006). Spectral analysis of large dimensional random matrices. Mathematics Monograph Series 2, Science Press, Beijing.
  9. 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.
  10. Costin, O. and Lebowitz, J. (1995). Gaussian fluctuations in random matrices. Physical Review Letters 75 69--72.
  11. 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 MR1084815 (91i:15001)
  12. Johansson, Kurt. On fluctuations of eigenvalues of random Hermitian matrices. Duke Math. J. 91 (1998), no. 1, 151--204. MR1487983 (2000m:82026)
  13. 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. MR1411619 (97j:82082)
  14. Collins, Benoˆıt; Mingo, James A.; Śniady, 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). MR2302524 (2009d:15057)
  15. Mingo, James A.; Śniady, Piotr; Speicher, Roland. Second order freeness and fluctuations of random matrices. II. Unitary random matrices. Adv. Math. 209 (2007), no. 1, 212--240. MR2294222 (2009c:15027)
  16. 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. MR2216446 (2007h:46080)
  17. 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.
  18. 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. MR1620151 (99f:60053)
  19. Wigner, Eugene P. Characteristic vectors of bordered matrices with infinite dimensions. Ann. of Math. (2) 62 (1955), 548--564. MR0077805 (17,1097c)
Bookmark and Share


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