Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
A CLT for a band matrix model
Download PDF
Download PDF
  • Published: 10 February 2005

A CLT for a band matrix model

  • Greg W. Anderson1 &
  • Ofer Zeitouni2 

Probability Theory and Related Fields volume 134, pages 283–338 (2006)Cite this article

  • 440 Accesses

  • 120 Citations

  • Metrics details

Abstract

A law of large numbers and a central limit theorem are derived for linear statistics of random symmetric matrices whose on-or-above diagonal entries are independent, but neither necessarily identically distributed, nor necessarily all of the same variance. The derivation is based on systematic combinatorial enumeration, study of generating functions, and concentration inequalities of the Poincaré type. Special cases treated, with an explicit evaluation of limiting variances, are generalized Wigner and Wishart matrices.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Bai, Z.D.: Methodologies in spectral analysis of large-dimensional random matrices, a review. Statist. Sinica 9, 611–677 (1999)

    MATH  MathSciNet  Google Scholar 

  2. Bai, Z.D., Yao, J.-F.: On the convergence of the spectral empirical process of Wigner matrices. Preprint, 2003

  3. Bai, Z.D., Silverstein, J.W.: CLT for linear spectral statistics of large-dimensional sample covariance matrices. Annals Probab. 32, 553–605 (2004)

    MATH  Google Scholar 

  4. Bobkov, S.G.: Remarks on Gromov-Milman's inequality. V. Syktyvkar Univ. 3, 15–22 (1999)(in Russian)

    MATH  MathSciNet  Google Scholar 

  5. Borovkov, A.A., Utev, S.A.: An inequality and a characterization of the normal distribution connected with it. Theor. Prob. Appl. 28, 209–218 (1983)

    MATH  MathSciNet  Google Scholar 

  6. Cabanal-Duvillard, T.: Fluctuations de la loi empirique de grande matrices aléatoires. Ann. Inst. H. Poincaré - Probab. Statist. 37, 373–402 (2001)

    MATH  MathSciNet  Google Scholar 

  7. Chatterjee, S., Bose, A.: A new method of bounding rate of convergence of empirical spectral distributions. Preprint, 2004. Available at www-stat.stanford.edu/∼ souravc/rateofconv.pdf

  8. Deift, P.: Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach. New York University-Courant Institute of Mathematical Sciences, AMS, 2000

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

    MATH  MathSciNet  Google Scholar 

  10. Girko, V.L.: Theory of Random Determinants. Kluwer, 1990

  11. Guionnet, A.: Large deviation upper bounds and central limit theorems for band matrices. Ann. Inst. H. Poincaré Probab. Statist 38, 341–384 (2002)

    MATH  MathSciNet  Google Scholar 

  12. Guionnet, A., Zeitouni, O.: Concentration of the spectral measure for large matrices. Elec. Commun. Probab. 5, 119–136 (2000)

    MATH  MathSciNet  Google Scholar 

  13. Hiai, F., Petz, D.: The Semicircle Law. Free Random Variables and Entropy. AMS, 2000

  14. Janson, S.: Gaussian Hilbert Spaces. Cambridge University Press, 1997

  15. Jonsson, D.: Some limit theorems for the eigenvalues of a sample covariance matrix. J. Mult. Anal. 12, 1–38 (1982)

    MATH  MathSciNet  Google Scholar 

  16. Johansson, K.: On fluctuations of eigenvalues of random Hermitian matrices. Duke J. Math. 91, 151–204 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  17. Khorunzhy, A.M., Khoruzhenko, B.A., Pastur, L.A.: Asymptotic properties of large random matrices with independent entries. J. Math. Phys. 37, 5033–5060 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  18. Ledoux, M.: The Concentration of Measure Phenomenon. American Mathematical Society, Providence, 2001

  19. Mehta, M.L.: Random Matrices. 2nd ed. Academic Press, 1991

  20. Mingo, J.A., Speicher, R.: Second Order Freeness and Fluctuations of Random Matrices: I. Gaussian and Wishart matrices and Cyclic Fock spaces. Preprint, available at arXiv:math.OA/0405191, 2004

  21. Molchanov, S.A., Pastur, L.A., Khorunzhii, A.M.: Distribution of the eigenvalues of random band matrices in the limit of their infinite order. Theoret. Math. Phys. 90, 108–118 (1992)

    MATH  MathSciNet  Google Scholar 

  22. Nica, A., Shlyakhtenko, D., Speicher, R.: Operator-valued distributions. I. Characterizations of freeness. Int. Math. Res. Notices 29, 1509–1538 (2002)

    MATH  MathSciNet  Google Scholar 

  23. Pastur, L., Lejay, A.: Matrices aléatoires: statistique asymptotique des valeurs propres. In: Lecture Notes in Mathematics 1801, 135–164 (2003)

    MATH  MathSciNet  Google Scholar 

  24. Pastur, L.A., Marčenko, V.A.: The distribution of eigenvalues in certain sets of random matrices. Math. USSR-Sbornik 1, 457–483 (1967)

    MATH  Google Scholar 

  25. Shlyakhtenko, D.: Random Gaussian band matrices and freeness with amalgamation. Int. Math. Res. Notices 20, 1013–1025 (1996)

    MATH  MathSciNet  Google Scholar 

  26. 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, 1–24 (1998)

    MATH  MathSciNet  Google Scholar 

  27. Stanley, R.: Enumerative Combinatorics, vol. II. Cambridge University press, 1999

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

    Article  MATH  MathSciNet  Google Scholar 

  29. Wishart, J.: The generalized product moment distribution in samples from a Normal multivariate population. Biometrika 20A, 32–52 (1928)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, University of Minnesota, 206 Church St, SE, Minneapolis, MN 55455, USA

    Greg W. Anderson

  2. Departments of Mathematics and of EE, Technion, Haifa, 32000, Israel

    Ofer Zeitouni

Authors
  1. Greg W. Anderson
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ofer Zeitouni
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

O.Z. was partially supported by NSF grant number DMS-0302230.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Anderson, G., Zeitouni, O. A CLT for a band matrix model. Probab. Theory Relat. Fields 134, 283–338 (2006). https://doi.org/10.1007/s00440-004-0422-3

Download citation

  • Received: 16 June 2004

  • Revised: 01 December 2004

  • Published: 10 February 2005

  • Issue Date: February 2006

  • DOI: https://doi.org/10.1007/s00440-004-0422-3

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Stochastic Process
  • Probability Theory
  • Limit Theorem
  • Matrix Model
  • Mathematical Biology
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature