Skip to main content
Log in

Limiting spectral distributions of some band matrices

  • Published:
Periodica Mathematica Hungarica Aims and scope Submit manuscript

Abstract

We use the method of moments to establish the limiting spectral distribution (LSD) of appropriately scaled large dimensional random symmetric circulant, reverse circulant, Toeplitz and Hankel matrices which have suitable band structures. The input sequence used to construct these matrices is assumed to be either i.i.d. with mean zero and variance one or independent and appropriate finite fourth moment. The class of LSD includes the normal and the symmetrized square root of chi-square with two degrees of freedom. In several other cases, explicit forms of the limit do not seem to be obtainable but the limits can be shown to be symmetric and their second and the fourth moments can be calculated with some effort. Simulations suggest some further properties of the limits.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

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

    MathSciNet  MATH  Google Scholar 

  2. A. Basak, Large dimensional random matrices, M. Stat. Project Report, May 2009, Indian Statistical Institute, Kolkata, 2009.

    Google Scholar 

  3. R. Bhatia, Matrix Analysis, Springer, New York, 1997.

    Book  Google Scholar 

  4. A. Bose and A. Basak, Limiting spectral distribution of some band matrices, Technical report No R16/2009, September 11, Stat-Math Unit, Indian Statistical Institute, Kolkata, 2009. Available at www.isical.ac.in/_statmath.

  5. A. Bose, S. Gangopadhyay and A. Sen, Limiting spectral distribution of XX′ matrices, Ann. Inst. H. Poincaré, 46 (2010), 677–707.

    Article  MathSciNet  MATH  Google Scholar 

  6. A. Bose and J. Mitra, Limiting spectral distribution of a special circulant, Statist. Probab. Lett., 60 (2002), 111–120.

    Article  MathSciNet  MATH  Google Scholar 

  7. A. Bose and A. Sen, Another look at the moment method for large dimensional random matrices, Electron. J. Probab., 13 (2008), 588–628.

    MathSciNet  MATH  Google Scholar 

  8. W. Bryc, A. Dembo and T. Jiang, Spectral measure of large random Hankel, Markov and Toeplitz matrices, Ann. Probab., 34 (2006), 1–38. Also available at http://arxiv.org/abs/math.PR/0307330

    Article  MathSciNet  MATH  Google Scholar 

  9. R. M. Gray, Toeplitz and Circulant Matrices: A review, Now Publishers, Norwell, Massachusetts, 2009.

    Google Scholar 

  10. U. Grenander and G. Szegő, Toeplitz forms and their applications, California Monographs in Mathematical Sciences, University of California Press, Berkeley - Los Angeles, 1958.

    MATH  Google Scholar 

  11. V. Kargin, Spectrum of random Toeplitz matrices with band structures, Electron. Comm. Probab., 14 (2009), 412–421.

    MathSciNet  MATH  Google Scholar 

  12. D.-Z. Liu and Z.-D. Wang, Limit distributions for eigenvalues for random Hankel and Toeplitz band matrices, J. Theoret. Probab., to appear. DOI: 10.1007/s10959-009-0260-4

  13. A. Massey, S. J. Miller and J. Sinsheimer, Distribution of eigenvalues of real symmetric palindromic Toeplitz matrices and circulant matrices, J. Theoret. Probab., 20 (2007), 637–662.

    Article  MathSciNet  MATH  Google Scholar 

  14. I. Popescu, General tridiagonal random matrix models, limiting distributions and fluctuations, Probab. Theory Related Fields, 144 (2009), 179–220.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Communicated by Bálint Tóth

Research supported by J. C. Bose National Fellowship, Department of Science and Technology, Government of India. Part of the work was done while the author was visiting Dept. of Economics, University of Cincinnati.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Basak, A., Bose, A. Limiting spectral distributions of some band matrices. Period Math Hung 63, 113–150 (2011). https://doi.org/10.1007/s10998-011-7113-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10998-011-7113-5

Mathematics subject classification number

Key words and phrases

Navigation