Another look at the moment method for large dimensional random matrices

Arup Bose (Indian Statistical Institute)
Arnab Sen (University of California, Berkeley)

Abstract


The methods to establish the limiting spectral distribution (LSD) of large dimensional random matrices includes the  well known moment method which invokes the trace formula. Its success has been demonstrated in several types of matrices such as the Wigner matrix and the sample variance covariance matrix. In a recent article Bryc, Dembo and Jiang (2006) establish the LSD for the random Toeplitz and Hankel matrices using the moment method.  They perform the necessary counting of terms in the trace by splitting the relevant sets into equivalent classes and relating the limits of the counts to certain volume calculations.

We build on their work and present a unified approach. This helps provide  relatively short and easy proofs for the LSD of common matrices while at the same time providing insight into the nature of different LSD and their interrelations. By extending these methods we are also able to deal with matrices with appropriate dependent entries.

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

Pages: 588-628

Publication Date: April 12, 2008

DOI: 10.1214/EJP.v13-501

References

  1. Arnold, L.  (1967). On the asymptotic distribution of the eigenvalues of random matrices. J. Math. Anal. Appl., 20, 262-268.
  2. Bai, Z. D. (1999) Methodologies in spectral analysis of large dimensional random matrices, a review. Statistica Sinica, 9, 611-677 (with discussions). MR1711663
  3. Bai, Z. D. and Silverstein, J. (2006). Spectral analysis of large dimensional random matrices. Science Press, Beijing.
  4. Bai, Z. D.  and Yin, Y. Q.  (1988). Convergence to the semicircle law. Ann. Probab.,  16, no. 2, 863-875. MR0929083
  5. Bhatia, R. (1997). Matrix Analysis. Springer, New York. MR1477662
  6. Bose, A. and Mitra, J. (2002). Limiting spectral distribution of a special circulant. Stat. Probab. Letters, 60, 1, 111-120. MR1945684
  7. Bryc, W.,  Dembo, A. and Jiang, T. (2006). Spectral measure of large random Hankel, Markov and Toeplitz matrices.  Ann. Probab., 34, no. 1, 1-38. MR2206341
  8. Chatterjee, S. (2005) A simple invariance theorem. Available at http://arxiv.org/abs/math.PR/0508213.
  9. Chow, Y. S. and Teicher, H. (1997). Probability theory: Independence, interchangeability, martingales. Third edition, Springer-Verlag, New York. MR1476912
  10. Feller, W. (1966). An Introduction to Probability Theory and Its Application. Vol. 2, Wiley, New York. MR0210154
  11. Grenander, U. (1963). Probabilities on algebraic structures. John Wiley & Sons, Inc., New York-London; Almqvist & Wiksell, Stockholm-Göteborg-Uppsala. MR0206994
  12. Hammond, C. and Miller, S. J. (2005) Distribution of eigenvalues for the ensemble of real symmetric Toeplitz matrices. J. Theoret. Probab. 18, no. 3, 537-566. MR2167641
  13. Jonsson, D. (1982). Some limit theorems for the eigenvalues of a sample covariance matrix. J. Multivariate Anal. 12, no. 1, 1-38. MR0650926
  14. Marčenko, V. A. and  Pastur, L. A. (1967). Distribution of eigenvalues in certain sets of random matrices,  (Russian) Mat. Sb. (N.S.), 72 (114), 507-536. MR0208649
  15. Massey, A., Miller, S. J. and  and Sinsheimer, J. (2007). Distribution of eigenvalues of real symmetric palindromic Toeplitz matrices and circulant matrices. J. Theoret. Probab. 20, 3, 637-662. MR2337145
  16. Pastur, L. (1972). The spectrum of random matrices. (Russian) Teoret. Mat. Fiz. 10 no. 1, 102–112. MR0475502
  17. Phillips P., Solo V. (1992) Asymptotics for linear processes, Ann. Stat. (2), 20, 971-1001. MR1165602
  18. Sen, A. (2006). Large dimensional random matrices. M. Stat. Project report., May 2006.  Indian Statistical Institute, Kolkata.
  19. Wachter, K.W. (1978). The strong limits of random matrix spectra for sample matrices of independent elements. Ann. Probab. 6, 1-18. MR0467894
  20. Wigner, E. P. (1955). Characteristic vectors of bordered matrices with infinite dimensions. Ann. of Math.,  (2),  62, 548-564. MR0077805
  21. Wigner, E. P. (1958). On the distribution of the roots of certain symmetric matrices. Ann. of Math., (2), 67, 325-327. MR0095527
  22. Yin, Y. Q.  (1986). Limiting spectral distribution for a class of random matrices. J. Multivariate Anal.,  20, no. 1, 50-68. MR0862241
  23. Yin, Y. Q.  and Krishnaiah, P. R.  (1985). Limit theorem for the eigenvalues of the sample covariance matrix when the underlying distribution is isotropic. Theory Probab. Appl., 30, no. 4,  810-816.  MR0816299
Bookmark and Share


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