Another look at the moment method for large dimensional random matrices

Electronic Journal of Probability

Volumn 13, Issue , 2008, Pages 588-628

  Bose, Arup ^a   Sen, Arnab ^b  

^a INDIAN STATISTICAL INSTITUTE   (India)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Almost sure convergence; Bounded Lipschitz metric; Catalan numbers; Circulant matrix; Convergence in distribution; Eigenvalues; Hankel matrix; Large dimensional random matrix; Limiting spectral distribution; Linear process; M dependent sequence; Mar enko Pastur law; Moment method; Palindromic matrix; Random probability; Reverse circulant matrix; Sample variance covariance matrix; Semicircular law; Spectral distribution and density; Symmetric circulant matrix; Toeplitz matrix; Volume method; Wigner matrix

Indexed keywords

EID: 42449107488     PISSN: None     EISSN: 10836489     Source Type: Journal    
DOI: 10.1214/EJP.v13-501     Document Type: Article

References (23)

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).
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.
5. Bhatia, R. (1997). Matrix Analysis. Springer, New York.
6. Bose, A. And Mitra, J. (2002). Limiting spectral distribution of a special circulant. Stat. Probab. Letters, 60, 1, 111-120.
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.
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.
10. Feller, W. (1966). An Introduction to Probability Theory and Its Applications. Vol. 2, Wiley, New York.
11. Grenander, U. (1963). Probabilities on Algebraic Structures. John Wiley & Sons, Inc., New York-London; Almqvist & Wiksell, Stockholm-Göteborg-Uppsala.
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.
13. Jonsson, D. (1982). Some limit theorems for the eigenvalues of a sample covariance matrix. J. Multivariate Anal. 12, no. 1, 1-38.
14. Marcenko, 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.
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.
16. Pastur, L. (1972). The spectrum of random matrices. (Russian) Teoret. Mat. Fiz. 10 no. 1, 102-112.
17. Phillips P., Solo V. (1992). Asymptotics for linear processes Ann. Stat. (2), 20, 971-1001.
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.
20. Wigner, E. P. (1955). Characteristic vectors of bordered matrices with infinite dimensions. Ann. Of Math., (2), 62, 548-564.
21. Wigner, E. P. (1958). On the distribution of the roots of certain symmetric matrices. Ann. Of Math., (2), 67, 325-327.
22. Yin, Y. Q. (1986). Limiting spectral distribution for a class of random matrices. J. Multivariate Anal., 20, no. 1, 50-68.
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.