Abstract
For sample covariance matrices with i.i.d. entries with sub-Gaussian tails, when both the number of samples and the number of variables become large and the ratio approaches one, it is a well-known result of Soshnikov that the limiting distribution of the largest eigenvalue is same that of Gaussian samples. In this paper, we extend this result to two cases. The first case is when the ratio approaches an arbitrary finite value. The second case is when the ratio becomes infinite or arbitrarily small.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bai Z.D. (1999). Methodologies in spectral analysis of large-dimensional random matrices, a review. Stat. Sin. 9(3): 611–677
Bai Z.D., Krishnaiah P.R. and Yin Y.Q. (1988). On the limit of the largest eigenvalue of the large-dimensional sample covariance matrix. Probab. Theor. Relat. Fields 78: 509–521
Baik J., Ben Arous G. and Péché S. (2005). Phase transition of the largest eigenvalue for non-null complex sample covariance matrices. Ann. Probab. 33(5): 1643–1697
Baik J. and Silverstein J. (2006). Eigenvalues of large sample covariance matrices of spiked population models. J. Mult. Anal. 97: 1382–1408
Bronk B.V. (1965). Exponential ensembles for random matrices. J. Math. Phys. 6: 228–237
Chen, W., Yan, S., Yang, L.: Identities from Weighted 2 Motzkin paths. http://www.billchen.org/publications/identit/identit.pdf
El Karoui, N.: On the largest eigenvalue of Wishart matrices with identity covariance when n, p and p/n tend to infinity. ArXiv math.ST/0309355 (2003)
El Karoui N. (2005). Recent results about the largest eigenvalue of random covariance matrices and statistical application. Acta Phys. Polon. B 36(9): 2681–2697
Féral, D., Péché, S.: The largest eigenvalue of some rank one deformation of large Wigner matrices. ArXiv math.PR/0605624. Comm. Math. Phys. (to appear) (2006)
Geman S. (1980). A limit theorem for the norm of random matrices. Ann. Prob. 8: 252–261
Hotelling H. (1933). Analysis of a complex of statistical variables into principal components. J. Educ. Psychol. 24: 417–441
Hoyle, D., Rattray, M.: Limiting form of the sample covariance eigenspectrum in PCA and kernel PCA. Advances in Neural Information Processing Systems NIPS 16 (2003)
James A. (1960). Distributions of matrix variates and latent roots derived from normal samples. Ann. Math. Stat. 31: 151–158
Johansson K. (2000). Shape fluctuations and random matrices. Comm. Math. Phys. 209: 437–476
Johnstone I.M. (2001). On the distribution of the largest principal component. Ann. Stat. 29: 295–327
Johnstone, I.M.: High dimensional statistical inference and random matrices. arXiv:math/0611589. In: Proceedings of the International Congress of Mathematicians, 2006 (to appear) (2007)
Jonsson D. (1982). Some limit theorems for the eigenvalues of sample covariance matrices. J. Mult. Anal. 12: 1–38
Laloux L., Cizeau P., Potters M. and Bouchaud J. (2000). Random matrix theory and financial correlations. Int. J. Theor. Appl. Financ. 3(3): 391–397
Malevergne Y. and Sornette D. (2004). Collective origin of the coexisence of apparent RMT noise and factors in large sample correlation matrices. Physica A 331(3–4): 660–668
Marcenko V.A. and Pastur L.A. (1967). Distribution of eigenvalues forsome sets of random matrices. Math. USSR-Sbornik 1: 457–486
Patterson, N., Price, A.L., Reich, D.: Population structure and eigenanalysis. PLoS Genet 2(12), e190 (2006) DOI: 10.1371/journal.pgen.0020190
Péché, S., Soshnikov, A.: Wigner random matrices with non-symmetrically distributed entries. arXiv:math/0702035. Applications of random matrices, determinants and pfaffians to problems in statistical mechanics. J. Stat. Phys (to appear) (2007)
Oravecz F. and Petz D. (1997). On the eigenvalue distribution of some symmetric random matrices. Acta Sci. Math. 63: 383–395
Plerous V., Gopikrishnan P., Rosenow B., Amaral L., Guhr T. and Stanley H. (2002). Random matrix approach to cross correlations in financial data. Phys. Rev. E 65(6): 66–126
Ruzmaikina A. (2006). Universality of the edge distribution of eigenvalues of Wigner random matrices with polynomially decaying distributions of entries. Commun. Math. Phys. 261: 277–296
Sear R. and Cuesta J. (2003). Instabilities in complex mixtures with a large number of components. Phys. Rev. Lett. 91(24): 245–701
Silverstein J. (1989). On the weak limit of the largest eigenvalue of a large-dimensional sample covariance matrix. J. Multivariate Anal. 30(2): 307–311
Sinai Y. and Soshnikov A. (1998). Central limit theorem for traces of large random symmetric matrices with independent matrix elements. Bol. Soc. Brasil. Mat. (N.S.) 29(1): 1–24
Sinai Y. and Soshnikov A. (1998). A refinement of Wigner’s semicircle law in a neighborhood of the spectrum edge for random symmetric matrices. Funct. Anal. Appl. 32: 114–131
Soshnikov A. (1999). Universality at the edge of the spectrum in Wigner random matrices. Commun. Math. Phys. 207: 697–733
Soshnikov A. (2002). A note on universality of the distribution of the largest eigenvalues in certain sample covariance matrices. J. Stat. Phys. 108: 1033–1056
Stanley R. (1999). Enumerative combinatorics, vol. 2. Cambridge Studies in Advanced Mathematics, 62. Cambridge University Press, Cambridge
Sulanke, R.: Counting lattice paths by Narayana polynomials. Electron. J. Combin. 7, 9p (electronic) (2000)
Sulanke R. (1998). Catalan path statistics having the Narayana distribution. In: Proceedings of the 7th Conference on Formal Power Series and Algebraic Combinatorics (Noisy-le-Grand, 1995). Discrete Math. 180(1–3): 369–389
Telatar E. (1999). Capacity of multi-antenna Gaussian channels. Eur. Trans. Telecomm. 10(6): 585–595
Tracy C. and Widom H. (1994). Level-spacing distribution and Airy kernel. Commun. Math. Phys. 159: 151–174
Tracy C. and Widom H. (1996). On orthogonal and symplectic matrix ensembles. Commun. Math. Phys. 177: 727–754
Wachter K. (1978). The strong limits of random matrix spectra for sample covariance matrices of independent elements. Ann. Probab. 6: 1–18
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Péché, S. Universality results for the largest eigenvalues of some sample covariance matrix ensembles. Probab. Theory Relat. Fields 143, 481–516 (2009). https://doi.org/10.1007/s00440-007-0133-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-007-0133-7