Concentration of measure and spectra of random matrices: Applications to correlation matrices, elliptical distributions and beyond

Annals of Applied Probability

Volumn 19, Issue 6, 2009, Pages 2362-2405

  Karoui, Noureddine El ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Concentration of measure; Correlation matrices; Covariance matrices; Eigenvalues of covariance matrices; Elliptical distributions; High dimensional inference; Multivariate statistical analysis; Random matrix theory

Indexed keywords

EID: 73949116713     PISSN: 10505164     EISSN: 10505164     Source Type: Journal    
DOI: 10.1214/08-AAP548     Document Type: Article

References (51)

1. ANDERSON, G. W. and ZEITOUNI, O. (2006). A CLT for a band matrix model. Probab. Theory Related Fields 134 283-338. MR2222385
2. ANDERSON, T. W. (2003). An Introduction to Multivariate Statistical Analysis, 3rd ed. Wiley Series in Probability and Statistics. Wiley, Hoboken, NJ. MR1990662
3. BAI, Z. D. (1999). Methodologies in spectral analysis of large-dimensional random matrices, a review. Statist. Sinica 9 611-677. MR1711663
4. BAI, Z. D. and SILVERSTEIN, J. W. (1998). No eigenvalues outside the support of the limiting spectral distribution of large-dimensional sample covariance matrices. Ann. Probab. 26 316-345. MR1617051
5. BAI, Z. D. and SILVERSTEIN, J. W. (2004). CLT for linear spectral statistics of largedimensional sample covariance matrices. Ann. Probab. 32 553-605. MR2040792
6. BAIK, J., BEN AROUS, G. and PÉCHÉ, S. (2005). Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices. Ann. Probab. 33 1643-1697. MR2165575
7. BHATIA, R. (1997). Matrix Analysis. Graduate Texts in Mathematics 169. Springer, New York. MR1477662
8. BICKEL, P. J. and LEVINA, E. (2008). Regularized estimation of large covariance matrices. Ann. Statist. 36 199-227. MR2387969
9. BOUTET DE MONVEL, A., KHORUNZHY, A. and VASILCHUK, V. (1996). Limiting eigenvalue distribution of random matrices with correlated entries. Markov Process. Related Fields 2 607-636. MR1431189
10. BOYD, S. and VANDENBERGHE, L. (2004). Convex Optimization. Cambridge Univ. Press, Cambridge. MR2061575
11. BURDA, Z., GÖRLICH, A., JAROSZ, A. and JURKIEWICZ, J. (2004). Signal and noise in correlation matrix. Phys. A 343 295-310. MR2094415
12. BURDA, Z., JURKIEWICZ, J. and WACłAW, B. (2005). Spectral moments of correlated Wishart matrices. Phys. Rev. E 71.
13. CAMPBELL, J., LO, A. and MACKINLAY, C. (1996). The Econometrics of Financial Markets. Princeton Univ. Press, Princeton, NJ.
14. EL KAROUI, N. (2003). On the largest eigenvalue of Wishart matrices with identity covariance when n, p and p/n ρ ∞. Available at arXiv:math.ST/0309355.
15. El Karoui, N. (2007a). The spectrum of kernel random matrices. Technical Report 748, Dept. Statistics, UC Berkeley. Ann. Statist. To appear.
16. EL KAROUI, N. (2007). Tracy-Widom limit for the largest eigenvalue of a large class of complex sample covariance matrices. Ann. Probab. 35 663-714. MR2308592
17. EL KAROUI, N. (2009). Spectrum estimation for large dimensional covariance matrices using random matrix theory. Ann. Statist. To appear. Available at arXiv:math.ST/0609418.
18. FANG, K. T., KOTZ, S. and NG, K. W. (1990). Symmetric Multivariate and Related Distributions. Monographs on Statistics and Applied Probability 36. Chapman & Hall, London. MR1071174
19. FORRESTER, P. J. (1993). The spectrum edge of random matrix ensembles. Nuclear Phys. B 402 709-728. MR1236195
20. FRAHM, G. and JAEKEL, U. (2005). Random matrix theory and robust covariance matrix estimation for financial data. Available at arXiv:physics/0503007.
21. GEMAN, S. (1980). A limit theorem for the norm of random matrices. Ann. Probab. 8 252-261. MR566592
22. GERONIMO, J. S. and HILL, T. P. (2003). Necessary and sufficient condition that the limit of Stieltjes transforms is a Stieltjes transform. J. Approx. Theory 121 54-60. MR1962995
23. GIRKO, V. L. (1990). Theory of Random Determinants. Mathematics and Its Applications (Soviet Series) 45. Kluwer Academic, Dordrecht. MR1080966
24. GRAY, R. M. (2002). Toeplitz and circulant matrices: A review. Available at http://ee.stanford.edu/~gray/toeplitz.pdf.
25. GRENANDER, U. and SZEGÖ, G. (1958). Toeplitz Forms and Their Applications. Univ. California Press, Berkeley. MR0094840
26. GUIONNET, A. and ZEITOUNI, O. (2000). Concentration of the spectral measure for large matrices. Electron. Comm. Probab. 5 119-136 (electronic). MR1781846
27. JIANG, T. (2004). The limiting distributions of eigenvalues of sample correlation matrices. Sankhyā 66 35-48. MR2082906
28. JOHANSSON, K. (2000). Shape fluctuations and random matrices. Comm. Math. Phys. 209 437-476. MR1737991
29. JOHNSTONE, I. M. (2001). On the distribution of the largest eigenvalue in principal components analysis. Ann. Statist. 29 295-327. MR1863961
30. JONSSON, D. (1982). Some limit theorems for the eigenvalues of a sample covariance matrix. J. Multivariate Anal. 12 1-38. MR650926
31. LALOUX, L., CIZEAU, P., BOUCHAUD, J.-P. and POTTERS, M. (1999). Random matrix theory and financial correlations. Int. J. Theor. Appl. Finance 3 391-397.
32. LEDOIT, O. and WOLF, M. (2004). A well-conditioned estimator for large-dimensional co-variance matrices. J. Multivariate Anal. 88 365-411. MR2026339
33. LEDOUX, M. (2001). The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs 89. Amer. Math. Soc., Providence, RI. MR1849347
34. LI, L., TULINO, A. M. and VERDÚ, S. (2004). Design of reduced-rank MMSE multiuser detectors using random matrix methods. IEEE Trans. Inform. Theory 50 986-1008. MR2094863
35. MARČENKO, V. A. and PASTUR, L. A. (1967). Distribution of eigenvalues in certain sets of random matrices. Mat. Sb. (N.S.) 72 507-536. MR0208649
36. MARDIA, K. V., KENT, J. T. and BIBBY, J. M. (1979). Multivariate Analysis. Academic Press, London. MR560319
37. MCNEIL, A. J., FREY, R. and EMBRECHTS, P. (2005). Quantitative Risk Management: Concepts, Techniques and Tools. Princeton Univ. Press, Princeton, NJ. MR2175089
38. NELSEN, R. B. (2006). An Introduction to Copulas, 2nd ed. Springer, New York. MR2197664
39. PAJOR, A. and PASTUR, L. (2007). On the limiting empirical measure of the sum of rank one matrices with log-concave distribution. Available at http://www.arxiv.org/abs/0710.1346.
40. PAUL, D. (2007). Asymptotics of sample eigenstructure for a large dimensional spiked covari-ance model. Statist. Sinica 17 1617-1642. MR2399865
41. PAUL, D. and SILVERSTEIN, J. (2007). No eigenvalues outside the support of the limiting empirical spectral distribution of a separable covariance matrix. Available at http://www4.ncsu.edu/~jack/pub.html.
42. RAO, N. R., MINGO, J., SPEICHER, R. and EDELMAN, A. (2007). Statistical eigen-inference from large Wishart matrices. Available at arXiv:math/0701314.
43. n ball. In Geometric Aspects of Functional Analysis. Lecture Notes in Math. 1745 245-256. Springer, Berlin. MR1796723
44. SILVERSTEIN, J. W. (1995). Strong convergence of the empirical distribution of eigenvalues of large-dimensional random matrices. J. Multivariate Anal. 55 331-339. MR1370408
45. SILVERSTEIN, J. W. and BAI, Z. D. (1995). On the empirical distribution of eigenvalues of a class of large-dimensional random matrices. J. Multivariate Anal. 54 175-192. MR1345534
46. TALAGRAND, M. (1995). Concentration of measure and isoperimetric inequalities in product spaces. Inst. Hautes Études Sci. Publ. Math. 81 73-205. MR1361756
47. TULINO, A. and VERDÚ, S. (2004). Random Matrix Theory and Wireless Communications. Foundations and Trends in Communications and Information Theory 1. Now Publishers, Boston.
48. VAN DER VAART, A. W. (1998). Asymptotic Statistics. Cambridge Series in Statistical and Probabilistic Mathematics 3. Cambridge Univ. Press, Cambridge. MR1652247
49. VOICULESCU, D. (2000). Lectures on free probability theory. In Lectures on Probability Theory and Statistics (Saint-Flour, 1998). Lecture Notes in Math. 1738 279-349. Springer, Berlin. MR1775641
50. YIN, Y. Q., BAI, Z. D. and KRISHNAIAH, P. R. (1988). On the limit of the largest eigenvalue of the large-dimensional sample covariance matrix. Probab. Theory Related Fields 78 509-521. MR950344
51. ZHANG, L. (2006). Spectral analysis of large dimensional random matrices. Ph.D. thesis, National Univ. Singapore.