The largest eigenvalues of sample covariance matrices for a spiked population: Diagonal case

Journal of Mathematical Physics

Volumn 50, Issue 7, 2009, Pages

  Féral, Delphine ^a   Péché, Sandrine ^b  

^a UNIVERSITÉ BORDEAUX 1   (France)

^b INSTITUT FOURIER   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 68749084936     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.3155785     Document Type: Article

References (32)

1. Amaral, L., Gopikrishnan, P., Guhr, T., Plerous, V., Rosenow, B., and Stanley, H., " Random matrix approach to cross correlations in financial data.," Phys. Rev. E 65, 066126 (2002). 10.1103/PhysRevE.65.066126
2. Bai, Z. D., " Methodologies in spectral analysis of large-dimensional random matrices, a review.," Stat. Sin. 9, 611 (1999).
3. Bai, Z. and Yao, J., " Central limit theorems for eigenvalues in a spiked population model.," Ann. Inst. Henri Poincare, Sect. A 44, 447 (2008).
4. Baik, J., " Painlev́ formulas of the limiting distributions for nonnull complex sample covariance matrices.," Duke Math. J. 133, 205 (2006). 10.1215/S0012-7094-06-13321-5 (Pubitemid 43988241)
5. Baik, J., Ben Arous, G., and Ṕch́, S., " Phase transition of the largest eigenvalue for non-null complex sample covariance matrices.," Ann. Probab. 33, 1643 (2005). 10.1214/009117905000000233 (Pubitemid 41560766)
6. Baik, J. and Silverstein, J., " Eigenvalues of large sample covariance matrices of spiked population models.," J. Multivariate Anal. 97, 1382 (2006). 10.1016/j.jmva.2005.08.003
7. Biroli, G., Bouchaud, J. P., and Potters, M., " On the top eigenvalue of heavy-tailed random matrices.," Europhys. Lett. 78, 10001 (2007). 10.1209/0295-5075/78/10001
8. Bouchaud, J., Cizeau, P., Laloux, L., and Potters, M., " Random matrix theory and financial correlations.," IEE Proc., Radar Sonar Navig. 3, 391 (2000).
9. Chen, W., Yan, S., and Yang, L., " Identities from Weighted 2 Motzkin paths.," available at www.billchen.org/publications/identit/ identit.pdf.
10. El Karoui, N., " Recent results about the largest eigenvalue of random covariance matrices and statistical application.," Acta Phys. Pol. B 36, 2681 (2005).
11. F́ral, D. and Ṕch́, S., " The largest eigenvalue of rank one deformation of large Wigner matrices.," Commun. Math. Phys. 272, 185 (2007). 10.1007/s00220-007-0209-3
12. Hoyle, D. and Rattray, M., " Limiting form of the sample covariance eigenspectrum in PCA and kernel PCA.," Adv. Neural Inf. Process. Syst. 16, 1181 (2004).
13. Johansson, K., " Shape fluctuations and random matrices.," Commun. Math. Phys. 209, 437 (2000). 10.1007/s002200050027
14. Johnstone, I. M., " On the distribution of the largest Principal Component.," Ann. Stat. 29, 295 (2001). 10.1214/aos/1009210544
15. Malevergne, Y. and Sornette, D., " Collective origin of the coexistence of apparent RMT noise and factors in large sample correlation matrices.," Physica A 331, 660 (2004). 10.1016/j.physa.2003.09.004
16. Marchenko, V. A. and Pastur, L. A., " Distribution of eigenvalues for some sets of random matrices.," Math. USSR. Sb. 1, 457 (1967). 10.1070/SM1967v001n04ABEH001994
17. Onatski, A., " The Tracy-Widom limit for the largest eigenvalues of singular complex Wishart matrices.," Ann. Appl. Probab. 18, 470 (2008). 10.1214/07-AAP454
18. Paul, D., " Asymptotics of the leading sample eigenvalues for a spiked covariance model.," Stat. Sin. 17, 1617 (2007).
19. Patterson, N., Price, A. L., and Reich, D., " Population structure and eigenanalysis.," PloS Genet 2, e190 (2006). 10.1371/journal.pgen. 0020190 (Pubitemid 46045188)
20. Ṕch́, S., " Universality results for largest eigenvalues of some sample covariance matrix ensembles.," Probab. Theory Relat. Fields 143, 481 (2008).
21. Ṕch́, S., " The edge of the spectrum of random matrices.," Habilitation thesis, Universit́ Joseph Fourier Grenoble, 2008.
22. Ṕch́, S. and Soshnikov, A., " Wigner random matrices with non symmetrically distributed entries.," J. Stat. Phys. 129, 857 (2007). 10.1007/s10955-007-9340-y
23. Sear, R. and Cuesta, J., " Instabilities in complex mixtures with a large number of components.," Phys. Rev. Lett. 91, 245701 (2003). 10.1103/PhysRevLett.91.245701
24. Sinai, Y. and Soshnikov, A., " Central limit theorem for traces of large random symmetric matrices with independent matrix elements.," Bol. Soc. Bras. Mat. 29, 1 (1998). 10.1007/BF01245866
25. Sinai, Y. and Soshnikov, A., " A refinement of Wigner's semicircle law in a neighborhood of the spectrum edge for random symmetric matrices.," Funct. Anal. Appl. 32, 114 (1998). 10.1007/BF02482597
26. Soshnikov, A., " Universality at the edge of the spectrum in Wigner random matrices.," Commun. Math. Phys. 207, 697 (1999). 10.1007/s002200050743
27. Soshnikov, A., " A note on universality of the distribution of the largest eigenvalues in certain sample covariance matrices.," J. Stat. Phys. 108, 1033 (2002). 10.1023/A:1019739414239 (Pubitemid 37133779)
28. Telatar, E., " Capacity of multi-antenna Gaussian channels.," Europ. Trans. Telecom. 10, 585 (1999). 10.1002/ett.4460100604
29. Tracy, C. and Widom, H., " Level-spacing distribution and Airy kernel.," Commun. Math. Phys. 159, 151 (1994). 10.1007/BF02100489
30. Tracy, C. and Widom, H., " On orthogonal and symplectic matrix ensembles.," Commun. Math. Phys. 177, 727 (1996). 10.1007/BF02099545
31. Yin, Y. Q., Bai, Z. D., and Krishnaiah, P. R., " On the limit of the largest eigenvalue of the large-dimensional sample covariance matrix.," Probab. Theory Relat. Fields 78, 509 (1988). 10.1007/BF00353874
32. Wigner, E., " Characteristic vectors of bordered matrices with infinite dimensions.," Ann. Math. 62, 548 (1955). 10.2307/1970079