Tracy-Widom limit for the largest eigenvalue of a large class of complex sample covariance matrices. (English) Zbl 1117.60020
Summary: We consider the asymptotic fluctuation behavior of the largest eigenvalue of certain sample covariance matrices in the asymptotic regime where both dimensions of the corresponding data matrix go to infinity. More precisely, let \(X\) be an \(n\times p\) matrix, and let its rows be i.i.d. complex normal vectors with mean 0 and covariance \(\Sigma_p\). We show that for a large class of covariance matrices \(\Sigma_p\), the largest eigenvalue of \(X^*X\) is asymptotically distributed (after recentering and rescaling) as the Tracy-Widom distribution that appears in the study of the Gaussian unitary ensemble. We give explicit formulas for the centering and scaling sequences that are easy to implement and involve only the spectral distribution of the population covariance, \(n\) and \(p\).
The main theorem applies to a number of covariance models found in applications. For example, well-behaved Toeplitz matrices as well as covariance matrices whose spectral distribution is a sum of atoms (under some conditions on the mass of the atoms) are among the models the theorem can handle. Generalizations of the theorem to certain spiked versions of our models and a.s. results about the largest eigenvalue are given. We also discuss a simple corollary that does not require normality of the entries of the data matrix and some consequences for applications in multivariate statistics.
The main theorem applies to a number of covariance models found in applications. For example, well-behaved Toeplitz matrices as well as covariance matrices whose spectral distribution is a sum of atoms (under some conditions on the mass of the atoms) are among the models the theorem can handle. Generalizations of the theorem to certain spiked versions of our models and a.s. results about the largest eigenvalue are given. We also discuss a simple corollary that does not require normality of the entries of the data matrix and some consequences for applications in multivariate statistics.
Keywords:
random matrix theory; Wishart matrices; Tracy-Widom distributions; trace class operators; operator determinants; steepest descent analysis; Toeplitz matricesReferences:
| [1] | Anderson, T. W. (1963). Asymptotic theory for principal component analysis. Ann. Math. Statist. 34 122–148. · Zbl 0202.49504 · doi:10.1214/aoms/1177704248 |
| [2] | Anderson, T. W. (2003). An Introduction to Multivariate Statistical Analysis , 3rd ed. Wiley, Hoboken, NJ. · Zbl 1039.62044 |
| [3] | Bai, Z. D. (1999). Methodologies in spectral analysis of large-dimensional random matrices, a review. Statist. Sinica 9 611–677. · Zbl 0949.60077 |
| [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. · Zbl 0937.60017 · doi:10.1214/aop/1022855421 |
| [5] | Baik, J. (2006). Painleve formulas of the limiting distributions for nonnull complex sample covariance matrices. Duke Math. J. 133 205–235. · Zbl 1139.33006 · doi:10.1215/S0012-7094-06-13321-5 |
| [6] | 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 1643–1697. · Zbl 1086.15022 · doi:10.1214/009117905000000233 |
| [7] | Borodin, A. (1999). Biorthogonal ensembles. Nuclear Phys. B 536 704–732. · Zbl 0948.82018 · doi:10.1016/S0550-3213(98)00642-7 |
| [8] | Böttcher, A. and Silbermann, B. (1999). Introduction to Large Truncated Toeplitz Matrices . Springer, New York. · Zbl 0916.15012 |
| [9] | Desrosiers, P. and Forrester, P. J. (2006). Asymptotic correlations for Gaussian and Wishart matrices with external source. Int. Math. Res. Not. 2006 Article ID 27395. · Zbl 1104.62076 |
| [10] | Dieng, M. (2005). Distribution functions for edge eigenvalues in orthogonal and symplectic ensembles: Painlevé representations. Int. Math. Res. Not. 2005 2263–2287. · Zbl 1093.60009 · doi:10.1155/IMRN.2005.2263 |
| [11] | El Karoui, N. (2006). A rate of convergence result for the largest eigenvalue of complex white Wishart matrices. Ann. Probab. · Zbl 1108.62014 · doi:10.1214/009117906000000502 |
| [12] | El Karoui, N. (2003). On the largest eigenvalue of Wishart matrices with identity covariance when \(n,p\) and \(p/n\to\infty\). Available at |
| [13] | El Karoui, N. (2004). New results about random covariance matrices and statistical applications. Ph.D. dissertation, Stanford Univ. |
| [14] | Forrester, P. J. (1993). The spectrum edge of random matrix ensembles. Nuclear Phys. B 402 709–728. · Zbl 1043.82538 · doi:10.1016/0550-3213(93)90126-A |
| [15] | Forrester, P. J. (2006). Eigenvalue distributions for some correlated complex sample covariance matrices. Available at · Zbl 1126.15028 · doi:10.1088/1751-8113/40/36/009 |
| [16] | Gohberg, I., Goldberg, S. and Krupnik, N. (2000). Traces and Determinants of Linear Operators . Birkhäuser, Basel. · Zbl 0946.47013 |
| [17] | Gravner, J., Tracy, C. A. and Widom, H. (2001). Limit theorems for height fluctuations in a class of discrete space and time growth models. J. Statist. Phys. 102 1085–1132. · Zbl 0989.82030 · doi:10.1023/A:1004879725949 |
| [18] | Gray, R. M. (2006). Toeplitz and circulant matrices: A review. Foundations and Trends in Communications and Information Theory 2 155–239. Available at http://ee.stanford.edu/ gray/toeplitz.pdf. · Zbl 1115.15021 |
| [19] | Grenander, U. and Szegö, G. (1958). Toeplitz Forms and Their Applications . Univ. California Press, Berkeley. · Zbl 0080.09501 |
| [20] | Gross, K. and Richards, D. (1989). Total positivity, spherical series, and hypergeometric functions of matrix argument. J. Approx. Theory 59 224–246. · Zbl 0692.33010 · doi:10.1016/0021-9045(89)90153-6 |
| [21] | Guionnet, A. and Zeitouni, O. (2000). Concentration of the spectral measure for large matrices. Electron. Comm. Probab. 5 119–136. · Zbl 0969.15010 |
| [22] | Horn, R. and Johnson, C. (1990). Matrix Analysis . Cambridge Univ. Press. · Zbl 0704.15002 |
| [23] | James, A. T. (1964). Distributions of matrix variates and latent roots derived from normal samples. Ann. Math. Statist. 35 475–501. · Zbl 0121.36605 · doi:10.1214/aoms/1177703550 |
| [24] | Johansson, K. (2000). Shape fluctuations and random matrices. Comm. Math. Phys. 209 437–476. · Zbl 0969.15008 · doi:10.1007/s002200050027 |
| [25] | Johnstone, I. (2001). On the distribution of the largest eigenvalue in principal component analysis. Ann. Statist. 29 295–327. · Zbl 1016.62078 · doi:10.1214/aos/1009210544 |
| [26] | Ledoux, M. (2001). The Concentration of Measure Phenomenon . Amer. Math. Soc., Providence, RI. · Zbl 0995.60002 |
| [27] | Marčenko, V. A. and Pastur, L. A. (1967). Distribution of eigenvalues in certain sets of random matrices. Mat. Sb. (N.S.) 72 507–536. · Zbl 0152.16101 |
| [28] | Olver, F. W. J. (1974). Asymptotics and Special Functions . Academic Press, New York–London. · Zbl 0303.41035 |
| [29] | Paul, D. (2007). Asymptotics of sample eigenstructure for a large dimensional spiked covariance model. Statist. Sinica . · Zbl 1134.62029 |
| [30] | Reed, M. and Simon, B. (1972). Methods of Modern Mathematical Physics. I. Functional Analysis . Academic Press, New York. · Zbl 0459.46001 |
| [31] | Silverstein, J. W. and Choi, S.-I. (1995). Analysis of the limiting spectral distribution of large-dimensional random matrices. J. Multivariate Anal. 54 295–309. · Zbl 0872.60013 · doi:10.1006/jmva.1995.1058 |
| [32] | Simon, S., Moustakas, A. and Marinelli, L. (2005). Capacity and character expansions: Moment generating function and other exact results for mimo correlated channels. Available at · Zbl 1213.94068 · doi:10.1109/TIT.2006.885519 |
| [33] | Soshnikov, A. (2000). Determinantal random point fields. Russian Math. Surveys 55 923–975. · Zbl 0991.60038 · doi:10.1070/rm2000v055n05ABEH000321 |
| [34] | Soshnikov, A. (2002). A note on universality of the distribution of the largest eigenvalues in certain sample covariance matrices. J. Statist. Phys. 108 1033–1056. · Zbl 1018.62042 · doi:10.1023/A:1019739414239 |
| [35] | Tracy, C. and Widom, H. (1994). Level-spacing distribution and the Airy kernel. Comm. Math. Phys. 159 151–174. · Zbl 0789.35152 · doi:10.1007/BF02100489 |
| [36] | Tracy, C. and Widom, H. (1996). On orthogonal and symplectic matrix ensembles. Comm. Math. Phys. 177 727–754. · Zbl 0851.60101 · doi:10.1007/BF02099545 |
| [37] | Tracy, C. and Widom, H. (1998). Correlation functions, cluster functions and spacing distributions for random matrices. J. Statist. Phys. 92 809–835. · Zbl 0942.60099 · doi:10.1023/A:1023084324803 |
| [38] | Tulino, A. and Verdú, S. (2004). Random Matrix Theory and Wireless Communications . Foundations and Trends in Communications and Information Theory 1 . Now Publishers, Hanover, MA. · Zbl 1143.94303 · doi:10.1561/0100000001 |
| [39] | van der Vaart, A. W. (1998). Asymptotic Statistics . Cambridge Univ. Press. · Zbl 0910.62001 · doi:10.1017/CBO9780511802256 |
| [40] | Wachter, K. W. (1978). The strong limits of random matrix spectra for sample matrices of independent elements. Ann. Probab. 6 1–18. JSTOR: · Zbl 0374.60039 · doi:10.1214/aop/1176995607 |
| [41] | Widom, H. (1999). On the relation between orthogonal, symplectic and unitary matrix ensembles. J. Statist. Phys. 94 347–363. · Zbl 0935.60090 · doi:10.1023/A:1004516918143 |
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