## The Annals of Applied Probability

### Gaussian fluctuations for linear spectral statistics of large random covariance matrices

#### Abstract

Consider a $N\times n$ matrix $\Sigma_{n}=\frac{1}{\sqrt{n}}R_{n}^{1/2}X_{n}$, where $R_{n}$ is a nonnegative definite Hermitian matrix and $X_{n}$ is a random matrix with i.i.d. real or complex standardized entries. The fluctuations of the linear statistics of the eigenvalues

$\operatorname{Trace}f(\Sigma_{n}\Sigma_{n}^{*})=\sum_{i=1}^{N}f(\lambda_{i}),\qquad (\lambda_{i})\mbox{ eigenvalues of }\Sigma_{n}\Sigma_{n}^{*},$ are shown to be Gaussian, in the regime where both dimensions of matrix $\Sigma_{n}$ go to infinity at the same pace and in the case where $f$ is of class $C^{3}$, that is, has three continuous derivatives. The main improvements with respect to Bai and Silverstein’s CLT [Ann. Probab. 32 (2004) 553–605] are twofold: First, we consider general entries with finite fourth moment, but whose fourth cumulant is nonnull, that is, whose fourth moment may differ from the moment of a (real or complex) Gaussian random variable. As a consequence, extra terms proportional to

$|\mathcal{V}|^{2}=|\mathbb{E}(X_{11}^{n})^{2}|^{2}\quad\mbox{and}\quad\kappa=\mathbb{E}\vert X_{11}^{n}\vert^{4}-\vert\mathcal{V} \vert^{2}-2$ appear in the limiting variance and in the limiting bias, which not only depend on the spectrum of matrix $R_{n}$ but also on its eigenvectors. Second, we relax the analyticity assumption over $f$ by representing the linear statistics with the help of Helffer–Sjöstrand’s formula.

The CLT is expressed in terms of vanishing Lévy–Prohorov distance between the linear statistics’ distribution and a Gaussian probability distribution, the mean and the variance of which depend upon $N$ and $n$ and may not converge.

#### Article information

Source
Ann. Appl. Probab., Volume 26, Number 3 (2016), 1837-1887.

Dates
Revised: July 2015
First available in Project Euclid: 14 June 2016

https://projecteuclid.org/euclid.aoap/1465905021

Digital Object Identifier
doi:10.1214/15-AAP1135

Mathematical Reviews number (MathSciNet)
MR3513608

Zentralblatt MATH identifier
06618844

Subjects
Primary: 15A52
Secondary: 15A18: Eigenvalues, singular values, and eigenvectors 60F15: Strong theorems

#### Citation

Najim, Jamal; Yao, Jianfeng. Gaussian fluctuations for linear spectral statistics of large random covariance matrices. Ann. Appl. Probab. 26 (2016), no. 3, 1837--1887. doi:10.1214/15-AAP1135. https://projecteuclid.org/euclid.aoap/1465905021

#### References

• [1] Albeverio, S., Pastur, L. and Shcherbina, M. (2001). On the $1/n$ expansion for some unitary invariant ensembles of random matrices. Comm. Math. Phys. 224 271–305.
• [2] Anderson, G. W. and Zeitouni, O. (2006). A CLT for a band matrix model. Probab. Theory Related Fields 134 283–338.
• [3] Arharov, L. V. (1971). Limit theorems for the characteristic roots of a sample covariance matrix. Dokl. Akad. Nauk SSSR 199 994–997.
• [4] Bai, Z., Chen, Y. and Liang, Y.-C., eds. (2009). Random Matrix Theory and Its Applications: Multivariate Statistics and Wireless Communications. Lecture Notes Series. Institute for Mathematical Sciences. National University of Singapore 18. World Scientific, Hackensack, NJ.
• [5] Bai, Z. and Silverstein, J. W. (2010). Spectral Analysis of Large Dimensional Random Matrices, 2nd ed. Springer, New York.
• [6] Bai, Z., Wang, X. and Zhou, W. (2010). Functional CLT for sample covariance matrices. Bernoulli 16 1086–1113.
• [7] Bai, Z. D. and Silverstein, J. W. (2004). CLT for linear spectral statistics of large-dimensional sample covariance matrices. Ann. Probab. 32 553–605.
• [8] Bao, Z., Pan, G. and Zhou, W. (2013). Central limit theorem for partial linear eigenvalue statistics of Wigner matrices. J. Stat. Phys. 150 88–129.
• [9] Bao, Z., Pan, G. and Zhou, W. (2013). On the MIMO channel capacity for the general channels. Preprint.
• [10] Benaych-Georges, F., Guionnet, A. and Male, C. (2014). Central limit theorems for linear statistics of heavy tailed random matrices. Comm. Math. Phys. 329 641–686.
• [11] Billingsley, P. (1995). Probability and Measure, 3rd ed. Wiley, New York.
• [12] Bordenave, C. (2013). Personal communication.
• [13] Bordenave, C. (2013). A short course on random matrices (preliminary draft). Available at http://www.math.univ-toulouse.fr/~bordenave/coursRMT.pdf.
• [14] Cabanal-Duvillard, T. (2001). Fluctuations de la loi empirique de grandes matrices aléatoires. Ann. Inst. Henri Poincaré Probab. Stat. 37 373–402.
• [15] Capitaine, M. and Donati-Martin, C. (2007). Strong asymptotic freeness for Wigner and Wishart matrices. Indiana Univ. Math. J. 56 767–803.
• [16] Chatterjee, S. (2009). Fluctuations of eigenvalues and second order Poincaré inequalities. Probab. Theory Related Fields 143 1–40.
• [17] Couillet, R. and Debbah, M. (2011). Random Matrix Methods for Wireless Communications. Cambridge Univ. Press, Cambridge.
• [18] Dozier, R. B. and Silverstein, J. W. (2007). On the empirical distribution of eigenvalues of large dimensional information-plus-noise-type matrices. J. Multivariate Anal. 98 678–694.
• [19] Dudley, R. M. (2002). Real Analysis and Probability. Cambridge Studies in Advanced Mathematics 74. Cambridge Univ. Press, Cambridge.
• [20] Dyn’kin, E. M. (1972). An operator calculus based on the Cauchy–Green formula. Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 30 33–39.
• [21] Girko, V. L. (2001). Theory of Stochastic Canonical Equations. Vol. I. Mathematics and Its Applications 535. Kluwer Academic, Dordrecht.
• [22] Girko, V. L. (2001). Theory of Stochastic Canonical Equations. Vol. II. Mathematics and Its Applications 535. Kluwer Academic, Dordrecht.
• [23] Guédon, O., Lytova, A., Pajor, A. and Pastur, L. (2014). The central limit theorem for linear eigenvalue statistics of the sum of independent random matrices of rank one. In Spectral Theory and Differential Equations. Amer. Math. Soc. Transl. Ser. 2 233 145–164. Amer. Math. Soc., Providence, RI.
• [24] Guionnet, A. (2002). Large deviations upper bounds and central limit theorems for non-commutative functionals of Gaussian large random matrices. Ann. Inst. Henri Poincaré Probab. Stat. 38 341–384.
• [25] Haagerup, U. and Thorbjørnsen, S. (2005). A new application of random matrices: $\mathrm{Ext}(C_{\mathrm{red}}(F_{2}))$ is not a group. Ann. of Math. (2) 162 711–775.
• [26] Haagerup, U. and Thorbjørnsen, S. (2012). Asymptotic expansions for the Gaussian unitary ensemble. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 15 1250003, 41.
• [27] Hachem, W., Kharouf, M., Najim, J. and Silverstein, J. W. (2012). A CLT for information-theoretic statistics of non-centered Gram random matrices. Random Matrices Theory Appl. 1 1150010, 50.
• [28] Hachem, W., Loubaton, P. and Najim, J. (2007). Deterministic equivalents for certain functionals of large random matrices. Ann. Appl. Probab. 17 875–930.
• [29] Hachem, W., Loubaton, P. and Najim, J. (2008). A CLT for information-theoretic statistics of gram random matrices with a given variance profile. Ann. Appl. Probab. 18 2071–2130.
• [30] Hachem, W., Loubaton, P., Najim, J. and Vallet, P. (2013). On bilinear forms based on the resolvent of large random matrices. Ann. Inst. Henri Poincaré Probab. Stat. 49 36–63.
• [31] Helffer, B. (2013). Spectral Theory and Its Applications. Cambridge Studies in Advanced Mathematics 139. Cambridge Univ. Press, Cambridge.
• [32] Helffer, B. and Sjöstrand, J. (1989). Équation de Schrödinger avec champ magnétique et équation de Harper. In Schrödinger Operators (Sønderborg, 1988). Lecture Notes in Physics 345 118–197. Springer, Berlin.
• [33] Johansson, K. (1998). On fluctuations of eigenvalues of random Hermitian matrices. Duke Math. J. 91 151–204.
• [34] Johnstone, I. M. (2001). On the distribution of the largest eigenvalue in principal components analysis. Ann. Statist. 29 295–327.
• [35] Jonsson, D. (1982). Some limit theorems for the eigenvalues of a sample covariance matrix. J. Multivariate Anal. 12 1–38.
• [36] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. Springer, New York.
• [37] Kammoun, A., Kharouf, M., Hachem, W. and Najim, J. (2009). A central limit theorem for the SINR at the LMMSE estimator output for large-dimensional signals. IEEE Trans. Inform. Theory 55 5048–5063.
• [38] Khorunzhy, A. M., Khoruzhenko, B. A. and Pastur, L. A. (1996). Asymptotic properties of large random matrices with independent entries. J. Math. Phys. 37 5033–5060.
• [39] Laloux, L., Cizeau, P., Bouchaud, J.-P. and Potters, M. (1999). Noise dressing of financial correlation matrices. Phys. Rev. Lett. 83 1467.
• [40] Lytova, A. and Pastur, L. (2009). Central limit theorem for linear eigenvalue statistics of random matrices with independent entries. Ann. Probab. 37 1778–1840.
• [41] Marčenko, V. A. and Pastur, L. A. (1967). Distribution of eigenvalues in certain sets of random matrices. Mat. Sb. (N.S.) 72 507–536.
• [42] Münnix, M. C., Schäfer, R. and Guhr, T. (2014). A random matrix approach to credit risk. PLoS ONE 9 e98030.
• [43] O’Rourke, S., Renfrew, D. and Soshnikov, A. (2013). On fluctuations of matrix entries of regular functions of Wigner matrices with non-identically distributed entries. J. Theoret. Probab. 26 750–780.
• [44] Pan, G. (2014). Comparison between two types of large sample covariance matrices. Ann. Inst. Henri Poincaré Probab. Stat. 50 655–677.
• [45] Pan, G. M. and Zhou, W. (2008). Central limit theorem for signal-to-interference ratio of reduced rank linear receiver. Ann. Appl. Probab. 18 1232–1270.
• [46] Pastur, L. and Shcherbina, M. (2011). Eigenvalue Distribution of Large Random Matrices. Mathematical Surveys and Monographs 171. Amer. Math. Soc., Providence, RI.
• [47] Pizzo, A., Renfrew, D. and Soshnikov, A. (2012). Fluctuations of matrix entries of regular functions of Wigner matrices. J. Stat. Phys. 146 550–591.
• [48] Schultz, H. (2005). Non-commutative polynomials of independent Gaussian random matrices. The real and symplectic cases. Probab. Theory Related Fields 131 261–309.
• [49] Shcherbina, M. (2011). Central limit theorem for linear eigenvalue statistics of the Wigner and sample covariance random matrices. Zh. Mat. Fiz. Anal. Geom. 7 176–192, 197, 199.
• [50] Silverstein, J. W. (1995). Strong convergence of the empirical distribution of eigenvalues of large-dimensional random matrices. J. Multivariate Anal. 55 331–339.
• [51] 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.
• [52] Sinai, Ya. 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 1–24.
• [53] Soshnikov, A. (2000). The central limit theorem for local linear statistics in classical compact groups and related combinatorial identities. Ann. Probab. 28 1353–1370.
• [54] Tillmann, H.-G. (1953). Randverteilungen analytischer Funktionen und Distributionen. Math. Z. 59 61–83.
• [55] Vallet, P., Loubaton, P. and Mestre, X. (2012). Improved subspace estimation for multivariate observations of high dimension: The deterministic signals case. IEEE Trans. Inform. Theory 58 1043–1068.
• [56] Wishart, J. (1928). The generalised product moment distribution in samples from a normal multivariate population. Biometrika 20A 32–52.
• [57] Yao, J. (2013). Estimation et fluctuations de fonctionnelles de grandes matrices aléatoires. Ph.D. thesis, Télécom ParisTech.