Electronic Journal of Probability

Local law for random Gram matrices

Johannes Alt, László Erdős, and Torben Krüger

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We prove a local law in the bulk of the spectrum for random Gram matrices $XX^*$, a generalization of sample covariance matrices, where $X$ is a large matrix with independent, centered entries with arbitrary variances. The limiting eigenvalue density that generalizes the Marchenko-Pastur law is determined by solving a system of nonlinear equations. Our entrywise and averaged local laws are on the optimal scale with the optimal error bounds. They hold both in the square case (hard edge) and in the properly rectangular case (soft edge). In the latter case we also establish a macroscopic gap away from zero in the spectrum of $XX^*$.

Article information

Electron. J. Probab., Volume 22 (2017), paper no. 25, 41 pp.

Received: 22 July 2016
Accepted: 27 February 2017
First available in Project Euclid: 8 March 2017

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Zentralblatt MATH identifier

Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 15B52: Random matrices

capacity of MIMO channels Marchenko-Pastur law hard edge soft edge general variance profile

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Alt, Johannes; Erdős, László; Krüger, Torben. Local law for random Gram matrices. Electron. J. Probab. 22 (2017), paper no. 25, 41 pp. doi:10.1214/17-EJP42. https://projecteuclid.org/euclid.ejp/1488942016

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  • [1] O. Ajanki, L. Erdős, and T. Krüger, Local semicircle law with imprimitive variance matrix, Elect. Comm. Probab. 19 (2014), no. 33, 1–9.
  • [2] O. Ajanki, L. Erdős, and T. Krüger, Quadratic vector equations on complex upper half-plane, arXiv:1506.05095v4, 2015.
  • [3] O. Ajanki, L. Erdős, and T. Krüger, Singularities of solutions to quadratic vector equations on the complex upper half-plane, Comm. Pure Appl. Math. (2016), doi:10.1002/cpa.21639 (Online).
  • [4] O. Ajanki, L. Erdős, and T. Krüger, Universality for general Wigner-type matrices, Prob. Theor. Rel. Fields (2016), doi:10.1007/s00440-016-0740-2 (Online).
  • [5] Z. Bao, G. Pan, and W. Zhou, Tracy-Widom law for the extreme eigenvalues of sample correlation matrices, Elect. J. Probab. 17 (2012), 32 pp.
  • [6] R. B. Bapat and T. E. S. Raghavan, Nonnegative matrices and applications, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 1997.
  • [7] F. Benaych-Georges and R. Couillet, Spectral analysis of the Gram matrix of mixture models, ESAIM: PS 20 (2016), 217–237.
  • [8] F. A. Berezin, Some remarks on Wigner distribution, Theoret. Math. Phys. 3 (1973), no. 17, 1163–1175.
  • [9] A. Bloemendal, L. Erdős, A. Knowles, H.-T. Yau, and J. Yin, Isotropic local laws for sample covariance and generalized Wigner matrices, Elect. J. Probab. 19 (2014), no. 33, 1–53.
  • [10] P. Bourgade, H.-T. Yau, and J. Yin, Local circular law for random matrices, Prob. Theor. Rel. Fields 159 (2014), no. 3-4, 545–595.
  • [11] R. Brent Dozier and J. W. Silverstein, On the empirical distribution of eigenvalues of large dimensional information-plus-noise-type matrices, Jour. Mult. Anal. 98 (2007), no. 4, 678 – 694.
  • [12] C. Cacciapuoti, A. Maltsev, and B. Schlein, Local Marchenko-Pastur law at the hard edge of sample covariance matrices, J. Math. Phys. 54 (2013), no. 4.
  • [13] R. Couillet and M. Debbah, Random matrix methods for wireless communications, Cambridge University Press, 2011.
  • [14] R. Couillet and W. Hachem, Analysis of the limiting spectral measure of large random matrices of the separable covariance type, Random Matrices: Theory and Applications 03 (2014), no. 04, 1450016.
  • [15] L. Erdős, B. Schlein, and H.-T. Yau, Universality of random matrices and local relaxation flow, Invent. Math. 185 (2011), no. 1, 75–119.
  • [16] L. Erdős, B. Schlein, H.-T. Yau, and J. Yin, The local relaxation flow approach to universality of the local statistics for random matrices, Ann. Inst. H. Poincaré Probab. Statist. 48 (2012), no. 1, 1–46.
  • [17] L. Erdős and H.-T. Yau, Universality of local spectral statistics of random matrices, Bull. Amer. Math. Soc. 49 (2012), no. 3, 377–414.
  • [18] L. Erdős, H.-T. Yau, and J. Yin, Bulk universality for generalized Wigner matrices, Prob. Theor. Rel. Fields 154 (2011), no. 1-2, 341–407.
  • [19] L. Erdős, H.-T. Yau, and J. Yin, Universality for generalized Wigner matrices with Bernoulli distribution, J. Comb. 2 (2011), no. 1, 15–82.
  • [20] L. Erdős, H.-T. Yau, and J. Yin, Rigidity of eigenvalues of generalized Wigner matrices, Adv. Math. 229 (2012), no. 3, 1435–1515.
  • [21] O. N. Feldheim and S. Sodin, A universality result for the smallest eigenvalues of certain sample covariance matrices, Geom. Func. Anal. 20 (2010), no. 1, 88–123.
  • [22] G. J. Foschini, Layered space-time architecture for wireless communication in a fading environment when using multi-element antennas, Bell Labs Tech. J. 1 (1996), no. 2, 41–59.
  • [23] G.J. Foschini and M.J. Gans, On limits of wireless communications in a fading environment when using multiple antennas, Wireless Per. Commun. 6 (1998), no. 3, 311–335.
  • [24] V. L. Girko, Theory of stochastic canonical equations. Vol. I, Mathematics and its Applications, vol. 535, Kluwer Academic Publishers, Dordrecht, 2001.
  • [25] W. Hachem, M. Kharouf, J. Najim, and J. W. Silverstein, A CLT for information-theoretic statistics of non-centered Gram random matrices, Random Matrices: Theory Appl. 01 (2012), no. 02, 1150010.
  • [26] W. Hachem, O. Khorunzhiy, P. Loubaton, J. Najim, and L. Pastur, A new approach for mutual information analysis of large dimensional multi-antenna channels, IEEE Trans. Inf. Theory 54 (2008), no. 9, 3987–4004.
  • [27] W. Hachem, P. Loubaton, and J. Najim, The empirical distribution of the eigenvalues of a Gram matrix with a given variance profile, Ann. Inst. H. Poincaré Probab. Statist. 42 (2006), no. 6, 649 – 670.
  • [28] W. Hachem, P. Loubaton, and J. Najim, Deterministic equivalents for certain functionals of large random matrices, Ann. Appl. Probab. 17 (2007), no. 3, 875–930.
  • [29] W. Hachem, P. Loubaton, and J. Najim, A CLT for information-theoretic statistics of Gram random matrices with a given variance profile, Ann. Appl. Probab. 18 (2008), no. 6, 2071–2130.
  • [30] A. M. Khorunzhy and L. A. Pastur, On the eigenvalue distribution of the deformed Wigner ensemble of random matrices, Spectral operator theory and related topics, Adv. Soviet Math., 19, Amer. Math. Soc., Providence, RI, 1994, pp. 97–127.
  • [31] A. Knowles and J. Yin, Anisotropic local laws for random matrices, Prob. Theor. Rel. Fields (2016), doi:10.1007/s00440-016-0730-4 (Online).
  • [32] V. A. Marchenko and L. A. Pastur, Distribution of eigenvalues for some sets of random matrices, Mat. Sbornik 1 (1967), no. 4, 457–483.
  • [33] M. L. Mehta, Random matrices, Pure and Applied Mathematics, Elsevier Science, 2004.
  • [34] N. S. Pillai and J. Yin, Universality of covariance matrices, Ann. Appl. Probab. 24 (2014), no. 3, 935–1001.
  • [35] J. W. Silverstein, The smallest eigenvalue of a large dimensional Wishart matrix, Ann. Probab. 13 (1985), no. 4, 1364–1368.
  • [36] J. W. Silverstein and Z. D. Bai, On the empirical distribution of eigenvalues of a class of large dimensional random matrices, Jour. Mult. Anal. 54 (1995), no. 2, 175 – 192.
  • [37] T. Tao and V. Vu, Random matrices: Universality of local eigenvalue statistics, Acta Math. 206 (2011), no. 1, 127–204.
  • [38] T. Tao and V. Vu, Random covariance matrices: Universality of local statistics of eigenvalues, Ann. Probab. 40 (2012), no. 3, 1285–1315.
  • [39] E. Telatar, Capacity of multi-antenna Gaussian channels, Eur. Trans. Telecomm. 10 (1999), no. 6, 585–595.
  • [40] A. M. Tulino and S. Verdú, Random matrix theory and wireless communications, Found. Trends Commun. Inf. Theory 1 (2004), no. 1, 1–182.
  • [41] F. J. Wegner, Disordered system with $ n $ orbitals per site: $ n =\infty $ limit, Physical Review B 19 (1979).
  • [42] E. P. Wigner, Characteristic vectors of bordered matrices with infinite dimensions, Ann. of Math. 62 (1955), no. 3, 548–564.
  • [43] J. Wishart, The generalised product moment distribution in samples from a normal multivariate population, Biometrika 20A (1928), no. 1-2, 32–52.