The Annals of Applied Probability

A CLT for information-theoretic statistics of Gram random matrices with a given variance profile

Walid Hachem, Philippe Loubaton, and Jamal Najim

Full-text: Open access


Consider an N×n random matrix Yn=(Ynij) with entries given by $$Y_{ij}^{n}=\frac{\sigma_{ij}(n)}{\sqrt{n}}X_{ij}^{n},$$ the Xnij being centered, independent and identically distributed random variables with unit variance and (σij(n); 1≤iN, 1≤jn) being an array of numbers we shall refer to as a variance profile. In this article, we study the fluctuations of the random variable

log det(YnY*n+ρIN),

where Y* is the Hermitian adjoint of Y and ρ>0 is an additional parameter. We prove that, when centered and properly rescaled, this random variable satisfies a central limit theorem (CLT) and has a Gaussian limit whose parameters are identified whenever N goes to infinity and N/nc∈(0, ∞). A complete description of the scaling parameter is given; in particular, it is shown that an additional term appears in this parameter in the case where the fourth moment of the Xij’s differs from the fourth moment of a Gaussian random variable. Such a CLT is of interest in the field of wireless communications.

Article information

Ann. Appl. Probab., Volume 18, Number 6 (2008), 2071-2130.

First available in Project Euclid: 26 November 2008

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Random matrix empirical distribution of the eigenvalues Stieltjes transform


Hachem, Walid; Loubaton, Philippe; Najim, Jamal. A CLT for information-theoretic statistics of Gram random matrices with a given variance profile. Ann. Appl. Probab. 18 (2008), no. 6, 2071--2130. doi:10.1214/08-AAP515.

Export citation


  • [1] Anderson, G. W. and Zeitouni, O. (2006). A CLT for a band matrix model. Probab. Theory Related Fields 134 283–338.
  • [2] 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.
  • [3] Bai, Z. D. and Silverstein, J. W. (2004). CLT for linear spectral statistics of large-dimensional sample covariance matrices. Ann. Probab. 32 553–605.
  • [4] Billingsley, P. (1995). Probability and Measure, 3rd ed. Wiley, New York.
  • [5] Boutet de Monvel, A. and Khorunzhy, A. (1998). Limit theorems for random matrices. Markov Process. Related Fields 4 175–197.
  • [6] 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.
  • [7] Cabanal-Duvillard, T. (2001). Fluctuations de la loi empirique de grandes matrices aléatoires. Ann. Inst. H. Poincaré Probab. Statist. 37 373–402.
  • [8] Debbah, M. and Müller, R. (2003). On the limiting behavior of directional mimo models. In Proceedings of the Sixth Baiona Workshop on Signal Processing in Communication. Spain.
  • [9] Girko, V. L. (1975). Sluchainye Matritsy. Izdat. Obed. “Višča Škola” pri Kiev. Gosudarstv. Univ., Kiev.
  • [10] Girko, V. L. (1990). Theory of Random Determinants. Mathematics and Its Applications (Soviet Series) 45. Kluwer Academic, Dordrecht. Translated from the Russian.
  • [11] Girko, V. L. (2001). Theory of Stochastic Canonical Equations. I. Mathematics and Its Applications 535. Kluwer Academic, Dordrecht.
  • [12] Girko, V. L. (2003). Thirty years of the central resolvent law and three laws on the 1/n expansion for resolvent of random matrices. Random Oper. Stochastic Equations 11 167–212.
  • [13] Guionnet, A. (2002). Large deviations upper bounds and central limit theorems for non-commutative functionals of Gaussian large random matrices. Ann. Inst. H. Poincaré Probab. Statist. 38 341–384.
  • [14] Hachem, W., Khorunzhiy, O., Loubaton, P., Najim, J. and Pastur, L. (2008). A new approach for capacity analysis of large dimensional multi-antenna channels. IEEE Trans. Inform. Theory. 54(9).
  • [15] Hachem, W., Loubaton, P. and Najim, J. (2005). The empirical eigenvalue distribution of a gram matrix: From independence to stationarity. Markov Process. Related Fields 11 629–648.
  • [16] Hachem, W., Loubaton, P. and Najim, J. (2006). The empirical distribution of the eigenvalues of a Gram matrix with a given variance profile. Ann. Inst. H. Poincaré Probab. Statist. 42 649–670.
  • [17] Hachem, W., Loubaton, P. and Najim, J. (2007). Deterministic equivalents for certain functionals of large random matrices. Ann. Appl. Probab. 17 875–930.
  • [18] Horn, R. A. and Johnson, C. R. (1994). Topics in Matrix Analysis. Cambridge Univ. Press, Cambridge. Corrected reprint of the 1991 original.
  • [19] Johansson, K. (1998). On fluctuations of eigenvalues of random Hermitian matrices. Duke Math. J. 91 151–204.
  • [20] Jonsson, D. (1982). Some limit theorems for the eigenvalues of a sample covariance matrix. J. Multivariate Anal. 12 1–38.
  • [21] 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.
  • [22] Mingo, J. A. and Speicher, R. (2006). Second order freeness and fluctuations of random matrices. I. Gaussian and Wishart matrices and cyclic Fock spaces. J. Funct. Anal. 235 226–270.
  • [23] Moustakas, A. L., Simon, S. H. and Sengupta, A. M. (2003). MIMO capacity through correlated channels in the presence of correlated interferers and noise: A (not so) large N analysis. IEEE Trans. Inform. Theory 49 2545–2561.
  • [24] Sengupta, A. M. and Mitra, P. (2008). Capacity of multivariate channels with multiplicative noise: I. Random matrix techniques and large-n expansions for full transfer matrices. Available at
  • [25] Silverstein, J. W. (1995). Strong convergence of the empirical distribution of eigenvalues of large-dimensional random matrices. J. Multivariate Anal. 55 331–339.
  • [26] 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.
  • [27] Sinai, Y. 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.
  • [28] Smithies, F. (1958). Integral Equations. Cambridge Tracts in Mathematics and Mathematical Physics 49. Cambridge Univ. Press, New York.
  • [29] Soshnikov, A. (2000). The central limit theorem for local linear statistics in classical compact groups and related combinatorial identities. Ann. Probab. 28 1353–1370.
  • [30] Taricco, G. (2006). On the capacity of separately-correlated MIMO Rician fading channels. In Proc. 49th Annual IEEE Globecom Conference, 2006. San Francisco.
  • [31] Telatar, I. (1999). Capacity of multi-antenna Gaussian channel. European Transactions on Telecommunications.
  • [32] Tse, D. and Zeitouni, O. (2000). Linear multiuser receivers in random environments. IEEE Trans. Inform. Theory 46 171–188.
  • [33] Tse, D. N. C. and Hanly, S. V. (1999). Linear multiuser receivers: Effective interference, effective bandwidth and user capacity. IEEE Trans. Inform. Theory 45 641–657.
  • [34] Tulino, A. and Verdú, S. (2004). Random matrix theory and wireless communications. In Foundations and Trends in Communications and Information Theory 1 1–182. Now Publishers.
  • [35] Tulino, A. and Verdu, S. (2005). Asymptotic outage capacity of multiantenna channels. In Proceedings of IEEE ICASSP ’05 5 828–835.