## The Annals of Applied Probability

### Deterministic equivalents for certain functionals of large random matrices

#### Abstract

Consider an N×n random matrix Yn=(Ynij) where the entries are given by , the Xnij being independent and identically distributed, centered with unit variance and satisfying some mild moment assumption. Consider now a deterministic N×n matrix An whose columns and rows are uniformly bounded in the Euclidean norm. Let Σn=Yn+An. We prove in this article that there exists a deterministic N×N matrix-valued function Tn(z) analytic in ℂ−ℝ+ such that, almost surely,

Otherwise stated, there exists a deterministic equivalent to the empirical Stieltjes transform of the distribution of the eigenvalues of ΣnΣnT. For each n, the entries of matrix Tn(z) are defined as the unique solutions of a certain system of nonlinear functional equations. It is also proved that is the Stieltjes transform of a probability measure πn(), and that for every bounded continuous function f, the following convergence holds almost surely

where the (λk)1≤kN are the eigenvalues of ΣnΣnT. This work is motivated by the context of performance evaluation of multiple inputs/multiple output (MIMO) wireless digital communication channels. As an application, we derive a deterministic equivalent to the mutual information:

where σ2 is a known parameter.

#### Article information

Source
Ann. Appl. Probab., Volume 17, Number 3 (2007), 875-930.

Dates
First available in Project Euclid: 22 May 2007

https://projecteuclid.org/euclid.aoap/1179839177

Digital Object Identifier
doi:10.1214/105051606000000925

Mathematical Reviews number (MathSciNet)
MR2326235

Zentralblatt MATH identifier
1181.15043

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

#### Citation

Hachem, Walid; Loubaton, Philippe; Najim, Jamal. Deterministic equivalents for certain functionals of large random matrices. Ann. Appl. Probab. 17 (2007), no. 3, 875--930. doi:10.1214/105051606000000925. https://projecteuclid.org/euclid.aoap/1179839177

#### References

• 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.
• Bai, Z. D. and Silverstein, J. W. (2004). CLT for linear spectral statistics of large-dimensional sample covariance matrices. Ann. Probab. 32 553--605.
• Bolotnikov, V. (1997). On a general moment problem on the half axis. Linear Algebra and Its Applications 255 57--112.
• 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.
• 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. To appear.
• Dumont, J., Hachem, W., Loubaton, P. and Najim, J. (2006). On the asymptotic analysis of mutual information of mimo Rician correlated channels. Proceedings of the ISCCSP Conference. Marrakech, Morocco.
• Dumont, J., Loubaton, P., Lasaulce, S. and Debbah, M. (2005). On the asymptotic performance of mimo correlated Ricean channels. In ICASSP Proceedings 5 813--816.
• Gesztesy, F. and Tsekanovskii, E. (2000). On matrix-valued Herglotz functions. Math. Nachr. 218 61--138.
• Girko, V. L. (1990). Theory of Random Determinants. Kluwer, Dordrecht.
• Girko, V. L. (2001). Theory of Stochastic Canonical Equations. Kluwer, Dordrecht.
• Guionnet, A. and Zeitouni, O. (2000). Concentration of the spectral measure for large matrices. Electron. Comm. Probab. 5 119--136.
• 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.
• 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.
• Harville, D. A. (1997). Matrix Algebra from a Statistician's Perspective. Springer, New York.
• Horn, R. and Johnson, C. (1985). Matrix Analysis. Cambridge Univ. Press.
• Kailath, T., Sayed, A. H. and Hassibi, B. (2000). Linear Estimation. Prentice Hall, Englewood Cliffs, NJ.
• Khorunzhy, A., Khoruzhenko, B. and Pastur, L. (1996). Asymptotic properties of large random matrices with independent entries. J. Math. Phys. 37 5033--5060.
• Krein, M. and Nudelman, A. (1997). The Markov Moment Problem and Extremal Problems. Amer. Math. Soc., Providence, RI.
• Marčenko, V. A. and Pastur, L. A. (1967). Distribution of eigenvalues in certain sets of random matrices. Mat. Sb. (N.S.) 72 507--536.
• Rozanov, Y. A. (1967). Stationary Random Processes. Holden-Day, San Francisco, CA.
• Rudin, W. (1987). Real and Complex Analysis. McGraw-Hill, New York.
• Sengupta, A. M. and Mitra, P. (2000). Capacity of multivariate channels with multiplicative noise: I. Random matrix techniques and large-$n$ expansions for full transfer matrices. Available at http://arxiv.org/abs/physics/0010081.
• Shlyakhtenko, D. (1996). Random Gaussian band matrices and freeness with amalgamation. Internat. Math. Res. Notices 20 1013--1025.
• Silverstein, J. W. (1995). Strong convergence of the empirical distribution of eigenvalues of large-dimensional random matrices. J. Multivariate Anal. 55 331--339.
• 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.
• Tulino, A. and Verdú, S. (2004). Random matrix theory and wireless communications. Foundations and Trends in Communications and Information Theory 1 1--182.
• Verdu, S. and Shamai, S. (1999). Spectral efficiency of CDMA with random spreading. IEEE Trans. Inform. Theory 45 622--640.
• Yin, Y. Q. (1986). Limiting spectral distribution for a class of random matrices. J. Multivariate Anal. 20 50--68.