On bilinear forms based on the resolvent of large random matrices
Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 1, pp. 36-63.

Considérons une matrice 𝛴 n , non centrée, de taille N×n, avec un profil de variance séparable :

𝛴 n =D n 1/2 X n D ˜ n 1/2 n+A n .
Les matrices D n et D ˜ n sont déterministes, diagonales et non négatives ; la matrice A n est déterministe ; la matrice X n est une matrice aléatoire dont les entrées complexes sont des variables aléatoires indépendantes et identiquement distribuées, de moyenne nulle et de variance unité. On note Q n (z) la résolvante associée à 𝛴 n 𝛴 n * , i.e.
Q n (z)=𝛴 n 𝛴 n * - z I N -1 .
Étant données deux suites déterministes de vecteurs (u n ) et (v n ) de norme euclidienne bornée, on étudie le comportement asymptotique de la forme bilinéaire aléatoire :
u n * Q n (z)v n z- + ,
quand les dimensions de la matrice 𝛴 n tendent vers l’infini au même rythme. De telles quantités apparaissent dans l’étude de fonctionnelles de 𝛴 n 𝛴 n * ne dépendant pas uniquement des valeurs propres de 𝛴 n 𝛴 n * , et sont centrales dans l’étude de problèmes relatifs aux matrices de Gram non centrées tels que l’établissement de théorèmes de la limite centrale, le comportement des entrées individuelles et les problèmes de séparation des valeurs propres.

Consider a N×n non-centered matrix 𝛴 n with a separable variance profile:

𝛴 n =D n 1/2 X n D ˜ n 1/2 n+A n .
Matrices D n and D ˜ n are non-negative deterministic diagonal, while matrix A n is deterministic, and X n is a random matrix with complex independent and identically distributed random variables, each with mean zero and variance one. Denote by Q n (z) the resolvent associated to 𝛴 n 𝛴 n * , i.e.
Q n (z)=𝛴 n 𝛴 n * - z I N -1 .
Given two sequences of deterministic vectors (u n ) and (v n ) with bounded Euclidean norms, we study the limiting behavior of the random bilinear form:
u n * Q n (z)v n z- + ,
as the dimensions of matrix 𝛴 n go to infinity at the same pace. Such quantities arise in the study of functionals of 𝛴 n 𝛴 n * which do not only depend on the eigenvalues of 𝛴 n 𝛴 n * , and are pivotal in the study of problems related to non-centered Gram matrices such as central limit theorems, individual entries of the resolvent, and eigenvalue separation.

DOI : 10.1214/11-AIHP450
Classification : Primary 15A52, secondary, 15A18, 60F15
Mots clés : random matrix, empirical distribution of the eigenvalues, Stieltjes transform
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     title = {On bilinear forms based on the resolvent of large random matrices},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {36--63},
     publisher = {Gauthier-Villars},
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Hachem, Walid; Loubaton, Philippe; Najim, Jamal; Vallet, Pascal. On bilinear forms based on the resolvent of large random matrices. Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 1, pp. 36-63. doi : 10.1214/11-AIHP450. http://archive.numdam.org/articles/10.1214/11-AIHP450/

[1] C. Artigue and P. Loubaton. On the precoder design of flat fading MIMO systems equipped with MMSE receivers: A large-system approach. IEEE Trans. Inform. Theory 57 (2011) 4138-4155. | MR

[2] Z. D. Bai, B. Q. Miao and G. M. Pan. On asymptotics of eigenvectors of large sample covariance matrix. Ann. Probab. 35 (2007) 1532-1572. | MR | Zbl

[3] Z. D. Bai and J. W. Silverstein. No eigenvalues outside the support of the limiting spectral distribution of large-dimensional sample covariance matrices. Ann. Probab. 26 (1998) 316-345. | MR | Zbl

[4] Z. D. Bai and J. W. Silverstein. Exact separation of eigenvalues of large-dimensional sample covariance matrices. Ann. Probab. 27 (1999) 1536-1555. | MR | Zbl

[5] Z. Bai and J. W. Silverstein. No eigenvalues outside the support of the limiting spectral distribution of information-plus-noise type matrices. Random Matrices Theory Appl. 1 (2012) 1150004. | MR | Zbl

[6] F. Benaych-Georges and R. N. Rao. The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices. Preprint, 2009. Available at arXiv:0910.2120. | Zbl

[7] R. B. Dozier and J. W. Silverstein. On the empirical distribution of eigenvalues of large dimensional information-plus-noise-type matrices. J. Multivariate Anal. 98 (2007) 678-694. | MR | Zbl

[8] M. Capitaine, C. Donati-Martin and D. Féral. The largest eigenvalues of finite rank deformation of large Wigner matrices: Convergence and nonuniversality of the fluctuations. Ann. Probab. 37 (2009) 1-47. | MR | Zbl

[9] J. Dumont, W. Hachem, S. Lasaulce, P. Loubaton and J. Najim. On the capacity achieving covariance matrix for Rician MIMO channels: An asymptotic approach. IEEE Trans. Inform. Theory 56 (2010) 1048-1069. | MR

[10] L. Erdös, H.-T. Yau and J. Yin. Rigidity of eigenvalues of generalized Wigner matrices. Unpublished manuscript, 2010. Available at[4] http://arxiv.org/pdf/1007.4652. | MR | Zbl

[11] V. L. Girko. An Introduction to Statistical Analysis of Random Arrays. VSP, Utrecht, 1998. | MR | Zbl

[12] U. Haagerup and S. Thorbjørnsen. A new application of random matrices: Ext(C red * (F 2 )) is not a group. Ann. of Math. (2) 162 (2005) 711-775. | MR | Zbl

[13] W. Hachem, M. Kharouf, J. Najim and J. Silverstein. A CLT for information-theoretic statistics of non-centered Gram random matrices. Random Matrices Theory Appl. 1 (2012) 1150010. | MR | Zbl

[14] W. Hachem, P. Loubaton and J. Najim. The empirical distribution of the eigenvalues of a Gram matrix with a given variance profile. Ann. Inst. Henri Poincaré Probab. Stat. 42 (2006) 649-670. | EuDML | Numdam | MR | Zbl

[15] W. Hachem, P. Loubaton and J. Najim. Deterministic equivalents for certain functionals of large random matrices. Ann. Appl. Probab. 17 (2007) 875-930. | MR | Zbl

[16] R. A. Horn and C. R. Johnson. Topics in Matrix Analysis. Cambridge Univ. Press, Cambridge, 1994. | MR | Zbl

[17] A. Kammoun, M. Kharouf, W. Hachem, J. Najim and A. El Kharroubi. On the fluctuations of the mutual information for non centered MIMO channels: The non Gaussian case. In Proc. IEEE International Workshop on Signal Processing Advances in Wireless Communications (SPAWC), 2010.

[18] V. A. Marčenko and L. A. Pastur. Distribution of eigenvalues in certain sets of random matrices. Mat. Sb. (N.S.) 72 (1967) 507-536. | MR | Zbl

[19] X. Mestre. Improved estimation of eigenvalues and eigenvectors of covariance matrices using their sample estimates. IEEE Trans. Inform. Theory 54 (2008) 5113-5129. | MR | Zbl

[20] X. Mestre. On the asymptotic behavior of the sample estimates of eigenvalues and eigenvectors of covariance matrices. IEEE Trans. Signal Process. 56 (2008) 5353-5368. | MR

[21] X. Mestre and M. A. Lagunas. Modified subspace algorithms for DOA estimation with large arrays. IEEE Trans. Signal Process. 56 (2008) 598-614. | MR

[22] W. Rudin. Real and Complex Analysis, 3rd edition. McGraw-Hill, New York, 1986. | Zbl

[23] J. W. Silverstein. Strong convergence of the empirical distribution of eigenvalues of large-dimensional random matrices. J. Multivariate Anal. 55 (1995) 331-339. | MR | Zbl

[24] J. W. Silverstein and Z. D. Bai. On the empirical distribution of eigenvalues of a class of large-dimensional random matrices. J. Multivariate Anal. 54 (1995) 175-192. | MR | Zbl

[25] P. Vallet, P. Loubaton and X. Mestre. Improved subspace estimation for multivariate observations of high dimension: The deterministic signals case. IEEE Trans. Inform. Theory 58 (2012) 1043-1068. | MR

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