On bilinear forms based on the resolvent of large random matrices

Annales de l'institut Henri Poincare (B) Probability and Statistics

Volumn 49, Issue 1, 2013, Pages 36-63

  Hachem, Walid ^a   Loubaton, Philippe ^b   Najim, Jamal ^a   Vallet, Pascal ^b  

^a CNRS   (France)

^b UNIVERSITÉ PARIS EST   (France)

Author keywords

Empirical distribution of the eigenvalues; Random matrix; Stieltjes transform

Indexed keywords

EID: 84879224041     PISSN: 02460203     EISSN: None     Source Type: Journal    
DOI: 10.1214/11-AIHP450     Document Type: Article

References (25)

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.
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. MR2330979
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. MR1617051
4. Z. D. Bai and J. W. Silverstein. Exact separation of eigenvalues of large-dimensional sample covariance matrices. Ann. Probab. 27 (1999) 1536-1555. MR1733159
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. MR2930382
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.
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. MR2322123
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. MR2489158
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. MR2723661
10. L. Erdös, H.-T. Yau and J. Yin. Rigidity of eigenvalues of generalized Wigner matrices. Unpublished manuscript, 2010. Available at http://arxiv.org/pdf/1007.4652.
11. V. L. Girko. An Introduction to Statistical Analysis of Random Arrays. VSP, Utrecht, 1998. MR1694087
12. 2)) is not a group. Ann. of Math. (2) 162 (2005) 711-775. MR2183281
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. MR2934716
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. MR2269232
15. W. Hachem, P. Loubaton and J. Najim. Deterministic equivalents for certain functionals of large random matrices. Ann. Appl. Probab. 17 (2007) 875-930. MR2326235
16. R. A. Horn and C. R. Johnson. Topics in Matrix Analysis. Cambridge Univ. Press, Cambridge, 1994. MR1288752
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. MR0208649
19. X. Mestre. Improved estimation of eigenvalues and eigenvectors of covariance matrices using their sample estimates. IEEE Trans. Inform. Theory 54 (2008) 5113-5129. MR2589886
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. MR2472837
21. X. Mestre and M. A. Lagunas. Modified subspace algorithms for DOA estimation with large arrays. IEEE Trans. Signal Process. 56 (2008) 598-614. MR2445537
22. W. Rudin. Real and Complex Analysis, 3rd edition. McGraw-Hill, New York, 1986.
23. J.W. Silverstein. Strong convergence of the empirical distribution of eigenvalues of large-dimensional random matrices. J. Multivariate Anal. 55 (1995) 331-339. MR1370408
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. MR1345534
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.