On the Marchenko-Pastur and Circular Laws for some Classes of Random Matrices with Dependent Entries

Radoslaw Adamczak (University of Warsaw)


In the first part of the article we prove limit theorems of Marchenko-Pastur type for the average spectral distribution of random matrices with dependent entries satisfying a weak law of large numbers, uniform bounds on moments and a martingale like condition investigated previously by Goetze and Tikhomirov. Examples include log-concave unconditional distributions on the space of matrices. In the second part we specialize to random matrices with independent isotropic unconditional log-concave rows for which (using the Tao-Vu replacement principle) we prove the circular law.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1065-1095

Publication Date: June 2, 2011

DOI: 10.1214/EJP.v16-899


  1. Adamczak, Radosław; Guédon, Olivier; Litvak, Alexander; Pajor, Alain; Tomczak-Jaegermann, Nicole. Condition number of a square matrix with i.i.d. columns drawn from a convex body. To appear in Proc. Amer. Math. Soc.
  2. Adamczak, Radosław; Guédon, Olivier; Litvak, Alexander; Pajor, Alain; Tomczak-Jaegermann, Nicole. Smallest singular value of random matrices with independent columns. C. R. Math. Acad. Sci. Paris 346 (2008), no. 15-16, 853--856. MR2441920
  3. Adamczak, Radosław; Litvak, Alexander E.; Pajor, Alain; Tomczak-Jaegermann, Nicole. Quantitative estimates of the convergence of the empirical covariance matrix in log-concave ensembles. J. Amer. Math. Soc. 23 (2010), no. 2, 535--561. MR2601042
  4. Anderson, Greg W.; Zeitouni, Ofer. A law of large numbers for finite-range dependent random matrices. Comm. Pure Appl. Math. 61 (2008), no. 8, 1118--1154. MR2417889
  5. Ané, Cécile, Blachère; Sébastien; Chafaï, Djalil; Fougères; Pierre, Gentil, Ivan; Malrieu, Florent; Roberto, Cyril and Scheffer, Grégory. Sur les inégalités de Sobolev logarithmiques, volume 10 of Panoramas et Synthèses [Panoramas and Syntheses]. Société Mathématique de France, Paris, Paris, 2000. MR1845806
  6. Aubrun, Guillaume. Random points in the unit ball of $lsp nsb p$. Positivity 10 (2006), no. 4, 755--759. MR2280648
  7. Bai, Zhidong; Silverstein, Jack W. Spectral analysis of large dimensional random matrices. Second edition. Springer Series in Statistics. Springer, New York, 2010. xvi+551 pp. ISBN: 978-1-4419-0660-1 MR2567175
  8. Bai, Z. D. Circular law. Ann. Probab. 25 (1997), no. 1, 494--529. MR1428519
  9. Bordenave, Charles; Caputo, Pietro; Chafaï, Djalil. Circular law theorem for random Markov matrices. Available at http://arxiv.org/abs/0808.1502 . To appear in Probab. Theory Related Fields.
  10. Bordenave, Charles; Caputo, Pietro; Chafaï, Djalil. Spectrum of non-hermitian heavy tailed random matrices. Available at http://arxiv.org/abs/1006.1713. To appear in Comm. Math. Phys.
  11. Borell, Christer. Convex measures on locally convex spaces. Ark. Mat. 12 (1974), 239--252. MR0388475
  12. Bryc, Włodzimierz; Dembo, Amir; Jiang, Tiefeng. Spectral measure of large random Hankel, Markov and Toeplitz matrices. Ann. Probab. 34 (2006), no. 1, 1--38. MR2206341
  13. Dozier, R. Brent; Silverstein, Jack W. On the empirical distribution of eigenvalues of large dimensional information-plus-noise-type matrices. J. Multivariate Anal. 98 (2007), no. 4, 678--694. MR2322123
  14. El Karoui, Noureddine. Concentration of measure and spectra of random matrices: applications to correlation matrices, elliptical distributions and beyond. Ann. Appl. Probab. 19 (2009), no. 6, 2362--2405. MR2588248
  15. Erdős, László; Schlein, Benjamin; Yau, Horng-Tzer; Yin Jun. The local relaxation flow approach to universality of the local statistics for random matrices. Preprint. Available at http://arxiv.org/abs/0911.3687.
  16. Girko, V. L. The circle law. Teor. Veroyatnost. i Mat. Statist. No. 28, (1983), 15--21. MR0727271
  17. Girko, V. L. The strong circular law. Twenty years later. I. Random Oper. Stochastic Equations 12 (2004), no. 1, 49--104. MR2046403
  18. Götze, F.; Tikhomirov, A. Limit theorems for spectra of positive random matrices under dependence. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 311 (2004), Veroyatn. i Stat. 7, 92--123, 299; translation in J. Math. Sci. (N. Y.) 133 (2006), no. 3, 1257--1276 MR2092202
  19. Götze, Friedrich; Tikhomirov, Alexander. The circular law for random matrices. Ann. Probab. 38 (2010), no. 4, 1444--1491. MR2663633
  20. Götze, F.; Tikhomirov, A. N. Limit theorems for spectra of random matrices with martingale structure. Stein's method and applications, 181--193, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., 5, Singapore Univ. Press, Singapore, 2005. MR2205336
  21. Hensley, Douglas. Slicing convex bodies—bounds for slice area in terms of the body's covariance. Proc. Amer. Math. Soc. 79 (1980), no. 4, 619--625. MR0572315
  22. Ibragimov, I. A.; Linnik, Yu. V. Independent and stationary sequences of random variables. With a supplementary chapter by I. A. Ibragimov and V. V. Petrov. Translation from the Russian edited by J. F. C. Kingman. Wolters-Noordhoff Publishing, Groningen, 1971. 443 pp. MR0322926
  23. Kannan, R.; Lovász, L.; Simonovits, M. Isoperimetric problems for convex bodies and a localization lemma. Discrete Comput. Geom. 13 (1995), no. 3-4, 541--559. MR1318794
  24. Klartag, B. Power-law estimates for the central limit theorem for convex sets. J. Funct. Anal. 245 (2007), no. 1, 284--310. MR2311626
  25. Litvak, A. E.; Pajor, A.; Rudelson, M.; Tomczak-Jaegermann, N. Smallest singular value of random matrices and geometry of random polytopes. Adv. Math. 195 (2005), no. 2, 491--523. MR2146352
  26. Marčenko, V. A.; Pastur, L. A. Distribution of eigenvalues in certain sets of random matrices. (Russian) Mat. Sb. (N.S.) 72 (114) 1967 507--536. MR0208649 (34 #8458)
  27. Pajor, A.; Pastur, L. On the limiting empirical measure of eigenvalues of the sum of rank one matrices with log-concave distribution. Studia Math. 195 (2009), no. 1, 11--29. MR2539559
  28. Pan, Guangming; Zhou, Wang. Circular law, extreme singular values and potential theory. J. Multivariate Anal. 101 (2010), no. 3, 645--656. MR2575411
  29. Paouris, Grigoris. Small ball probability estimates for log-concave measures. To appear in Trans. Amer. Math. Soc.
  30. Péché, Sandrine. Universality results for the largest eigenvalues of some sample covariance matrix ensembles. Probab. Theory Related Fields 143 (2009), no. 3-4, 481--516. MR2475670
  31. Rudelson, Mark; Vershynin, Roman. The Littlewood-Offord problem and invertibility of random matrices. Adv. Math. 218 (2008), no. 2, 600--633. MR2407948
  32. Samson, Paul-Marie. Concentration of measure inequalities for Markov chains and $Phi$-mixing processes. Ann. Probab. 28 (2000), no. 1, 416--461. MR1756011
  33. Soshnikov, Alexander. Universality at the edge of the spectrum in Wigner random matrices. Comm. Math. Phys. 207 (1999), no. 3, 697--733. MR1727234
  34. Tao, Terence; Vu, Van. Random covariance matrices: Universality of local statistics of eigenvalues. Preprint. Available at http://arxiv.org/abs/0912.0966.
  35. Tao, Terence; Vu, Van. The condition number of a randomly perturbed matrix. STOC'07—Proceedings of the 39th Annual ACM Symposium on Theory of Computing, 248--255, ACM, New York, 2007. MR2402448
  36. Tao, Terence; Vu, Van. Random matrices: universality of ESDs and the circular law. With an appendix by Manjunath Krishnapur. Ann. Probab. 38 (2010), no. 5, 2023--2065. MR2722794
  37. Yin, Y. Q.; Krishnaiah, P. R. Limit theorem for the eigenvalues of the sample covariance matrix when the underlying distribution is isotropic. Teor. Veroyatnost. i Primenen. 30 (1985), no. 4, 810--816. MR0816299

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.