A note on the Marchenko-Pastur law for a class of random matrices with dependent entries

Sean O'Rourke (Rutgers University)

Abstract


We consider a class of real random matrices with dependent entries and show that the limiting empirical spectral distribution is given by the Marchenko-Pastur law. Additionally, we establish a rate of convergence of the expected empirical spectral distribution.

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

Pages: 1-13

Publication Date: July 17, 2012

DOI: 10.1214/ECP.v17-2020

References

  • Adamczak, Radosław. On the Marchenko-Pastur and circular laws for some classes of random matrices with dependent entries. Electron. J. Probab. 16 (2011), no. 37, 1068--1095. MR2820070
  • 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
  • Anderson, Greg W.; Guionnet, Alice; Zeitouni, Ofer. An introduction to random matrices. Cambridge Studies in Advanced Mathematics, 118. Cambridge University Press, Cambridge, 2010. xiv+492 pp. ISBN: 978-0-521-19452-5 MR2760897
  • Aubrun, Guillaume. Random points in the unit ball of $l^ n_ p$. Positivity 10 (2006), no. 4, 755--759. MR2280648
  • Bai, Z. D. Methodologies in spectral analysis of large-dimensional random matrices, a review. With comments by G. J. Rodgers and Jack W. Silverstein; and a rejoinder by the author. Statist. Sinica 9 (1999), no. 3, 611--677. MR1711663
  • Bai, Zhidong; Hu, Jiang; Pan, Guangming; Zhou, Wang. A note on rate of convergence in probability to semicircular law. Electron. J. Probab. 16 (2011), no. 88, 2439--2451. MR2861680
  • Bai, Z. D. Convergence rate of expected spectral distributions of large random matrices. I. Wigner matrices. Ann. Probab. 21 (1993), no. 2, 625--648. MR1217559
  • Z. D. Bai, J. Silverstein, Spectral analysis of large dimensional random matrices, Mathematics Monograph Series 2, Science Press, Beijing 2006.
  • Bordenave, B., Caputo, P., Chafaï, D., phCircular law theorem for random Markov matrices, Probability Theory and Related Fields, DOI 10.1007/s00440-010-0336-1 (2011).
  • Chafaï, Djalil. The Dirichlet Markov ensemble. J. Multivariate Anal. 101 (2010), no. 3, 555--567. MR2575404
  • Chatterjee, S., Diaconis P., Sly A., phProperties of Uniform Doubly Stochastic Matrices, arXiv:1010.6136v1 [math.PR]
  • Durrett, Rick. Probability: theory and examples. Fourth edition. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 2010. x+428 pp. ISBN: 978-0-521-76539-8 MR2722836
  • 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
  • Götze, F.; Tikhomirov, A.; phOn the Rate of Convergence to the Marchenko--Pastur Distribution, arXiv:1110.1284v1 [math.PR].
  • 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
  • Horn, Roger A.; Johnson, Charles R. Topics in matrix analysis. Cambridge University Press, Cambridge, 1991. viii+607 pp. ISBN: 0-521-30587-X MR1091716
  • V. Marchenko, L. Pastur, phDistribution of eigenvalues of some sets of random matrices, Math USSR-Sb. 1, (1967), 457--486.
  • Mehta, Madan Lal. Random matrices. Third edition. Pure and Applied Mathematics (Amsterdam), 142. Elsevier/Academic Press, Amsterdam, 2004. xviii+688 pp. ISBN: 0-12-088409-7 MR2129906
  • Mendelson, Shahar; Pajor, Alain. On singular values of matrices with independent rows. Bernoulli 12 (2006), no. 5, 761--773. MR2265341
  • Pajor, A.,, Pastur, L., ph On the Limiting Empirical Measure of the sum of rank one matrices with log-concave distribution, arXiv:0710.1346v1 [math.PR].
  • Wachter, Kenneth W. The strong limits of random matrix spectra for sample matrices of independent elements. Ann. Probability 6 (1978), no. 1, 1--18. MR0467894
  • 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
Bookmark and Share


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