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From the first rigorous proof of the Circular Law in 1984 to the Circular Law for block random matrices under the generalized Lindeberg condition

  • Vyacheslav L. Girko EMAIL logo

Abstract

The Circular Law under Lindeberg’s condition for the independent blocks of random matrices having zero expectations and double stochastic matrix of covariances of their array is proven.

MSC 2010: 15A52; 45B85; 60F99

Dedicated to the twenty-fifth anniversary of the Journal Random Operators and Stochastic Equations



Communicated by Anatoly F. Turbin


References

[1] J. Aljadeff, D. Renfrew and M. Stern, Eigenvalues of block structured asymmetric random matrices, J. Math. Phys. 56 (2015), no. 10, Article ID 103502. 10.1063/1.4931476Search in Google Scholar

[2] J. Alt, L. Erdős, T. Krüger and Y. Nemish, Location of the spectrum of Kronecker random matrices, preprint (2018), https://arxiv.org/abs/1706.08343v3. 10.1214/18-AIHP894Search in Google Scholar

[3] V. L. Girko, Theory of Random Determinants, Math. Appl. (Soviet Series) 45, Kluwer Academic, Dordrecht, 1990. 10.1007/978-94-009-1858-0Search in Google Scholar

[4] V. L. Girko, An Introduction to Statistical Analysis of Random Arrays, VSP, Utrecht, 1998. 10.1515/9783110916683Search in Google Scholar

[5] V. L. Girko, Theory of Stochastic Canonical Equations. Vol. I and II, Math. Appl., Kluwer Academic, Dordrecht, 2001. 10.1007/978-94-010-0989-8Search in Google Scholar

[6] V. L. Girko, The Circular Law. Thirty years later, Random Oper. Stoch. Equ. 20 (2012), no. 2, 143–187. 10.1515/rose-2012-0007Search in Google Scholar

[7] V. L. Girko, The circle law. Girko’s circular law: Let λ be eigenvalues of a set of random n×n matrices. Then λ/n is uniformly distributed on the disk, CRC Concise Encyclopedia of Mathematics on CD-ROM, 1996-9 Eric W. Weisstein. Search in Google Scholar

[8] F. Juhsz, On the structural eigenvalues of block random matrices, Linear Algebra Appl. 246 (1996), 225–231. 10.1016/0024-3795(94)00356-4Search in Google Scholar

[9] V. A. Marchenko and L. A. Pastur, Distribution of the eigenvalues in certain sets of random matrices (in Russian), Mat. Sb. 1 (1967), 457–483. 10.1070/SM1967v001n04ABEH001994Search in Google Scholar

[10] H. Nguyen and S. O’Rourke, On the concentration of random multilinear forms and the universality of random block matrices, Probab. Theory Related Fields 162 (2015), no. 1–2, 97–154. 10.1007/s00440-014-0567-7Search in Google Scholar

Received: 2015-12-10
Accepted: 2018-4-4
Published Online: 2018-5-19
Published in Print: 2018-6-1

© 2018 Walter de Gruyter GmbH, Berlin/Boston

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