Strong Circular Law

Some explanations and improved formulas for the proof of Strong Circular Law, published in [6], 1984, are given: let Ξ = (ξ ij) n i,j=1 be a random complex matrix, whose entries ξ (n) ij are independent for every n, Eξ (n) ij =0, E|ξ (n) ij| 2 = n -1, given on the common probability space, there exist densities p (n) ij (x) of the real parts or densities q (n) ij (x) of the imaginary parts of random entries √nξ (n)ij and for some δ > 0, β > 1 {equation presented} is the normalized spectral function, and λ k are the eigenvalues of a complex random matrix Ξ = (ξ (n) ij) n i,j=1. © VSP 1997.52173196Burkholder, D.L., Distribution function inequalities for martingales (1973) Ann. Probab., 1 (1), pp. 19-42Ginibre, J., Statistical ensembles of complex, quaternion and real matrices (1965) J. Math. Phys., 6, pp. 440-449Girko, V.L., (1975) Random Matrices. Kiev, , Published by Kiev University, (in Russian)Girko, V.L., F-Transform (1982) Reports AN USSR, Series of Mathematics and Mechanics, 3, pp. 5-6On the circle law (1983) Theory of Probability and Mathematical Statistics, 28, pp. 15-23. , V. L. GirkoGirko, V.L., The circle law (1984) Theory of Probabilities and Their Applications, 29 (4), pp. 669-679Regularized V-transform (1984) Theory of Probability and Its Applications, 29 (2), pp. 416-417Girko, V.L., The elliptic law (1985) Reports of the Fourth Vilnius Conference on the Theory of Probability and Mathematical Statistics, Vilnius, pp. 171-173Girko, V.L., The elliptic law (1985) Doclady AN USSR, Ser. Matematika i Mehanika, 1, pp. 3-7Girko, V.L., The elliptic law (1985) Theory of Probability and Their Applications, 30 (4), pp. 640-651Girko, V.L., The elliptic law and elements of g-analysis (1987) Proceedings of the Fourth Vilnius Conference, 1, pp. 489-509. , VNU Science Press BVGirko, V.L., (1990) Theory of Random Determinants, , Kluwer Academic PublishersGirko, V.L., The circular law: Ten years later (1994) Random Oper, and Stoch. Equ., 2 (3), pp. 235-277Girko, V.L., The elliptic law: Ten years later I (1995) Random Operators and Stochastic Equations, 3 (3), pp. 257-302Girko, V.L., The elliptic law: Ten years later II (1995) Random Operators and Stochastic Equations, 3 (4), pp. 377-398Girko, V.L., (1996) Theory of Linear Algebraic Equations with Random Coefficients, 320p. , Allerton Press, Inc. New YorkMehta, M.L., (1967) Random Matrices and the Statistical Theory of Energy Levels, , Academic Press, New York, LondonPastur, L.A., The spectra of random selfadjoint operators (1973) Russian Math. Surveys, 28 (1), pp. 1-6