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Analysis of the limiting spectral distribution of large dimensional random matrices. (English) Zbl 0872.60013
Summary: Results on the analytic behavior of the limiting spectral distribution of matrices of sample covariance type, studied by V. A. Marchenko and L. A. Pastur [Math. USSR, Sb. 1, 457-483 (1967); translation from Mat. Sb., n. Ser. 72(114), 507-536 (1967; Zbl 0152.16101)] and Y. Q. Yin [J. Multivariate Anal. 20, 50-68 (1986; Zbl 0614.62060)], are derived. Through an equation defining its Stieltjes transform, it is shown that the limiting distribution has a continuous derivative away from zero, the derivative being analytic wherever it is positive, and resembles \(\sqrt {|x-x_0 |}\) for most cases of \(x_0\) in the boundary of its support. A complete analysis of a way to determine its support, originally outlined by Marchenko and Pastur (loc. cit.), is also presented.

60E99 Distribution theory
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