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SOME REMARKS ON THE DOZIER–SILVERSTEIN THEOREM FOR RANDOM MATRICES WITH DEPENDENT ENTRIES

    https://doi.org/10.1142/S2010326312500177Cited by:12 (Source: Crossref)

    The Dozier–Silverstein theorem asserts the almost sure convergence of the empirical spectral distribution of information plus noise matrices, i.e. perturbations of deterministic matrices whose spectral distribution converges (information matrices) by random matrices with i.i.d. entries (noise matrices). We show that a modification of the original proof given by Dozier and Silverstein allows to extend this result to more general noise matrices, in particular matrices with independent columns satisfying a natural concentration inequality for quadratic forms, matrices with independent entries, satisfying a Lindeberg-type condition (recovering a recent result by Xie), matrices with heavy-tailed entries in the domain of attraction of the Gaussian distribution and certain classes of matrices with dependencies among columns (generalizing those investigated recently by O'Rourke and containing matrices with exchangeable entries). As a corollary we obtain the circular law for random matrices with independent log-concave isotropic rows.

    AMSC: 60B20

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