On Asymptotic Behavior of Multilinear Eigenvalue Statistics of Random Matrices |
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Authors: | A Lytova L Pastur |
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Institution: | (1) Mathematical Division, B. Verkin Institute for Low Temperature Physics, 47 Lenin Ave., 61103 Kharkov, Ukraine |
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Abstract: | We prove the Law of Large Numbers and the Central Limit Theorem for analogs of U- and V- (von Mises) statistics of eigenvalues of random matrices as their size tends to infinity. We show first that for a certain
class of test functions (kernels), determining the statistics, the validity of these limiting laws reduces to the validity
of analogous facts for certain linear eigenvalue statistics. We then check the conditions of the reduction statements for
several most known ensembles of random matrices. The reduction phenomenon is well known in statistics, dealing with i.i.d.
random variables. It is of interest that an analogous phenomenon is also the case for random matrices, whose eigenvalues are
strongly dependent even if the entries of matrices are independent. |
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Keywords: | Random matrices Multilinear eigenvalue statistics Central limit theorem |
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