Abstract
Let x be a complex random variable such that \( {\mathbf{E}}x = 0,\,{\mathbf{E}}{\left| x \right|^2} = 1 \), and \( {\mathbf{E}}{\left| x \right|^4} < \infty \). Let \( {x_{ij}},i,j \in \left\{ {1,2, \ldots } \right\} \), be independent copies of x. Let \( {\mathbf{X}} = \left( {{N^{ - 1/2}}{x_{ij}}} \right) \), 1≤i,j≤N, be a random matrix. Writing X ∗ for the adjoint matrix of X, consider the product X m X ∗m with some m ∈{1,2,...}. The matrix X m X ∗m is Hermitian positive semidefinite. Let λ1,λ2,...,λ N be eigenvalues of X m X ∗m (or squared singular values of the matrix X m). In this paper, we find the asymptotic distribution function \( {G^{(m)}}(x) = {\lim_{N \to \infty }}{\mathbf{E}}F_N^{(m)}(x) \) of the empirical distribution function \( F_N^{(m)}(x) = {N^{ - 1}}\sum\nolimits_{k = 1}^N {\mathbb{I}\left\{ {{\lambda_k} \leqslant x} \right\}} \), where \( \mathbb{I}\left\{ A \right\} \) stands for the indicator function of an event A. With m=1, our result turns to a well-known result of Marchenko and Pastur [V. Marchenko and L. Pastur, The eigenvalue distribution in some ensembles of random matrices, Math. USSR Sb., 1:457–483, 1967].
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Partially supported by the RF grant of the Leading Scientific Schools (No. NSH-4472.2010.1), RFBR (grant No. 09-01-12180), RFBR–DFG (grant No. 09-01-91331), and CRC 701 “Spectral Structures and Topological Methods in Mathematics”, Bielefeld.
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Alexeev, N., Götze, F. & Tikhomirov, A. Asymptotic distribution of singular values of powers of random matrices. Lith Math J 50, 121–132 (2010). https://doi.org/10.1007/s10986-010-9074-4
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DOI: https://doi.org/10.1007/s10986-010-9074-4