We consider powers of random matrices with independent entries. Let X ij , i, j ≥ 1, be independent complex random variables with E X ij = 0 and E|X ij |2 = 1, and let X denote an n × n matrix with |X|ij − X ij for 1 ≤ i, j ≤ n. Denote by \( s_1^{(m)}\geq \ldots \geq s_n^{(m) } \) the singular values of the random matrix \( \mathbf{W}:={n^{{-\frac{m}{2}}}}{{\mathbf{X}}^m} \) and define the empirical distribution of the squared singular values by
where I {B} denotes the indicator of an event B. We prove that the expected spectral distribution \( F_n^{(m) }(x)=\mathbf{E}\mathcal{F}_n^{(m) }(x) \) converges under the Lindeberg condition to the distribution function G (m)(x) defined by its moments
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Translate from Zapiski Nauchnykh Seminarov POMI, Vol. 408, 2012, pp. 9–42.
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Alexeev, N.V., Götze, F. & Tikhomirov, A.N. On the Asymptotic Distribution of Singular Values of Powers of Random Matrices. J Math Sci 199, 68–87 (2014). https://doi.org/10.1007/s10958-014-1834-y
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DOI: https://doi.org/10.1007/s10958-014-1834-y