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Lithuanian Mathematical Journal

, Volume 50, Issue 2, pp 121–132 | Cite as

Asymptotic distribution of singular values of powers of random matrices

  • N. AlexeevEmail author
  • F. Götze
  • A. Tikhomirov
Article

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,jN, 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 λ12,...,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].

Keywords

random matrices Fuss–Catalan numbers semi-circular law Marchenko–Pastur distribution 

MSC

60F05 15B52 

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Copyright information

© Springer Science+Business Media, Inc. 2010

Authors and Affiliations

  1. 1.Faculty of Mathematics and MechanicsSt. Petersburg State UniversitySt. PetersburgRussia
  2. 2.Faculty of MathematicsUniversity of BielefeldBielefeldGermany
  3. 3.Department of Mathematics, Komi Research Center of Ural Branch of RASSyktyvkar State UniversitySyktyvkarRussia

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