Lithuanian Mathematical Journal

, Volume 50, Issue 2, pp 121–132

# Asymptotic distribution of singular values of powers of random matrices

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

60F05 15B52

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