Asymptotic distribution of singular values of powers of random matrices

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].