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].