On the asymptotic distribution of the singular values of powers of random matrices

Abstract

We consider powers of random matrices with independent entries. Let $X_{ij},i,j\ge 1$, be independent complex random variables with $\E X_{ij}=0$ and $\E |X_{ij}|^2=1$ and let $\mathbf{X}$ denote an $n\times n$ matrix with $[\mathbf{X}{]}_{ij}=X_{ij}$, for $1\le i,j\le n$. Denote by $s_1^{(m)}\ge ...\ge s_n^{(m)}$ 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 $$ \mathcal F_n^{(m)}(x)=\frac1n\sum_{k=1}^nI_{\{s_k^{(m)}^2\le x\}}, $$ where $I_{\{B\}}$ denotes the indicator of an event $B$. We prove that under a Lindeberg condition for the fourth moment that the expected spectral distribution $F_n^{(m)}(x)=\E \mathcal F_n^{(m)}(x)$ converges to the distribution function $G^{(m)}(x)$ defined by its moments $${\alpha }_k(m):={\int }_{\mathbb{R}}{x}^{k}dG(x)=\frac{1}{mk+1}\left(\genfrac{}{}{0ex}{}{km+k}{k}\right).$$