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

We consider powers of random matrices with independent entries. Let $X_{ij}, i,j\ge 1$,beindependentcomplexrandomvariableswith$\E X_{ij}=0$and$\E |X_{ij}|^2=1$andlet$\mathbf X$denotean$n\times n$matrixwith$[\mathbf X]{ij}=X{ij}$,for$1\le i, j\le n$\mathrm{.}Denoteby$s_1^{(m)}\ge…\ge s_n^{(m)}$thesingularvaluesoftherandommatrix$\mathbf W:={n^{-\frac m2}} \mathbf X^m$anddefinetheempiricaldistributionofthesquaredsingularvaluesby$${\mathcal{F}}_n^{(m)}(x)=\frac{1}{n}{\sum }_{k=1}^n{I}_{\{{s_k^{(m)}}^2\le x\}},$$where$I_{{B}}$denotestheindicatorofanevent$B$. We prove that under a Lindeberg condition for the fourth moment that the expected spectral distribution $F_n^{(m)}(x)=\text{E}{\mathcal{F}}_n^{(m)}(x)$ converges to the distribution function $G^{(m)}(x)$ defined by its moments $${\alpha }_k(m):={\int }_{\mathbb{R}}{x}^{k}dG(x)=\frac{1}{mk+1}\left(\genfrac{}{}{0px}{}{km+k}{k}\right)\mathrm{.}$$