  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 $\mathbf W:={n^{-\frac m2}} \mathbf
X^m$ 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 $$
\alpha_k(m):=\int_{\mathbb R}x^k\,d\,G(x)=\frac {1}{mk+1}\binom{km+k}{k}. $$
