Asymptotic distribution of singular values of powers of random matrices

Let $x$ be a complex random variable such that $\text{E}x=0$, $\text{E}\mathrm{\mid }x{\mathrm{\mid }}^2=1$, $\text{E}\mathrm{\mid }x{\mathrm{\mid }}^4<\mathrm{\infty }$. Let $x_{ij}$, $i,j\in \{1,2,\mathrm{.}\mathrm{.}\mathrm{.}\}$ be independet copies of $x$. Let $\text{Xb}=(N^{-1\mathrm{/}2}{x}_{ij})$, $1\le i,j\le N$ be a random matrix. Writing ${\text{Xb}}^{\ast }$ for the adjoint matrix of $\text{Xb}$, consider the product ${\text{Xb}}^m{{\text{Xb}}^{\ast }}^m$ with some $m\in \{1,2,\mathrm{.}\mathrm{.}\mathrm{.}\}$. The matrix ${\text{Xb}}^m{{\text{Xb}}^{\ast }}^m$ is Hermitian positive semi-definite. Let ${\lambda }_1,{\lambda }_2,\mathrm{.}\mathrm{.}\mathrm{.},{\lambda }_N$ be eigenvalues of ${\text{Xb}}^m{{\text{Xb}}^{\ast }}^m$ (or squared singular values of the matrix ${\text{Xb}}^m$). In this paper we find the asymptotic distribution function $$G^{(m)}(x)=\underset{N\to \mathrm{\infty }}{\lim }\text{E}{F}_N^{(m)}(x)$$ of the empirical distribution function $$F_N^{(m)}(x)=N^{-1}\sum _{k=1}^N\mathbb{I}\{{\lambda }_k\le x\},$$ where $\mathbb{I}\{A\}$ stands for the indicator function of event $A$. The moments of $G^{(m)}$ satisfy $$M_p^{(m)}={\int }_{\mathbb{R}}{x}^{p}d{G}^{(m)}(x)=\frac{1}{mp+1}\left(\genfrac{}{}{0px}{}{mp+p}{p}\right)\mathrm{.}$$ In Free Probability Theory $M_p^{(m)}$ are known as Fuss–Catalan numbers. With $m=1$ our result turns to a well known result of Marchenko–Pastur 1967.