Abstract:
Consider $n=p_0\le p_1\le\cdots\le p_m$ for some fixed $m\ge 1$. Let $\mathbf X^{(\nu)}=\frac1{\sqrt {p_{\nu-1}}}(X_{jk}^{(\nu)})_{1\le j\le p_{\nu-1},\,1\le k\le p_{\nu}}$ be random matrices with mutually independent random entries $X_{jk}^{(\nu)}$ such that $\mathbf{E}\, X_{jk}=0$ and $\mathbf{E}\,|X_{jk}|^2=1$. Consider the matrix $\mathbf W=\prod_{\nu=1}^m\mathbf X^{(\nu)}$. Let $\mathbf{\Sigma}=\mathbf W\mathbf W^*$. Denote by $\mathcal F_n(x)$ the empirical spectral distribution of the matrix $\mathbf{\Sigma}$ and put $F_n(x)=\mathbf{E}\,\mathcal F_n(x)$.
Assuming that $\lim_{n\to\infty}\frac{p_{\nu}}n=y_{\nu}$ it is shown that $F_n(x)$ converges to the limit distribution $G(x)$ with Stieltjes transform $S(z)$ satisfying the equation
\begin{equation}\notag
1+zS(z)-S(z)\prod_{\nu=1}^{m}(1-y_{\nu}-y_{\nu}zS(z))=0.
\end{equation}
In the case $y_{\nu}=1$, for $\nu=1,\ldots,m$, the moments of the distribution $G(x)$ are Fuss–Catalan numbers with parameter $m$, $M_m(p)={{mp+p}\choose{p}}$.
Let $y_1=\cdots=y_m=1$. Denote by $\mu_n$ the empirical spectral measure of the matrix $\mathbf W$. It is shown that $\mu_n$ converges to the distribution on the unit disc on the complex plane with the density
$p(x,y)=\frac1{\pi m(x^2+y^2)^{\frac{m-1}m}}$ under assumptions that the entries $X_{jk}^{(\nu)}$ are i.i.d and have the finite second moment.
Furthermore, we shall discuss the limit distributions of spectrum of products of rectangular random matrices.
|
