Distribution of Linear Statistics of Singular Values of the Product of Random Matrices

Abstract

In this paper we consider the product of two independent random matrices ${\mathbb{X}}^{(1)}$ and ${\mathbb{X}}^{(2)}$. Assume that $X_{jk}^{(q)},1\le j,k\le n,q=1,2,$ are i.i.d. random variables with $\mathbb{E}{X}_{jk}^{(q)}=0,\mathbb{E}(X_{jk}^{(q)}{)}^2=1$. Denote by $s_1,...,s_n$ the singular values of $\mathbb{W}:=\frac{1}{n}{\mathbb{X}}^{(1)}{\mathbb{X}}^{(2)}$. We prove the central limit theorem for linear statistics of the squared singular values $s_1^2,...,s_n^2$ showing that the limiting variance depends on ${\kappa }_4:=\mathbb{E}(X_{11}^1{)}^4-3$.

arXiv comment40 pages