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

In this paper we consider the product of two independent random matrices ${\mathbb{X}}^{(1)}$ and ${\mathbb{X}}^{(2)}$. Assume that $X_{jk}^{(q)}, 1 \le j,k \le n, q = 1, 2,$arei\mathrm{.}i\mathrm{.}d\mathrm{.}randomvariableswith$\mathbb E X_{jk}^{(q)} = 0, \mathbb E (X_{jk}^{(q)})^2 = 1$\mathrm{.}Denoteby$s_1, …, s_n$ 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 $s_1^2,\mathrm{.}\mathrm{.}\mathrm{.},s_n^2$ showing that the limiting variance depends on $\kappa_4: = \mathbb E (X_{11}^{1})^4 - 3$.