Semicircle Law for Hadamard Products

In this paper, assuming $p/n\rightarrow 0$ as $n\rightarrow\infty$, we will prove the weak and strong convergence to the semicircle law of the empirical spectral distribution of the Hadamard product of a normalized sample covariance matrix and a sparsing matrix, which is of the form $A_p=\frac{1}{\sqrt{np}}(X_{m,n}X_{m,n}^*-\sigma^2nI_m)\circ D_{m}$, where the matrices $X_{m,n}$ and $D_m$ are independent and the entries of $X_{m,n}$ $(m\times n)$ are independent, the matrix $D_{m}$ $(m\times m)$ is Hermitian with independent entries above and on the diagonal, $p$ is the sum of the second moments of the row (and column) entries of $D_m$, and “$\circ$” denotes the Hadamard product of matrices.