Semicircle Law for Hadamard Products

In this paper, assuming p/n → 0 as n → ∞, 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 Ap = 1 √ np (Xm,nX ∗ m,n − σnIm) ◦ Dm, where the matrices Xm,n and Dm are independent and the entries of Xm,n (m × n) are independent, the matrix Dm (m ×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 Dm, and “◦” denotes the Hadamard product of matrices.