Semicircle law for Hadamard products

Abstract

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(X m, nXm, n - σ2nIm) o D m, 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 "o" denotes the Hadamard product of matrices. © 2007 Society for Industrial and Applied Mathematics.