Local Laws for Sparse Sample Covariance Matrices

We proved the local Marchenko-Pastur law for sparse sample covariance matrices that corresponded to rectangular observation matrices of order n x m with n/m -> y (where y > 0) and sparse probability np(n) > log(beta) n (where beta > 0). The bounds of the distance between the empirical spectral distribution function of the sparse sample covariance matrices and the Marchenko-Pastur law distribution function that was obtained in the complex domain z is an element of D with Im z > v(0) > 0 (where v(0)) were of order log(4) n/n and the domain bounds did not depend on p(n) while np(n) > log(beta) n.