Local Laws for Sparse Sample Covariance Matrices

Abstract

We proved the local Marchenko–Pastur law for sparse sample covariance matrices that corresponded to rectangular observation matrices of order n×m with n/m→y (where y>0) and sparse probability npn>logβn (where β>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∈D with Imz>v0>0 (where v0) were of order log4n/n and the domain bounds did not depend on pn while npn>logβn.

Keywords

• sparse sample covariance matrices
• local Marchenko–Pastur law
• Stieltjes transformation