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 $n\times m$ with $n/m\to y$ (where $y>0$) and sparse probability $n{p}_n>{log}^{\beta }n$ (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 $z\in \mathcal{D}$ with $Imz>v_0>0$ (where $v_0$) were of order ${log}^{4}n/n$ and the domain bounds did not depend on $p_n$ while $n{p}_n>{log}^{\beta }n$.