Local Laws for Sparse Sample Covariance Matrices Without the Truncation Condition

We consider sparse sample covariance matrices \(\frac{1}{n{p}_{n}}{\mathbf{X}\mathbf{X}}^{*},\) $\frac{1}{n{p}_n}\mathbf{X}\mathbf{X}_{}^{\ast },$ where X is a sparse matrix of order n × m with the sparse probability p_n. We prove the local Marchenko–Pastur law in some complex domain assuming that np_n > log^β n, β > 0, and some (4+δ)-moment condition is fulfilled, δ > 0.