Local Laws for Sparse Sample Covariance Matrices without the truncation condition

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