Local Laws for Sparse Sample Covariance Matrices without the truncation condition

TL;DR

This paper establishes the local Marchenko--Pastur law for sparse sample covariance matrices under certain sparsity and moment conditions, extending understanding without requiring truncation assumptions.

Contribution

It proves the local Marchenko--Pastur law for sparse covariance matrices with minimal sparsity and moment conditions, removing the need for truncation.

Findings

Validates the local Marchenko--Pastur law under specified conditions

Extends results to sparser matrices without truncation

Provides theoretical foundation for sparse covariance matrix analysis

Abstract

We consider sparse sample covariance matrices $\frac1{np_n}\mathbf X\mathbf X^*$, 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 $np_n>\log^{\beta}n$, $\beta>0$ and some $(4+\delta)$-moment condition is fulfilled, $\delta>0$.