Limiting spectral distribution for large sample covariance matrices with graph-dependent elements

TL;DR

This paper derives the limiting spectral distribution for large sample covariance matrices with graph-dependent entries, providing conditions under which the Marchenko-Pastur law applies when dependencies grow with sample size.

Contribution

It extends spectral distribution results to covariance matrices with graph-dependent entries where interdependence increases with sample size.

Findings

Derived the limiting spectral distribution for graph-dependent covariance matrices

Established necessary and sufficient conditions for Marchenko-Pastur law in this setting

Provided tight bounds and conditions for m-dependent orthonormal elements

Abstract

We obtain the limiting spectral distribution for large sample covariance matrices associated with random vectors having graph-dependent entries under the assumption that the interdependence among the entries grows with the sample size n. Our results are tight. In particular, they give necessary and sufficient conditions for the Marchenko-Pastur theorem for sample covariance matrices with m-dependent orthonormal elements when m = o(n).