Lower bounds on the smallest eigenvalue of a sample covariance matrix

TL;DR

This paper establishes tight lower bounds on the smallest eigenvalue of sample covariance matrices for centered isotropic random vectors, even with minimal assumptions on their components.

Contribution

It provides novel, tight lower bounds on the smallest eigenvalue under weak or no assumptions, advancing understanding of covariance matrix behavior.

Findings

Derived tight lower bounds for the smallest eigenvalue.

Applicable under weak/no assumptions on vector components.

Enhances theoretical understanding of covariance matrices.

Abstract

We derive tight lower bounds on the smallest eigenvalue of a sample covariance matrix of a centred isotropic random vector under weak or no assumptions on its components.