On the overestimation of the largest eigenvalue of a covariance matrix

TL;DR

This paper demonstrates that in high-dimensional settings, the sample covariance matrix's largest eigenvalue consistently overestimates the true value, with similar results for the smallest eigenvalue, using a novel proof approach.

Contribution

Introduces a new proof method showing the almost sure overestimation of the largest eigenvalue of sample covariance matrices in infinite dimensions.

Findings

Largest eigenvalue overestimation with probability 1 in infinite dimensions

Similar overestimation result for the smallest eigenvalue

Provides a new proof technique for eigenvalue analysis

Abstract

In this paper, we use a new approach to prove that the largest eigenvalue of the sample covariance matrix of a normally distributed vector is bigger than the true largest eigenvalue with probability 1 when the dimension is infinite. We prove a similar result for the smallest eigenvalue.