Universality results for largest eigenvalues of some sample covariance matrix ensembles

TL;DR

This paper extends universality results for the largest eigenvalues of sample covariance matrices with iid sub-Gaussian entries to cases where the ratio of samples to variables approaches any finite value, infinity, or zero.

Contribution

It generalizes existing universality results to a broader range of ratios between samples and variables in sample covariance matrices.

Findings

Largest eigenvalue distribution matches Gaussian case for various ratios.

Universality holds when ratio approaches any finite value.

Results include cases where ratio tends to infinity or zero.

Abstract

For sample covariance matrices with iid entries with sub-Gaussian tails, when both the number of samples and the number of variables become large and the ratio approaches to one, it is a well-known result of A. Soshnikov that the limiting distribution of the largest eigenvalue is same as the of Gaussian samples. In this paper, we extend this result to two cases. The first case is when the ratio approaches to an arbitrary finite value. The second case is when the ratio becomes infinity or arbitrarily small.