Tail sums of Wishart and GUE eigenvalues beyond the bulk edge

TL;DR

This paper provides quantitative bounds on the tail sums of eigenvalues beyond the bulk edge for Wishart and GUE ensembles, with implications for high-dimensional covariance matrix estimation.

Contribution

It introduces new bounds on eigenvalue tail sums outside the bulk edge for Wishart and GUE ensembles, aiding high-dimensional covariance analysis.

Findings

Expected number of eigenvalues outside the bulk edge is less than one.

Tail sums of eigenvalues are quantitatively bounded.

Results are independent of the aspect ratio in the Wishart case.

Abstract

Consider the classical Gaussian unitary ensemble of size $N$ and the real Wishart ensemble $W_N(n,I)$. In the limits as $N \to \infty$ and $N/n \to \gamma > 0$, the expected number of eigenvalues that exit the upper bulk edge is less than one, 0.031 and 0.170 respectively, the latter number being independent of $\gamma$. These statements are consequences of quantitative bounds on tail sums of eigenvalues outside the bulk which are established here for applications in high dimensional covariance matrix estimation.