Partial Linear Eigenvalue Statistics for Wigner and Sample Covariance Random Matrices

TL;DR

This paper investigates the asymptotic fluctuations of partial sums of eigenvalues for Wigner and sample covariance matrices, extending understanding of eigenvalue statistics in large random matrices.

Contribution

It introduces new results on the fluctuation behavior of partial linear eigenvalue statistics for Wigner and sample covariance matrices, covering fixed and growing truncation sizes.

Findings

Fluctuations characterized for fixed k.

Results extend to k growing with n.

Provides insights into eigenvalue distribution behavior.

Abstract

Let $M_n$ be a $n \times n$ Wigner or sample covariance random matrix, and let $\mu_1(M_n), \mu_2(M_n), ..., \mu_n(M_n)$ denote the unordered eigenvalues of $M_n$. We study the fluctuations of the partial linear eigenvalue statistics $$ \sum_{i=1}^{n-k} f(\mu_i(M_n)) $$ as $n \rightarrow \infty$ for sufficiently nice test functions $f$. We consider both the case when $k$ is fixed and when $\min{k,n-k}$ tends to infinity with $n$.