Partial Linear Eigenvalue Statistics for Wigner and Sample Covariance Random Matrices

Abstract

Let $M_n$ be a $n \times n$ Wigner or sample covariance random matrix, and let $μ_1(M_n), μ_2(M_n), ..., μ_n(M_n)$ denote the unordered eigenvalues of $M_n$. We study the fluctuations of the partial linear eigenvalue statistics $$ \sum_{i=1}^{n-k} f(μ_i(M_n)) $$ 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 $n$.

20 pages; incorporated the referee's comments and suggestions