Partial Linear Eigenvalue Statistics for Wigner and Sample Covariance Random Matrices

Let $M_n$ be a $n\times n$ Wigner or sample covariance random matrix, and let ${\mu }_1(M_n),{\mu }_2(M_n),\mathrm{.}\mathrm{.}\mathrm{.},{\mu }_n(M_n)$ 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 $n\to \mathrm{\infty }$ 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 $n$.