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 ${\mu }_1(M_n),{\mu }_2(M_n),...,{\mu }_n(M_n)$ 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 $n\to \infty$ 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

References in corpus (5)

• Central limit theorem for linear eigenvalue statistics of random matrices with independent entries
• Central Limit Theorem for linear eigenvalue statistics of the Wigner and sample covariance random matrices
• Regularity conditions in the CLT for linear eigenvalue statistics of Wigner matrices
• Central limit theorem for partial linear eigenvalue statistics of Wigner matrices
• Local Circular Law for Random Matrices