Central limit theorem for partial linear eigenvalue statistics of Wigner matrices

Abstract

In this paper, we study the complex Wigner matrices $M_n=\frac{1}{\sqrt{n}}{W}_n$ whose eigenvalues are typically in the interval $[-2,2]$. Let ${\lambda }_1\le {\lambda }_2...\le {\lambda }_n$ be the ordered eigenvalues of $M_n$. Under the assumption of four matching moments with the Gaussian Unitary Ensemble(GUE), for test function $f$ 4-times continuously differentiable on an open interval including $[-2,2]$, we establish central limit theorems for two types of partial linear statistics of the eigenvalues. The first type is defined with a threshold $u$ in the bulk of the Wigner semicircle law as ${\mathcal{A}}_n[f;u]={\sum }_{l=1}^{n}f({\lambda }_l)1_{\{{\lambda }_l\le u\}}$. And the second one is ${\mathcal{B}}_n[f;k]={\sum }_{l=1}^{k}f({\lambda }_l)$ with positive integer $k=k_n$ such that $k/n\to y\in (0,1)$ as $n$ tends to infinity. Moreover, we derive a weak convergence result for a partial sum process constructed from ${\mathcal{B}}_n[f;\lfloor nt\rfloor ]$.

39 pages

References in corpus (5)

• Central limit theorem for linear eigenvalue statistics of random matrices with independent entries
• Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices
• Central Limit Theorem for linear eigenvalue statistics of the Wigner and sample covariance random matrices
• Random matrices: Sharp concentration of eigenvalues
• A central limit theorem for the determinant of a Wigner matrix

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• Central Limit Theorem for Linear Statistics of Eigenvalues of Band Random Matrices
• Convergence rate of eigenvector empirical spectral distribution of large Wigner matrices