Convergence rate of eigenvector empirical spectral distribution of large Wigner matrices

TL;DR

This paper analyzes the convergence rates of the eigenvector empirical spectral distribution (VESD) of large Wigner matrices, establishing optimal bounds and demonstrating their independence from specific vectors through numerical studies.

Contribution

It derives the optimal convergence bounds for VESD of Wigner matrices to the semicircle law, under finite moment conditions, and shows these bounds are independent of the choice of vectors.

Findings

Expected VESD converges at rate O(n^{-1/2})

Convergence in probability at rate O(n^{-1/4})

Almost sure convergence at rate O(n^{-1/6})

Abstract

In this paper, we adopt the eigenvector empirical spectral distribution (VESD) to investigate the limiting behavior of eigenvectors of a large dimensional Wigner matrix W_n. In particular, we derive the optimal bound for the rate of convergence of the expected VESD of W_n to the semicircle law, which is of order O(n^{-1/2}) under the assumption of having finite 10th moment. We further show that the convergence rates in probability and almost surely of the VESD are O(n^{-1/4}) and O(n^{-1/6}), respectively, under finite 8th moment condition. Numerical studies demonstrate that the convergence rate does not depend on the choice of unit vector involved in the VESD function, and the best possible bound for the rate of convergence of the VESD is of order O(n^{-1/2}).