A Note on Rate of Convergence in Probability to Semicircular Law

TL;DR

This paper establishes that for Wigner matrices with finite sixth moments, the empirical spectral distribution converges to the semicircular law at a rate of O(n^{-1/2}) in probability as the matrix size increases.

Contribution

It proves a specific convergence rate for the spectral distribution of Wigner matrices under finite sixth moment assumptions.

Findings

Convergence rate of O(n^{-1/2}) in probability for spectral distribution.

Finite sixth moment condition is sufficient for this convergence rate.

Results apply as matrix dimension tends to infinity.

Abstract

In the present paper, we prove that under the assumption of the finite sixth moment for elements of a Wigner matrix, the convergence rate of its empirical spectral distribution to the Wigner semicircular law in probability is $O(n^{-1/2})$ when the dimension $n$ tends to infinity.