The Berry-Esseen Theorem for Circular $\beta$-ensemble

TL;DR

This paper establishes the Berry-Esseen theorem for the circular beta-ensemble, leading to a central limit theorem for point counts on the circle and extending results to the Sine beta process.

Contribution

It proves the Berry-Esseen theorem for the circular beta-ensemble and derives a CLT for linear statistics, extending to the Sine beta process.

Findings

Berry-Esseen bounds for CβE point counts

Central limit theorem for arcs on the circle

Normality of linear statistics for certain test functions

Abstract

We will prove the Berry-Esseen theorem for the number counting function of the circular $\beta$-ensemble (C$\beta$E), which will imply the central limit theorem for the number of points in arcs of the unit circle in mesoscopic and macroscopic scales. We will prove the main result by estimating the characteristic functions of the Pr\"ufer phases and the number counting function, which will imply the uniform upper and lower bounds of their variance. We also show that the similar results hold for the Sine$_\beta$ process. As a direct application of the uniform variance bound, we can prove the normality of the linear statistics when the test function $f(\theta)\in W^{1,p}(S^1)$ for some $p\in(1,+\infty)$.