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

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 Prüfer 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 $p\in (1,+\infty )$.

the title has been changed for this version, which was 'Normality of Circular $\beta$-ensemble'

References in corpus (1)

• Uniform point variance bounds in classical beta ensembles

Cited by in corpus (1)

• Pair Dependent Linear Statistics for Circular Beta Ensemble