Normality of Circular $beta$-ensemble

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. We will prove the main result by estimating the characteristic functions of the Prufer phases and the number counting function, which will imply the 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 $pin(1,+infty)$.