Linear statistics of the circular $\beta $-ensemble, Stein’s method, and circular Dyson Brownian motion

Abstract

We study the linear statistics of the circular $\beta $-ensemble with a Stein’s method argument, where the exchangeable pair is generated through circular Dyson Brownian motion. This generalizes previous results obtained in such a way for the CUE and provides a novel approach for studying linear statistics of $\beta $-ensembles. This approach allows studying simultaneously a collection of linear statistics whose number grows with the dimension of the ensemble. Also this approach requires estimating only low order moments of the linear statistics.