  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.
