Empirical characteristic process (${\mathcal{C}}_n(u)$)

The centered process ${\mathcal{C}}_n(u)=\sqrt{n}({\phi }_n(u)-\phi (u))$ describes fluctuations of the empirical characteristic function. At a fixed frequency, its real and imaginary parts satisfy a multivariate central limit theorem. A process-level limit requires further control over the frequency index.