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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.

The empirical characteristic function replaces the expected value in a characteristic function by the sample average:Its centered fluctuation at the central limit theorem scale is the empirical characteristic process:In particular, ${\phi }_n(0)=1$ and ${\mathcal{C}}_n(0)=0$.