Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 209 / 2 / a / Solution

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 ${\mathcal{C}}_n(0)=0$.