= Solution
The <empirical characteristic function> replaces the <expected value> in a <characteristic function> by the sample average:
$$
\boxed{\varphi_n(u)=\frac1n\sum_{j=1}^n e^{iuX_j},\qquad u\in\mathbb R.}
$$
Its centered fluctuation at the <central limit theorem> scale is the <empirical characteristic process>:
$$
\boxed{\mathcal C_n(u)=\sqrt n\bigl(\varphi_n(u)-\varphi(u)\bigr),\qquad
\varphi(u)=\mathbb E e^{iuX_1}.}
$$
In particular, $\varphi_n(0)=1$ and $\mathcal C_n(0)=0$.
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