= Solution
For real $v$, $\overline{\varphi(v)}=\varphi(-v)$. Expand the <complex covariance> of the sample averages. Terms indexed by distinct observations vanish because the observations are <independent>; each diagonal term equals $\mathbb E e^{i(u-v)X_1}-\varphi(u)\overline{\varphi(v)}$. Hence
$$
\boxed{\operatorname{Cov}_{\mathbb C}\bigl(\varphi_n(u),\varphi_n(v)\bigr)
=\frac1n\bigl(\varphi(u-v)-\varphi(u)\varphi(-v)\bigr).}
$$
Taking $v=u$ and using $\varphi(0)=1$ gives the exact <variance>:
$$
\boxed{\operatorname{Var}_{\mathbb C}(\varphi_n(u))
=\frac{1-|\varphi(u)|^2}{n}\leq\frac1n.}
$$
The upper bound follows from $|\varphi(u)|\leq\mathbb E|e^{iuX_1}|=1$, and holds without any moment assumptions on $X_1$.
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