= Solution
Fix $u$. Put $m=\varphi(u)$ and form the centered real vectors
$$
Y_j=\begin{pmatrix}\cos(uX_j)-\operatorname{Re}m\\ \sin(uX_j)-\operatorname{Im}m\end{pmatrix}.
$$
They are bounded <independent and identically distributed random variables>, so the <multivariate central limit theorem> applies to $n^{-1/2}\sum_jY_j=(\operatorname{Re}\mathcal C_n(u),\operatorname{Im}\mathcal C_n(u))^T$.
To identify its limiting <covariance matrix>, put $Z=e^{iuX_1}-m$. Its <complex covariance> and <pseudo-covariance> are respectively
$$
V=\mathbb E|Z|^2=1-|m|^2,\qquad
P=\mathbb E Z^2=\varphi(2u)-m^2.
$$
For $Z=A+iB$, $|Z|^2=A^2+B^2$ and $Z^2=A^2-B^2+2iAB$. Thus
$$
\boxed{\Sigma(u)=\frac12\begin{pmatrix}V+\operatorname{Re}P&\operatorname{Im}P\\
\operatorname{Im}P&V-\operatorname{Re}P\end{pmatrix}.}
$$
The two conditions on $\Gamma(u)$ identify exactly this real <covariance matrix>: the second is $\operatorname{Cov}_{\mathbb C}(\Gamma(u),\overline{\Gamma(u)})=\mathbb E\Gamma(u)^2$, with a conjugate in its second argument. A centered <multivariate normal distribution> is determined by its <covariance matrix>, including when that matrix is singular. Therefore
$$
\boxed{(\operatorname{Re}\mathcal C_n(u),\operatorname{Im}\mathcal C_n(u))
\xrightarrow{d}(\operatorname{Re}\Gamma(u),\operatorname{Im}\Gamma(u)).}
$$
This proves <convergence in distribution> at each fixed $u$; it does not assert convergence of the entire <empirical characteristic process>.
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