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For real $v$, $\overline{\phi (v)}=\phi (-v)$. Expand the complex covariance of the sample averages. Terms indexed by distinct observations vanish because the observations are independent; each diagonal term equals $\mathbb{E}{e}^{i(u-v)X_1}-\phi (u)\overline{\phi (v)}$. HenceTaking $v=u$ and using $\phi (0)=1$ gives the exact variance:The upper bound follows from $|\phi (u)|\le \mathbb{E}|e^{iu{X}_1}|=1$, and holds without any moment assumptions on $X_1$.

Fix $u$. Put $m=\phi (u)$ and form the centered real vectorsThey are bounded independent and identically distributed random variables, so the multivariate central limit theorem applies to $n^{-1/2}{\sum }_j{Y}_j=(\mathrm{Re}{\mathcal{C}}_n(u),\mathrm{Im}{\mathcal{C}}_n(u){)}^T$.

To identify its limiting covariance matrix, put $Z=e^{iu{X}_1}-m$. Its complex covariance and pseudo-covariance are respectivelyFor $Z=A+iB$, $|Z{|}^2=A^2+B^2$ and $Z^2=A^2-B^2+2iAB$. ThusThe two conditions on $\Gamma (u)$ identify exactly this real covariance matrix: the second is ${\mathrm{Cov}}_{\mathbb{C}}(\Gamma (u),\overline{\Gamma (u)})=\mathbb{E}\Gamma (u{)}^2$, with a conjugate in its second argument. A centered multivariate normal distribution is determined by its covariance matrix, including when that matrix is singular. ThereforeThis proves convergence in distribution at each fixed $u$; it does not assert convergence of the entire empirical characteristic process.