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

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 $X_1$.