Suppose that $X_1,\ldots,X_n$ are iid random variables such that $\operatorname EX_1=0$ and $\operatorname E|X_1|^s<\infty$ with some $s>2$. I am interested in the growth rate of $$ \operatorname E\max_{1\le j\le q}\Bigl|n^{-1/2}\sum_{t=1}^nX_te^{-it\omega_j}\Bigr|^2 $$ as $n\to\infty$, where $q=\lfloor(n-1)/2\rfloor$, $i=\sqrt{-1}$ and $\omega_j=2\pi j/n$ with $1\le j\le q$.
Is it possible to assume that $X_t$'s are symmetric without loss of generality if I want to establish that the expected value is, say, $O(a_n)$ as $n\to\infty$, where $a_n\to\infty$ as $n\to\infty$?
Suppose that $X_1',\ldots,X_n'$ is an independent copy of $X_1,\ldots,X_n$. We have that \begin{align*} \operatorname E\max_{1\le j\le q}\Bigl|n^{-1/2}\sum_{t=1}^nX_te^{-it\omega_j}\Bigr|^2 &\le2\operatorname E\max_{1\le j\le q}\Bigl|n^{-1/2}\sum_{t=1}^n(X_t-X_t')e^{-it\omega_j}\Bigr|^2+2\operatorname E\max_{1\le j\le q}\Bigl|n^{-1/2}\sum_{t=1}^nX_t'e^{-it\omega_j}\Bigr|^2, \end{align*} but this does not lead anywhere.
Any help is much appreciated!