Let $X_j$, $1 \leq j \leq n$ be chosen i.i.d. uniform over $[0,2\pi)$. Denote $S_j \triangleq X_1 +X_2+\cdots+X_j$ and suppose that $c_j$, $1 \leq j \leq n$ are some constants such that $|c_j|=1$.

I'm interested in the growth rate of $$g(n)=\sum \limits_{j=1}^n c_je^{i\frac{S_j}{n^{\alpha}}},$$ where $i$ is the imaginary number and $\alpha$ is a fixed number $0 <\alpha<1$. Is it of order $\sqrt{n}$?

Maybe the simple case to start from is when $c_j=1$. The simulations suggest the order $\sqrt{n}$.