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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 $\sqrt{n}$.

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  • $\begingroup$ If you write down $g(n)\overline{g(n)}$ and expand, you obtain $g(n)\overline g(n)=n+2\sum_{j<k}\text{Re}\big(c_k\bar c_j\exp(\frac 1{n^\alpha}S_j^k)\big)$, where $S_j^k$ is $S_k-S_j$. If $X_i$ is mean 0, I don't think the variance of $g(n)$ is finite. For the case where the mean is positive, you're probably OK. $\endgroup$ Oct 27, 2020 at 0:24
  • $\begingroup$ The case of $\alpha=0$ is the common random walk and for this case it trivially holds. When $C_i=1$ also the simulations suggest that it holds also. Actually in this case, $\frac{1}{n}g(n)$ would have some drift around the curve $\frac{1}{\sqrt{n}}$. $\endgroup$
    – Mini
    Oct 27, 2020 at 9:39

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