# Growth rate of exponential sum of $S_j$

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 $$\sqrt{n}$$.

• 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. Oct 27, 2020 at 0:24
• 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 $\frac{1}{\sqrt{n}}$.
– Mini
Oct 27, 2020 at 9:39