Define $e_{st}=\int_{-\lambda}^\lambda e^{i(t-s)x}dx$ for some $\lambda\in(0,1)$. Do we have the result that $\sum_{s=1}^n\sum_{t=n+1}^{2n}e_{st}=o(\sqrt{n})$ or $\sum_{s=1}^n\sum_{t=n+1}^{2n}e_{st}=o(n)$?
This comes from the proof of Proposition 4.1 in Testing and estimating in the change-point problem of the spectral function by Giraitis and Leipus. But the original proof may contain some typos and skipped some steps. Any hint will be appreciated!