Anti-concentration with respect to random permutations

Classical anti-concentration results focus on the random sum $S:={\sum }_{i=1}^n{\xi }_i{v}_i$, where ${\xi }_i$ are independent random variables and $v_i$ are real numbers. In this paper, we prove new concentration results concerning the random sum $S:={\sum }_{i=1}^n{w}_{{\pi }_i}{v}_i$, where $w_i,v_i$ are real numbers and $\pi$ is a random permutation.