Anti-concentration with respect to random permutations

TL;DR

This paper extends anti-concentration results to sums involving random permutations, providing new bounds for sums where the order of terms is randomized, which differs from traditional independent-variable models.

Contribution

It introduces novel anti-concentration bounds for sums with permuted weights, expanding the scope of classical results to permutation-based randomness.

Findings

New concentration bounds for permutation-based sums

Extension of classical anti-concentration results

Applicable to sums with permuted weights

Abstract

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.