Probability Mass of Rademacher Sums Beyond One Standard Deviation

Abstract

Let $a_1,\dots ,a_n\in \mathbb{R}$ satisfy ${\sum }_i{a}_i^2=1$, and let ${\epsilon }_1,\dots ,{\epsilon }_n$ be uniformly random $\pm 1$ signs and $X={\sum }_{i=1}^n{a}_i{\epsilon }_i$. It is conjectured that $X={\sum }_{i=1}^n{a}_i{\epsilon }_i$ has $\mathrm{Pr}[X\ge 1]\ge 7/64$. The best lower bound so far is $1/20$, due to Oleszkiewicz. In this paper we improve this to $\mathrm{Pr}[X\ge 1]\ge 6/64$.

arXiv comment43 pages; minor changes