Year: 2026
Type: preprint
Abstract: Let $\varepsilon_1,\ldots,\varepsilon_n$ be independent Rademacher random variables. We settle the best constant $c$ such that \[ \mathbb{E}\Bigl|\sum_{j=1}^n a_j\varepsilon_j\Bigr| \ge \frac{1}{\sqrt{2}} +c\Bigl|a-\frac{e_1+e_2}{\sqrt{2}}\Bigr| \] holds for every $n\ge 2$ and every unit vector $a\in\mathbb{R}^n$ with $a_1\ge\cdots\ge a_n\ge 0$. We prove that the optimal constant is $c=(2-\sqrt{2})^{3/2}/8$, with equality attained at $a=\frac{1}{2}(1,1,1,1,0,\ldots,0)$ for $n\ge 4$. This completes a line of research by De-Diakonikolas-Servedio, Eskenazis-Nayar-Tkocz and Fang-Wang.
Source: arXiv (Cornell University)
Author: Xie, Xinyuan
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221044908