Some explorations on two conjectures about Rademacher sequences

Abstract

In this paper, we explore two conjectures about Rademacher sequences. Let $({\epsilon }_i)$ be a Rademacher sequence, i.e., a sequence of independent $\{-1,1\}$-valued symmetric random variables. Set $S_n=a_1{\epsilon }_1+\cdots +a_n{\epsilon }_n$ for $a=(a_1,\dots ,a_n)\in {\mathbb{R}}^n$. The first conjecture says that $P(|S_n|\le \Vert a\Vert )\ge \frac{1}{2}$ for all $a\in {\mathbb{R}}^n$ and $n\in \mathbb{N}$. The second conjecture says that $P(|S_n|\ge \Vert a\Vert )\ge \frac{7}{32}$ for all $a\in {\mathbb{R}}^n$ and $n\in \mathbb{N}$. Regarding the first conjecture, we present several new equivalent formulations. These include a topological view, a combinatorial version and a strengthened version of the conjecture. Regarding the second conjecture, we prove that it holds true when $n\le 7$.

19 pages

References in corpus (1)

• On a sum of centered random variables with nonreducing variances