A note on the Hitczenko-Kwapien conjecture about a Rademacher sequence

Abstract

Let $(\epsilon_i)$ be a Rademacher sequence, i.e., a sequence of independent and identically distributed random variables satisfying $P(\epsilon_i=1)=P(\epsilon_i=-1)=1/2$. Set $S_n=a_1\epsilon_1+\cdots+a_n\epsilon_n$ for $a=(a_1,\dots,a_n)\in \mathbb{R}^n$. The Hitczenko-Kwapien conjecture says that $P\left(\left|S_n\right|\geq\|a\|\right)\geq {7}/{32}$ for all $a\in \mathbb{R}^n$ and $n\in \mathbb{N}$. Up to now, we know that it holds when $n\leq 7$. In this note, we show that it holds when $n=8$.