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

Let $({\epsilon }_i)$ be a Rademacher sequence, i.e., a sequence of independent and identically distributed random variables satisfying $P({\epsilon }_i=1)=P({\epsilon }_i=-1)=1\mathrm{/}2$. 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 $P(\mid S_n\mid \ge \mathrm{\parallel }a\mathrm{\parallel })\ge 7\mathrm{/}32$ for all $a\in {\mathbb{R}}^n$ and $n\in \mathbb{N}$. Up to now, we know that it holds when $n\le 7$. In this note, we show that it holds when $n=8$.