For independent Rademacher random variables $\epsilon\_i, i=1,2, \cdots, n$, i.e. $P(\epsilon\_i=-1)=P(\epsilon\_i=1)=\frac{1}{2}$, do we have 


$$max\_{0\le a\_i\le b, i=1,2,\cdots,n}P(|\sum\_{i=1}^n\epsilon\_ia\_i^2|>x)\le max\_{0\le a\_i\le b, i=1,2,\cdots,n}P(|\sum\_{i=1}^n\epsilon\_ia\_i|>x), \forall x>0,$$
where $0\le b<1$.



a simple probability inequality
No. Let $a\_1 = 1, a\_2 = 1/2, x=2/3$. 


$P(|\epsilon\_1 + 1/4 \epsilon\_2| > 2/3) = 1$


$P(|\epsilon\_1 + 1/2 \epsilon\_2| > 2/3) = 1/2$


