# Inequality for square of the subgaussian distributions

Hi all,

For my research I am trying to bound some exponential moments of subgaussian r.v.’s. And I am stuck with proving one of such inequalities. More specifically:

Let $a$ be unit vector in $\mathcal{R}^{{n}}$ and $w_{{i}}$, $i=1,2,...,n$, be $n$ i.i.d *Rademacher* rv’s. Also let $v=\sum_{{i}}^{{n}}a_{{i}}w_{{i}}$. I know that $\forall 0, where $z$ is standard normal r.v. and independent of $w_{{i}}$’s.

Now my question is: would this inequality also works if we change the sign on $t$? i.e.:

 $\forall t>0,\;{\mathbb{E}}(e^{{-tv^{{2}}}})\leq{\mathbb{E}}(e^{{-tz^{{2}}}})$

I have run many numerical experiments and it seems to be correct, but I am yet to prove it.

What I have done so far is as follows:

 ${\mathbb{E}_{v}}(e^{{-tv^{{2}}}})={\mathbb{E}_{{z}}}{\mathbb{E}_{{v}}}(e^{{i% \sqrt{2t}vz}})={\mathbb{E}_{{z}}}\prod_{{i=1}}^{{n}}{\mathbb{E}_{{w_{{i}}}}}(e% ^{{i\sqrt{2t}a_{{i}}w_{{i}}z}})={\mathbb{E}_{{z}}}\prod_{{i=1}}^{{n}}cos(\sqrt% {2t}a_{{i}}z)$ (1)

but I am stuck here (not even sure if what I have done is going to get me anywhere at all). This must be something that someone out there should know about, I am hoping.

Any help, suggestion or pointers would be greatly appreciate it.