Let $\epsilon\_1,...,\epsilon\_n$ be i.i.d. random signs, $\mathbf{u}\_1,...,\mathbf{u}\_n$ i.i.d. uniform random vectors on the unit sphere $\mathbb{S}^{d-1}$, assuming $d$ even, and $\mathbf{v}\_1,...,\mathbf{v}\_n$ be their half-truncations, that is $\mathbf{v}\_i[j] = \mathbf{u}\_i[j]$ for all $j \in \left \{1,...,d/2\right\}$, else $\mathbf{v}\_i[j] = 0$, where $\mathbf{v}\_i[j]$ denotes the $j$-th entry of $\mathbf{v}\_i$. We assume that the $\epsilon\_i$'s are independent from the $\mathbf{u}\_i$'s.   

I am looking for nontrivial lower bounds on the following quantity.
$$\mathbb{E}\left |\sum\_{i = 1}^n \epsilon\_i \| \mathbf{v}\_i\|^2\_2 \right |$$
Any help would be greatly appreciated!



Lower bounds on random process
$\newcommand\ep\epsilon\newcommand\v{\mathbf v}$Conditioning on the $\ep\_i$'s and using Jensen's and [Szarek's](http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.bwnjournal-article-smv58i1p13bwm) inequalities, we have
$$E\Big|\sum\_1^n\ep\_i\|\v\_i\|\_2^2\Big|
\ge E\Big|\sum\_1^n\ep\_ic\_i\Big|\ge\frac1{\sqrt2}\sqrt{\sum\_1^nc\_i^2},$$
where
$$c\_i:=E\|\v\_i\|\_2^2=1/2,$$
by symmetry. So,
$$E\Big|\sum\_1^n\ep\_i\|\v\_i\|\_2^2\Big|
\ge \frac{\sqrt n}2.$$




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This lower bound is of course sharp, up a universal constant factor, because
$$E\Big|\sum\_1^n\ep\_i\|\v\_i\|\_2^2\Big|
\le\sqrt{E\Big(\sum\_1^n\ep\_i\|\v\_i\|\_2^2\Big)^2}
=\sqrt{\sum\_1^n E\|\v\_i\|\_2^4}
= \sqrt{n E\|\v\_1\|\_2^4}
\le\sqrt n.$$


