# Lower bounds on random process

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!

$$\newcommand\ep\epsilon\newcommand\v{\mathbf v}$$Conditioning on the $$\ep_i$$'s and using Jensen's and Szarek's 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, $$E\Big|\sum_1^n\ep_i\|\v_i\|_2^2\Big| \ge \frac{\sqrt n}2.$$
This lower bound is of course sharp, up a universal constant factor, because $$E\Big|\sum_1^n\ep_i\|\v_i\|_2^2\Big| \le\sqrt{E\Big(\sum_1^n\ep_i\|\v_i\|_2^2\Big)^2} =\sqrt{\sum_1^n E\|\v_i\|_2^4} = \sqrt{n E\|\v_1\|_2^4} \le\sqrt n.$$