In proving the Jonhson-Lindenstrauss Lemma with $\mathrm{Uniform[-1,1]}$ random projections I used the following fact which I have been trying to prove:
$$ \mathbb{E}[e^{\lambda(3Z(w)^2-1)}] \leq e^{2\lambda^2}, \quad \forall \ \lambda \leq \frac{1}{4}$$
Where $Z(w)$ is defined as: $Z(w) = \sum_{j=1}^p a_j w_j$, with $w = (w_1, \dots, w_p) \in \mathbb{R}^p$ being a unit vector and $a_j$ ~ $\mathrm{Uniform}[-1,1]$ i.i.d for $j = 1,\dots, p$.
I have also been given the following facts which "may be useful" to prove this:
- For all $x \leq 0$, $\quad e^x \leq 1 + x + \frac{x^2}{2}$
- For all $x > 0$, $\quad 1 + x \leq e^x$
- If $\zeta$ ~ $\mathcal{N}(0,1)$ then $\mathbb{E}[\zeta^{2\ell}] = \frac{(2\ell)!}{2^\ell \ell!}$ and $\mathbb{E}[\zeta^{2\ell -1}] = 0$ for all $\ell \in \mathbb{N}$
This is quite similar to the problem where we need to prove the same bound holds when each $a_j$ ~ $\mathcal{N}(0,1)$ (and the expression is $\mathbb{E}[e^{\lambda(Z(w)^2 - 1)}]$ rather than $\mathbb{E}[e^{\lambda(3Z(w)^2 - 1)}]$), which is fairly straightforward to prove but I have been unable to modify the proof for the case when $a_j$ ~ $\mathrm{Uniform}[-1,1]$, since to my knowledge the density of the sum of uniform random variables does not have a nice expression like with Normal random variables. I have tried to find some sort of upper bound for the density of this sum by a normal density but to no avail. I also have tried using Hoeffding's Lemma but this doesn't seem to work either. Does anyone have any clue how to prove this?
It should also be noted that $\mathbb{E}[Z(w)^2] = \frac{1}{3}$, if this is of any use.