Let $a_1, \ldots a_n$ be i.i.d random variables that take the value $1$ and $-1$ with equal probability. Fix a unit vector $(x_1,\ldots, x_n)$ (such that $\sum_j x_j^2 = 1 $).
Consider $v = \langle a, x\rangle$. It's easy to see that $E[v] = 0$. It's also easy to bound Pr(|v| > t) via standard Hoeffding bounds.
What I'm looking for is a more accurate analysis of the distribution of $|v|$. Heuristically, it seems that $E[|v|]$ should be around $\sqrt{n}$, and what I'd like are concentration bounds around $E[|v|]$ (specifically, bounds on lower and upper tails). My feeling is that this is either easy, well known, or both :), and I'm wondering if there's a quick reference or argument.