I am looking for good and explicit lower and upper bounds on the probability that $x^TGx\ge 0$, where $G$ is a symmetric matrix with zero trace and $x$ is a vector whose components are independent random variables uniformly distributed in either $[-1,1]$ or $\{-1,1\}$.
(What I really need are realistically general explicit sufficient conditions on $G\in R^{n\times n}$ for the probability to be $\ge p$ for some $p>0$ independent of the dimension $n$. Thus the bounds should be strong enough to deduce such conditions and to prove that the gap between sufficient and necessary is not too large.)
