# Lower-bound probability of non-centered quadratic form

Let $$X\sim N(\mu,\sigma^2I)\in \mathbb{R}^n$$ be a non-centered ($$\mu\neq 0$$) Gaussian vector with independent coordinates. I'm wondering if there is any sharp lower bound of the following probability: $$\mathbb{P}\left(X^TAX - \mathbb{E}(X^TAX) \geq \delta\right)$$ where $$A$$ is a symmetric matrix and $$\delta>0$$.

I know that when $$X$$ is centered, the upper bound is well-known, which is the Hanson-Wright inequality. However, in my case, $$X$$ is not centered and I'm interested in the lower bound.

• It seems like you are interested in anti-concentration inequalities, or small ball probabilities. This paper might be useful: arxiv.org/abs/1507.00829. – Ankitp Feb 15 '19 at 13:28

Yeah you are looking for the anti concentration inequality for Gaussian polynomials. For degree $$d$$ polynomials the probability that you lie in an $$\epsilon$$ interval is upper bounded by $$O(d\epsilon^{1/d})$$. Check out Carbery Wright 2001.