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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.

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  • $\begingroup$ It seems like you are interested in anti-concentration inequalities, or small ball probabilities. This paper might be useful: arxiv.org/abs/1507.00829. $\endgroup$
    – Ankitp
    Commented Feb 15, 2019 at 13:28
  • $\begingroup$ Do you mean you want the infimum over all symmetric matrices $A$? If so, that way of using the word "where" is not the clearest use of language. $\endgroup$
    – Michael Hardy
    Commented Nov 20, 2021 at 4:42

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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.

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