# Hanson-Wright inequality with random matrix

I'm interested in bounding the tail probabilities of a quadratic form $$x^t A x$$ where $$x\in \mathbb{R}^n$$ is a sub-Gaussian vector with independent entries. $$A\in \mathbb{R}^{n\times n}$$ is a matrix. So I'm exactly in the setup of the Hanson-Wright inequality. In fact, I wish I could use it because if it would apply, it would give me exactly the bounds I'm looking for.

My problem is that in my case, the matrix $$A$$ is random, too. Worse even, I don't have independence of $$A$$ and $$x$$. However, there are two special properties in my case which can be used:

1. $$A$$ and $$x$$ are uncorrelated, i.e. $$\mathbb{E}[Ax]=\mathbb{E}[A]\mathbb{E}[\mathbb{x}]=\mathbb{E}[A]0=0$$.
2. $$A$$ is an orthogonal projection of rank $$r < n$$.

So my question is: Does somebody know a generalization of the Hanson-Wright inequality which would apply in this case?

[I am asking this question is because I study the finite-sample performance of OLS and other linear estimators. In the case of OLS, one can think of $$A$$ as orthogonal projection on the column space of the regressors and of $$x$$ as the error term. If the regressors were fixed, then Hanson-Wright would do the job immediately, but I need to allow for the regressors to be random, too.]

• Uncorrelatedness alone is most likely insufficient for saying much. Jan 28, 2020 at 6:33
• @Puzzler Nice! Upvoted. Similar question here mathoverflow.net/q/385586/78539 Mar 5, 2021 at 8:12

Let $$x \sim N(0,I_n)$$. For any independent rank-1 projection $$A$$, conditioned on $$A$$, we have $$x^T A x \sim \chi^2_1.$$ So unconditionally, $$x^T A x = O(1)$$ with high probability.
Now, let $$A = \frac{x x^T}{\|x\|_2^2}$$. Then, $$A$$ is a rank-1 projection and we have $$\mathbb E [A x] = \mathbb E[x] = 0 = \mathbb E[A] \mathbb E[x]$$. So, $$A$$ and $$x$$ are uncorrelated (in the sense stated in the question). But $$x^T A x = \frac{x^T x x^T x}{\|x\|_2^2} =\|x\|_2^2 \sim \chi_n^2$$ so $$x^T Ax \approx n$$ with high probability. (This rank-1 projection behaves like the full rank projection when applied to $$x$$.)
Assuming independence of $$A$$ and $$x$$, one can condition on $$A$$ and apply the Hanson--Wright inequality. Since the bound does not depend on $$A$$ (it only depends on $$\|A\|_F = r$$ and $$\|A\| = 1$$), the same bound would hold unconditionally. It would be as if $$A$$ was deterministic.