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:

- $A$ and $x$ are uncorrelated, i.e. $\mathbb{E}[Ax]=\mathbb{E}[A]\mathbb{E}[\mathbb{x}]=\mathbb{E}[A]0=0$.
- $A$ 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.]