# Concentration inequity with Random Gaussian Matrix

The rows of $X_{n\times p}$ are $\it{i.i.d}\:\:\: p$ variate Gaussian vectors with distribution $\mathcal{N}_p(0,\Omega)$. Let $\varepsilon$ be another $n$-variate normal random variable independent of $X$ having distribution $\mathcal{N}_n(0,\Sigma)$. Then for any fixed $u \in \mathbb{R}^p$ with $\|u\|_2=1$, can we upperbound $$P\left(\left|\frac{\varepsilon'Xu}{n}\right|> \eta\right)$$

by exponentially small expression (in terms of $n, p, \lambda_{\max}(\Sigma)$ and $\lambda_{\max}(\Omega)$)?

I am aware of Hanson-Wright type concentration inequalities but I have not seen one with product of two Gaussian inner-product. If I get some suggested readings/paper that will be helpful as well and thanks in advance.

• Any progress on this one? – nemo Feb 13 '18 at 17:27