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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.

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

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