# high probability Gaussian concentration inequality for $\|X\|_{\Sigma^{-1}}$ where $X \sim \mathcal{N}(0,\alpha(\delta) \Sigma)$

In the appendix of this paper appendix, Lemma I.4 the following Gaussian concentration lemma is given.

Consider $$d$$- dimensional multivariate Gaussian distribution $$\overline{\xi}_t^k \sim \mathcal{N} (0, H \nu_k(\delta)(\Sigma^t_k)^{-1})$$. For any $$\delta > 0$$, with probability $$1-\delta$$

$$\begin{equation} \|\bar{\xi}_t^k\|_{\Sigma_t^k} \leq c\sqrt{H d \nu_k(\delta)\log(d/\delta)} \end{equation}$$ where c is some absolute constant.

Here $$\Sigma^t_k$$ is a positive definite matrix and $$\nu_k$$ is a function of $$\delta$$. In the paper they use a specific definition for $$\nu_k$$ but from the lemma description, it seems this lemma comes from more general statement. Could someone point me to a reference where I can find a more general statement of this? My guess there is something of the following form:

Consider random variable $$X \sim \mathcal{N}(0,\alpha(\delta) \Sigma)$$ where $$\alpha$$ is a scalar function of $$\delta$$ and $$\Sigma \in \mathbb{R}^{d\times d}$$. Then for any $$\delta > 0$$, with probability $$1-\delta$$

$$\begin{equation} \|X\|_{\Sigma^{-1}} \leq c\sqrt{d \alpha(\delta)\log(d/\delta)} \end{equation}$$ where c is some absolute constant.

If $$G$$ is a random vector with $$d$$ iid $$N(0,1)$$ entries then $$P(|\|G\|\le E[\|G\|] + \sqrt{2\log(1/\delta)} ) \ge 1-\delta$$; this follows by Gaussian concentration of $$\|G\|$$ which is a 1-Lipschitz of the entries of $$G$$, see for instance Theorems 5.5, 5.6 in Concentration Inequalities: A Nonasymptotic Theory of Independence by Boucheron Gabor Lugosi and Massart. Furthermore $$E[\|G\|]\le \sqrt d$$ by Jensen's inequality.
Now in your problem you may apply this result to $$G=E[XX^T]^{-1/2} X$$ where $$E[XX^T]=\Sigma \alpha(\delta)$$. It looks that the above concentration bound gives $$P\Big(|\|\Sigma^{-1/2}X\|\le \sqrt{\alpha(\delta)}\big( \sqrt{d}+ \sqrt{2\log(1/\delta) }\big) \Big) \ge 1-\delta$$.