Concentration of the norm of subGaussian random vectors

I will use the same notation and definitions in High Dimensional Probability, by Roman Vershynin.

I have a sub-Gaussian vector $$y$$, in $$\mathbb{R}^n$$ and sub-Gaussian norm $$C$$ non dependent on $$n$$. I also assume that $$y$$ respects the concentration of the norm property, which I define as

$$\frac{1}{\sqrt n} (|y|^2 - \mathbb{E}|y|^2)$$ is a sub-exponential random variable with sub-exponential norm $$C'$$ non dependent on $$n$$, where $$|\cdot|$$ is the Euclidean norm;

(note that this property holds, for example, for standard Gaussian vectors).

The question is the following: does such a vector respect also the following property?

$$\sum_i s_i \,(y_i^2 - \mathbb{E}y_i^2)$$ is a sub-exponential random variable with sub-exponential norm $$C''$$ non dependent on $$n$$, where $$|s| = 1$$, and $$s_i \geq 0 \; \forall i$$.

Note that we are not assuming the independence of the entries of $$y$$, and that both sub-Gaussianity and concentration of the norm are necessary properties for the thesis to hold. In particular, here is an example of a sub-Gaussian vector that does not respect the concentration of the norm property:

$$v = Z g$$, where $$g$$ is a standard Gaussian vector, and $$Z$$ is a scalar random variable uniform in $$[0, 1]$$.

However, I'm not sure the result is true in general, since I wasn't able neither to prove it nor to find a counterexample.

Your desired conclusion holds, even without the additional assumption that $$\frac1{\sqrt n}(|y|^2-E|y|^2)$$ is a sub-exponential random variable with a sub-exponential norm not dependent on $$n$$.
We only need to assume that $$y$$ is a sub-Gaussian vector $$y$$ in $$\mathbb{R}^n$$ with a sub-Gaussian norm $$C$$ not dependent on $$n$$: $$E\exp\Big(\sum_1^n y_i^2/C^2\Big)\le2.$$ Then, by Jensen's inequality, $$\exp\Big(\sum_1^n Ey_i^2/C^2\Big)\le E\exp\Big(\sum_1^n y_i^2/C^2\Big)\le2.$$ So, for any $$s\in\mathbb{R}^n$$ with $$|s|=1$$, \begin{aligned} E\,&\exp\Big(\Big|\sum_1^n s_i(y_i^2-Ey_i^2)\Big|/(2C^2)\Big) \\ &\le E\exp\Big(\sum_1^n (y_i^2+Ey_i^2)/(2C^2)\Big) \\ &\le\sqrt{E\exp\Big(\sum_1^n (y_i^2+Ey_i^2)/C^2\Big)} \\ &=\sqrt{E\exp\Big(\sum_1^n y_i^2/C^2\Big) \;\exp\Big(\sum_1^n Ey_i^2/C^2\Big)} \\ &\le\sqrt{2\times2}=2. \end{aligned} So, the sub-exponential norm of $$\sum_1^n s_i(y_i^2-Ey_i^2)$$ is $$\le2C^2$$.
• Thanks for your reply. However, I believe there is a misunderstanding on the definition of the sub-Gaussian norm in your solution. In your first equation, you assumed the sub-Gaussianity of the squared norm of $y$, which is not our hypothesis. For a vector to be sub-Gaussian with norm $C$, the following needs to hold: $C = \sup_{|u| = 1} | \langle u, y \rangle |_{\psi_2}$, where $|\cdot |_{\psi_2}$ is the usual scalar sub-Gaussian norm. Commented Jul 6, 2021 at 15:32