Concentration of the norm of subGaussian random vectors

Question

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.

Answer 1

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 $\le2C^2$.