Let $X\in{}\mathbb{R}^N$, with independent sub-gaussian coordinates s.t. $E[X_i^2]=1, E[X_i]=0$.
W.T.S:
$\text{Var}(\|X\|_2)\le{C'K^4}$ with $C'>0$ and $K:=\max_{1\le{i\le{N}}}{\|X_i\|_{\psi^2}}$,
I don't know where to start, ive tried using various properties of sub-gaussian r.v. with no luck, any hints?
The concentration of the norm theorem:
Let $X\in{}\mathbb{R}^N$, with independent sub-gaussian coordinates s.t. $E[X_i^2]=1$.
$$\|\|X\|_2-\sqrt{n}\|_{\Psi^2}\le{CK^2},\space{}C>0,\space{}K:=\max_{1\le{i\le{N}}}{\|X_i\|_{\psi^2}}$$
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P.S. I have shown the fact that:
$$\sqrt{n}-CK^2\le{}\mathbb{E}[\|X\|_2]\le{}\sqrt{n}+CK^2$$
I essentially used the theorem, found a lower bound of the subgaussian norm of $\|X\|_2-\sqrt{n}$ with the Lp norm, set p to 1. and used Jensens inequality since $f(x)=|x|$ is a convex function.
I am using the following book and believe the question is similar to ex3.1.4:
https://www.math.uci.edu/~rvershyn/papers/HDP-book/HDP-book.pdf
Best Answer
Start with $\text{Var}\|X\|_2 = \operatorname{E}(\|X\|_2 - \operatorname{E}\|X\|_2)^2 \leq \operatorname{E}(\|X\|_2 - \sqrt{n})^2$ because the mean $\text{E}\|X\|_2$ minimizes the mean squared error. It is used for a similar question.
Further, by using the inequality for the $L_1$ norm $|\|X\|_2 - \sqrt{n}| \leq CK^2$ you obtained we have the upper bound as follows, $$\text{Var}\|X\|_2 \leq \text{E}(\|X\|_2 - \sqrt{n})^2 \leq C^2K^4 .$$