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1.We know that if $X=(X_1,...,X_n)$ be a random vector with independent sub-gaussian coordinates $X_i$ that satisfy $EX_i^2=1$, then

$$||||X||_2-\sqrt{n}||_{\psi_2}\leq CK^2$$

where $K=max||X_i||_{\psi_2}$. Additionally, $-CK^2\leq \mathbb{E}||X||_2-\sqrt{n}\leq CK^2$.

Can we conclude that $\mathbb{E}||X||_2$ converges to $\sqrt{n}$ as $n \rightarrow \infty$?

I know that $\frac{||X||_2}{\sqrt{n}}-1$ converges to $0$ in probability. But I don't know if it converges in expectation or not.

  1. I want to show that $Var(||X||_2)\leq CK^4$.

It suffices to show that $(\mathbb{E}||X||_2)^2\geq n-CK^4$. I want to exploit above inequalities to drive the desired bound.

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