Let $X=(X_1 \dots X_n)\in \mathbb{R}^n,$ be a subgaussian random vector so that $X_i$'s are independent, $\mathbb{E}X_i = 0, \mathbb{E}X_i^2=1.$ Before we pose our question, *let's state the following*:

**Definition notation and background:** *A random variable $V \in \mathbb{R}$ is called subgaussian, if $P[|V|\ge t]\le 2 e^{-c(V)t^2}, c(V)>0$ depends on $V$ only. One can show that: $\mathbb{E}\left[exp({\frac{V^2}{t^2}})\right ]\le 2$ for some $t>0,$ which makes us define the Orlicz norm: $||V||_{\psi_2}:= inf_{t > 0}\mathbb{E}\left[exp({\frac{V^2}{t^2}})\right ]\le 2.$ Note that: then we have: $P[|V|\ge t]\le 2 e^{-ct^2/||V||_{\psi_2}^2}, c>0$ is an absolute constant.
A random vector $X\in \mathbb{R}^n$ is called subgaussian if $<X,x>_{\mathbb{R}^n}$ is subgaussian random variable for every constant $x \in \mathbb{R}^n.$*

**With the above, let's state the question, $c, C$ below are absolute:**

We know the following about the concentration of $ ||X||_2,(\mathbb{E}X_i=0, \mathbb{E}X_i^2=1$)

$$P[ |\hspace{1mm}{||X||_2 - \sqrt{n}} \hspace{1mm}| \ge t] \le 2e^{-ct^2/K^4}, K=max_{1 \le i \le n}||X_i||_{\psi_2}.$$

*Or equivalently:*

$$||\hspace{1mm}{||X||_2 - \sqrt{n}} \hspace{1mm}||_{\psi_2}\le CK^2 > 0, K=max_{1 \le i \le n}||X_i||_{\psi_2}$$

where ${\psi_2}$ denotes the subgaussian norm. *But we note that the right side of the inequality is dimension-free, i.e. there's no function of $n$ on the right that goes to zero as $n \to \infty.$* So **my question is: do we necessarily have tighter concentration of the norm around $\sqrt{n}$ as $n \to \infty?$** That is: do we have:

$$(1a) \hspace{1mm} lim_{n \to \infty} |\mathbb{E}||X||_2 - \sqrt{n}|=0? $$
$$(1b) \hspace{1mm} lim_{n \to \infty} | \frac{\mathbb{E}||X||_2}{\sqrt{n}} - 1|=0? $$
$$ (2a) \hspace{1mm} lim_{n \to \infty}||\hspace{1mm}{||X||_2 - \sqrt{n}} \hspace{1mm}||_{\psi_2}=0?$$
$$ (2b) \hspace{1mm} lim_{n \to \infty}||\hspace{1mm}{\frac{||X||_2}{\sqrt{n}} - 1 } \hspace{1mm}||_{\psi_2}=0?$$
**Next, if we don't assume that the co-ordinates are independent, are (1a,b) and (2a,b) still true?**