I apologize if this question is vague or trivial. I have a random unit vector $\mathbf u$ in $\mathbb R^n$ and the following facts about it are true when $n\to \infty$:
- $u_i \to \mathcal N(0,1)$ in distribution as $n\to\infty$ for all $i$
- For any finite $k,$ $u_{i_1},u_{i_2}..\ldots u_{i_k} \to \mathcal N(0,\mathbf I_k)$ in distribution as $n\to\infty$
- For any sequence of unit deterministic vectors $\mathbf a_n,$ $\mathbf a^T \mathbf u \to \mathcal N(0,1)$ in distibution as $n\to\infty$
Now I am having a hard time saying anything about the distribution of $\mathbf u$ as a whole. Is it close some how to $\mathcal N(0,\mathbf I_n)$ for a large enough $n$? Can we it converges to the latter in distribution. Is it possible to say something about the difference $\|\mathbf v - \mathbf z\|_2$ where $\mathbf z$ is some realization of the standard gaussian vector?