probability theory – Nature of Distribution of a Random Vector whose finite dimensional distributions and distribution of inner products are known

I apologize if this question is vague or trivial. I have a random vector $mathbf u$ in $mathbb R^n$ and the following facts about it are true when $nto infty$:

  • $u_i to mathcal N(0,1)$ in distribution as $ntoinfty$
  • 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 $ntoinfty$
  • For any unit deterministic vector $mathbf a,$ $mathbf a^T mathbf u to mathcal N(0,1)$ in distibution

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?