# 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?

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