This is a relatively vague question since it occurred during the class couple of semesters back when the Professor tried to explain the "intuition" of Gaussian distribution. The information is in incomplete pieces, but I still wish to fully understand this intuition:
Given $X_1, X_2, \cdots, X_m$ are i.i.d. random variables, zero mean, unit variance. It was mentioned to us we can look at the normalized sum as $\frac{1}{\sqrt{m}}\sum_{i = 1} ^m X_i \langle (X_1, \cdots, X_m), (\frac{1}{\sqrt{m}}, \cdots, \frac{1}{\sqrt{m}}) \rangle$ and hence as a projection of a vector onto "some kind" of shell of radius $\sqrt{m}$-I remember this shell was related to the "norm" of the Gaussian vectors:
Then he mentioned some kind of "typical" normal vector direction, and the projection being fairly close to the typical normal vector. Hence the "uniqueness" of Gaussian distribution etc.
This is all I have so far. I am sorry for the incomplete pieces of information, but I wonder how is this related to the Central Limit Theorem?