Imagine I have a vector $\mathbf{x} = [x_1, \dots, x_N]^\top$ where each of the $x_n \sim \mathcal{N}(0, 1)$. Its $\ell_2$-norm is $||\mathbf{x}||_2 = \sqrt{x_1^2 +\dots + x_N^2}$. I am interested in the distribution of the marginals of the normalized $\mathbf{y} = \Big[\frac{x_1}{||\mathbf{x}||_2}, \dots, \frac{x_N}{||\mathbf{x}||_2}\Big]^\top$. Is there a way to specify the distribution of the individual $\frac{x_n}{||\mathbf{x}||_2}, n\in[N]$ at all? By simulation, it seems that at least when $N$ grows, the marginals behave like Gaussian random variables with $\mathcal{N}(0, N^{-1})$. Or is that effect solely due to an approximate independence between numerator and denominator for large $N$, which makes the marginals act like a student-t random variable?
So far, I have found plenty on the distribution of the entire vector, for example here: On the distribution of a normalized Gaussian vector.