I have a sequence of $N$ dependent random variables
$$y_i = \frac{x_i}{||\vec x||_2} \quad \mathrm{for} \quad \vec x \sim \mathcal N(0,\mathbb{1}_N),$$
where the $x_i$ are the iid elements of $\vec x$ and $||.||_2$ is the Euclidean norm. Note that $\sum_i y_i^2 = 1$.
For large $N$, by the CLT the distribution for $||\vec x||_2$ is closely approximated by $\mathcal N(\sqrt{N},1/\sqrt{2})$ and the distribution for $y_i$ goes to $\mathcal N(0,1/\sqrt{N})$, although this clearly does not enforce the constraint.
Because this fits pretty well and is independent of the correlation between $x_i$ and $||\vec x||_2$, I thought this meant I could approximate the $y_i$ as if they were independent as $N \to \infty$. However, when I try to use the iid approximation, $y_i \sim \mathcal N(0,1/\sqrt{N})$, and the CLT to compute things like $\sum_i^N y_i^4$, I get a value for the variance which is too large. This obviously results from neglecting the constraint, since if I draw the $y_i \sim \mathcal N(0,1/\sqrt{N})$, the fit is good although the two $y_i$ histograms are nearly identical.
My question: Can I get a better approximation for the variance of these sums $\sum_i^N |y_i|^m$ as $N\to \infty$, since the results appear to be normally distributed? In particular, I don't understand why
$$\langle | y|^m \rangle \sim \int | y|^m \mathcal N(0,1/\sqrt{N}) dy $$
gives a bad approximation to the moments when the distributions for $y_i$ are very closely approximated by $\mathcal N(0,1/\sqrt{N})$.