Let $X\sim\mathcal{N}(0,\Sigma)$ be a $n$-dimensional Gaussian random vector $X\in\Re^n$ where $\Sigma_{ii}=1, \forall i\in n$. Define $Y$ to be its projection onto $\mathbb{S}^{n-1}(\sqrt{n})$: $Y:= \frac{X}{ \sqrt{\frac{1}{n}\sum_i X_i^2}}$ Can we prove $E \|Y Y^\top \|_F^2 < \alpha_n \|\Sigma\|_F^2$ where $\alpha_n$ is a constant only dependent on $n$?
The $\sqrt{n}$ factor ensures that the $\ell^2$ norm of the two vectors is equal in expectation $E \|Y\|_2^2 = E\|X\|_2^2 = n$, and hence they are of comparable length. Therefore, a result with $a_n<1$ can be interpreted as, the normalization makes the distribution more "orthogonal" in expectation.
We know that law of $X$ is heavily concentrated around $\mathbb{S}_{n-1}(\sqrt{n})$ when $n\to\infty$, and therefore in the limit the law of $X$ converges to the law of $Y$. This question concerns the non-asymptotic properties of $Y$.