Let $\mathbf{X} = \{X_1,X_2,...,X_n\}$ be iid normal random variables with mean $\mu=0$ and variance $\sigma^2$. In the limit of large $n$, is there a closed form for the following expected value and variance?

$$ \mathrm{E}(\sum_i X_i^2 \,|\, \sum_i |X_i|=\alpha) $$ $$ \mathrm{Var}(\sum_i X_i^2 \,|\, \sum_i |X_i|=\alpha) $$

Here's my thought. From the spherical symmetry of the joint pdf, taking the expected value and variance conditioned on $\sum_i |X_i|=\alpha$ is equivalent to taking them on the $n-1$ dimensional simplex $\Delta^{n-1}$ in the first quadrant:

$$ \{ \mathbf{x}\in \mathbb{R}^n : \sum_i x_i = \alpha, x_i \geq 0, i = 0,...,n \} $$

I understand that in the limit of large $n$, the Gaussian measure on the simplex is dominant around $r=\sqrt{n-1}\sigma$, where $r$ is the distance from the centroid. On the other hand, the inradius of the simplex is $\frac{\alpha}{\sqrt{n(n-1)}}$, so if $\alpha > (n-1)\sqrt{n}\sigma$, then the dominant region of the probability measure is completely contained within the simplex, and finding the expected value and variance is trivial. However, I'm at lost as to what to do when $\alpha < (n-1)\sqrt{n}\sigma$.



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