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Is there an analytical expression for the distribution of variances of MVN RVs? I mean if $X=[x_1, \dots, x_D]\sim \mathcal{N}(0, \Sigma)$ where $\Sigma$ is a $D$-dimensional covariance matrix, is there an analytical expression for the distribution of $V(X) = \frac{1}{D}\sum_d(x_d-\bar{X})^2$ where $\bar{X}=\frac{1}{D}\sum_d x_d$?

Even better would be an expression for the distribution of $\frac{V(X)}{V'(X)}$ where $V'(X)$ is the variance of a subset of the $D$ components of $X$. That is $V'(X) = \frac{1}{D}\sum_{d \in \mathcal{D}}(x_d-\bar{X}')^2$, where $\mathcal{D} \subseteq \{1, \dots, D\}$ and $\bar{X}'$ is the mean over these same dimensions. I'm guessing there is no such expression but it never hurts to ask.

There is a related question Second moment of draws from a multivariate normal covariance matrix but there they just ask for the moments not the full distribution.

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  • $\begingroup$ If I were you I would start with a diagonal covariance matrix to get some intuition for the problem. At that point the components of $X$ are independent, and you should have a lot of room to work with. Then maybe work on 2x2 no correlated variables. $\endgroup$ – jlimahaverford Mar 26 '15 at 20:56

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