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What is the sampling distribution of the variance of a collection of variables that follow a multivariate normal distribution? Specifically, assume that the $n-$dimensional vector $\boldsymbol{x} \sim \mathcal{N}(\boldsymbol{\mu},\boldsymbol{\Sigma})$, where $\boldsymbol{\mu}$ and $\boldsymbol{\Sigma}$ are known. Denote the sample mean by $\bar{x} = \sum_{i=1}^n x_i/n$. What is the distribution of $s^2_x \equiv \sum_{i=1}^n (x_i - \bar{x})^2/(n-1)$?

I know that the sample variance of a collection of independent and identically distributed normal variables follows a chi-squared distribution, but have been unable to find an extension to the case of correlated normal variables.

I have posed this question more generally, but I am specifically interested in the case of exchangeable variables $x_i$ which marginally have the same variance but are positively correlated with each other. I have simulated the problem with various variance and correlation parameters and suspect that the sample variance is chi-squared in this instance as well, but would like a reliable reference for this result if true.

EDIT: It seems that a transformation of a multivariate normal distribution would be useful here. As far as I understand, we can write $\boldsymbol{x} = \boldsymbol{\mu} + \boldsymbol{A}\boldsymbol{z}$, where $\boldsymbol{z}$ is an $n-$dimensional vector of independent and identically distributed unit normal variables and $\boldsymbol{A}$ is the Cholesky decomposition of $\boldsymbol{\Sigma}$ such that $\boldsymbol{A}\boldsymbol{A}' = \boldsymbol{\Sigma}$.

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