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I have a situation where I have a bunch of noisy observations with known (and normal) sampling error and a non-trivial co-variance structure between the observations. In an uncorrelated setting, I'd generate weights by taking the inverse variance of the observations and calculate the standard error of the mean by taking the reciprocal of the sum of weights.

How does that generalize in the face of an arbitrary correlation matrix?

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You say you have known sampling error. If you have multinormal data $(X_1, \dots, X_n)=X$ with commom mean $\mu$ and known covariance matrix $\Sigma$, then you can use the result that if $A$ is a matrix of constants, and $X \sim MN(\mu 1_n, \Sigma)$ then $AX \sim MN(A\mu, A \Sigma A^T)$. Just use the result with $A= n^{-1} 1_n$. Here $1_n$ is a constant (column) vector with all components $1$.

Just plug in your known $\Sigma$. However, if you give more details of your real problem, then maybe we can indicate a better statistical solution!

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