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Given the following system of equations:

$\textbf{y} = \textbf{x}_{1} + \sum_{l=2}^{L}\textbf{x}_{l} + \textbf{w}$

where $\textbf{y}$, $\textbf{x}_{l}, \; \forall l$ and $\textbf{w}$ are Gaussian random vectors with zero mean and variances $ \gt 0$.

How can we check if $\textbf{y}$ and $\textbf{x}_{1}$ are jointly Gaussian?

This question arises from the fact that the linear MMSE estimator of $\textbf{x}_{1}$ given $\textbf{y}$ becomes the Bayesian estimator when $\textbf{y}$ and $\textbf{x}_{1}$ are jointly Gaussian, however, I don't know if they are jointly Gaussian, and if so, how can I prove/check that?

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You cannot check if $y$ and $x_1$ are jointly Gaussian. If all the $x_i$ and $w$ are known or assumed to be jointly Gaussian, not just individually Gaussian, then $y$ and $x_1$ are jointly Gaussian. Without the joint Gaussianity assumption following If in the previous sentence, $y$ and $x_1$ too are not jointly Gaussian except in very unusual circumstances. For a simplified version of such unusual circumstances, see the answers to this question of mine. But bear in mind that these are the exception, not the rule. The sum of Gaussian random variables that are not jointly Gaussian shouldn't be assumed to be a Gaussian random variable; it hardly ever is.

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