If $(X,Y)$, $(X,Z)$, and $(Y,Z)$ are all Gaussian, does it follow that $(X,Y,Z)$ is also Gaussian? I'm having trouble coming up with a counterexample...

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    $\begingroup$ Possible duplicate of Is it possible to have a pair of Gaussian random variables for which the joint distribution is not Gaussian? A straightforward counterexample is to extend the example in the upper right figure in @whuber's answer to three dimensions. $\endgroup$ Commented Nov 19, 2019 at 20:13
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    $\begingroup$ Why do three normal random variable necessarily have to have a Gaussian joint distribution? They're not necessarily independent. @Konstantin $\endgroup$
    – Bindiya12
    Commented Nov 19, 2019 at 20:18
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    $\begingroup$ @Konstantin - Just being able to construct a covariance matrix does not mean the three variables are jointly Gaussian. $\endgroup$
    – jbowman
    Commented Nov 19, 2019 at 21:14
  • $\begingroup$ @jbowman Thank you, you are so right, I've learned from the post Jarle Tufto ponted at. $\endgroup$
    – Konstantin
    Commented Nov 19, 2019 at 21:20
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    $\begingroup$ The example in the duplicate was posted by @Cardinal, not by me. However, at stats.stackexchange.com/a/434151/919 I recently posted an example of a triviariate distribution that has uniform 2D marginals but is not uniform. By treating this as a copula one obtains an (intriguing) example involving pairwise Gaussian variables that are not jointly Gaussian, but in a subtle way. This approach extends to any univariate distribution whatsoever. $\endgroup$
    – whuber
    Commented Nov 19, 2019 at 21:33


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