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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...

Bindiya12
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    Possible duplicate of [Is it possible to have a pair of Gaussian random variables for which the joint distribution is not Gaussian?](https://stats.stackexchange.com/questions/30159/is-it-possible-to-have-a-pair-of-gaussian-random-variables-for-which-the-joint-d) A straightforward counterexample is to extend the example in the upper right figure in @whuber's answer to three dimensions. – Jarle Tufto Nov 19 '19 at 20:13
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    Why do three normal random variable necessarily have to have a Gaussian joint distribution? They're not necessarily independent. @Konstantin – Bindiya12 Nov 19 '19 at 20:18
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    @Konstantin - Just being able to construct a covariance matrix does not mean the three variables are jointly Gaussian. – jbowman Nov 19 '19 at 21:14
  • @jbowman Thank you, you are so right, I've learned from the post Jarle Tufto ponted at. – Konstantin Nov 19 '19 at 21:20
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    The example in the duplicate was posted by @Cardinal, not by me. However, at https://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. – whuber Nov 19 '19 at 21:33
  • Not the one I was looking for, but e.g. see whuber's answer here: https://stats.stackexchange.com/questions/396687/derive-multivariate-from-bivariate-normal-distribution ... or my answer here: https://stats.stackexchange.com/questions/305067/get-joint-distribution-from-pairwise-marginal-distribution/305069#305069 ... or see this one: https://stats.stackexchange.com/questions/81469/if-any-two-variables-follow-a-normal-bivariate-distribution-does-it-also-have-a/81471#81471 ... any of those answers directly responds to this question. The third might be a good duplicate to add above – Glen_b Nov 20 '19 at 00:28

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