# vector of three RVs that are pairwise Gaussian is Gaussian [duplicate]

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...

• 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. Commented Nov 19, 2019 at 20:13
• Why do three normal random variable necessarily have to have a Gaussian joint distribution? They're not necessarily independent. @Konstantin Commented Nov 19, 2019 at 20:18
• @Konstantin - Just being able to construct a covariance matrix does not mean the three variables are jointly Gaussian. Commented Nov 19, 2019 at 21:14
• @jbowman Thank you, you are so right, I've learned from the post Jarle Tufto ponted at. Commented Nov 19, 2019 at 21:20
• 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.
– whuber
Commented Nov 19, 2019 at 21:33