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## Gaussian random vectors [closed]

If $(X,Y)$, $(Z,X)$ and $(Y,Z)$ are all Gaussian random vectors, is $(X,Y,Z)$ a Gaussian random vector?

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In other words, the question is to what extent the measure in $\mathbb R^3$ is determined by its coordinate plane projections. You certainly have the freedom to add the sum of alternaing Dirac masses at the vertices of any coordinate parallelepiped (as well as any integral combination of those) and that is enough to make some monster triples that are very different from jointly Gaussian. Also, this question is a bit too "homeworky" for MO. Voting to close. – fedja Dec 30 2010 at 1:53
@fedja: The definition of Gaussian in this case is about projections: an $n$-dimensional Gaussian random vector is defined as one whose dot-product with any fixed (i.e. non-random) vector is a 1-dimensional Gaussian random variable. – Michael Hardy Dec 30 2010 at 3:37