I was just contacted by some researchers willing to test if a multivariate copula is – or not – Gaussian. They use a test proposed in an unpublished paper by Malevergne and Sornette, stating that one should simply test for pairwise normality. In this paper, the following result is mentioned

In other words, it is stated that (or at least something closed to this)

Unfortunately, this result is (probably) not correct (and if it is valid, it is nontrivial). It should be possible to construct a counterexample (thanks to Roger Nelsen (web) for the idea) by letting the correlation in each pair be close to -1 (i.e. global pairwise contercomonotonicity). Then the correlation matrix of the triplet would fail to be positive definite (which is a requierment for Gaussian vectors). This idea can be related to the “compatibility” condition in Joe (1997).

My colleague Mihai Gradinaru (web) mentioned the book by Wlodzimierz Bryc, Normal distribution: characterizations with applications, but I did not find any result in it (I looked very quickly). Anyway, I am still working on this… any comment is welcome !