From Maxwell's theorem, standard normal distribution is the only rotation-invariant distribution with fully-factorized marginals. What if we do not ask the marginals to be independent, but instead require them to be univariate standard normal? Will multivariate standard Gaussian distribution still be the only rotation-invariant distribution?
2026-05-05 12:05:44.1777982744
Is normal the only rotation-invariant distribution whose marginals are normal?
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Actually, this holds as a corollary to the Cramer-Wold theorem. The statement can be even stronger: Normal distribution is the only distribution whose marginals in all possible directions are normal.