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Are the marginals of the Multivariate t distribution with $\nu$ degrees of freedom univariate Student t distributions with $\nu$ degrees of freedom?

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Yes.

The proof is simple:

  1. The definition of a multivariate-t that you give is that it is a random variable that is the ratio of a multivariate Gaussian and the square-root of a Gamma variable
  2. The marginal distribution of that is the marginal distribution of the ratio of the Gaussian and the square-root of a gamma
  3. The distribution of a ratio of normal over square-root of Gamma is a univariate-t
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