Are the marginals of the multivariate t distribution univariate Student t distributions?

Question

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

Answer 1

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