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Let $x$ and $y$ be 2 random variables with t-distributions with mean = 0 but different degrees of freedom. Can their joint distribution be elliptical?

dayum
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No, this is not possible.

The marginals in an elliptical distribution are all scaled versions of one another (this is part of the definition). Thus, whenever one marginal has an absolute moment of order $\kappa$ (which may be a fraction), so does the other. But a $t$ distribution of $\nu$ degrees of freedom, whose PDF decays asymptotically like $|x|^{-\nu+1},$ has finite absolute moments for $\kappa \lt \nu$ and infinite absolute moments for $\kappa \ge \nu.$ Consequently, when the marginals have different values of $\nu,$ there will exist $\kappa$ for which one has an infinite moment and the other has a finite one, completing the proof.

whuber
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    Thanks @whuber . I also looked at your answer to this question: https://stats.stackexchange.com/questions/192526/multivariate-t-distribution-with-different-degrees-of-freedom-per-dimension where you state that copula can accomodate different degrees of freedom for different vairables in a multivariate distribution. Do you mean that even with using copula, the resulting distribution wont be elliptical? – dayum Sep 16 '20 at 01:16
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    That's correct. – whuber Sep 16 '20 at 12:31