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I am wondering about the relation between the univariate and multivariate t-distribution. How is the product of univariate t-distributions with respect to $x_i$ related to the multivariate t-distribution with respect to $X$ and what is the difference between the independent ($x_i$ iid $\forall i$) and dependent case?

Or is there no relation? (Since I wound no publication dealing with these situations)

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