This is a cross-posting of a question I asked at CrossValidated. It hasn't generated much activity so I'm trying here:

Suppose $X\sim \operatorname{InvWishart}(\nu, \Sigma_0)$. I'm interested in the marginal distribution of the diagonal elements $\operatorname{diag}(X) = (x_{11}, \dots, x_{pp})$. There are a few simple results on the distribution of submatrices of $X$. From these I can figure that the marginal distribution of any single element on the diagonal is inverse gamma. But I've been unable to deduce the joint distribution. I suspect that I'm missing something simple; it seems like this "ought" to be known but I haven't been able to find/show it.