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I am interested in finding expressions for the marginal distributions of the off-diagonal terms in a Wishart-distributed random variable.

More specifically, suppose $X$ is an $n \times p$ matrix, each row of which is independently drawn from a $p$-variate normal distribution with zero mean: $X_{(i)}{=}(x_i^1,\dots,x_i^p)^T\sim N_p(0,V)$. Then $S=X^T X$ is a Wishart-distributed random variable, denoted $S\sim W_p(V,n)$.

Expressions exist for the marginal distributions of the diagonal terms, $S_{i,i}$ (e.g., page 4, third bullet point in this document). But I haven't been able to find similar expressions for the off-diagonal marginal distributions, $S_{i,j}$, where $i \neq j$.

Do they exist and, if so, what are they?

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  • $\begingroup$ I wonder if you can rotate the problem away. That is, consider the distribution of $QSQ^{\top}$ for orthonormal $Q$, where you have somehow rotated an off diagonal onto a diagonal? $\endgroup$
    – shabbychef
    Sep 18, 2014 at 16:19

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I believe the answer to this can actually be found on the wikipedia article about the Wishart distribution, which indicates that it is variance-gamma distributed.

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