# The multivariate normal distribution has the same relationship with the Wishart distribution as the multivariate t-distribution with the …?

Is there a name for the distribution resulting from the sum of outer products of t-distributed random vectors?

Alternatively, is there a matrix-valued distribution with the support of positive definite matrices that exhibits a higher degree of tail dependence than the Wishart distribution?

Let the p-dimensional random vectors $X_1, \ldots, X_n$ be i.i.d. having a multivariate t-distribution given by $$f\left(x_1,x_2,\ldots,x_N\right) = K\left(N,p\right) |\Lambda|^{-N/2} \{\nu + \sum_{j=1}^N \left(x_j - \theta\right)' \Lambda^{-1} \left(x_j - \theta\right) \}^{-(\nu + Np)/2},$$ where $\nu>0$ and $K\left(N,p\right) = \nu^{\nu/2} \Gamma\{\left(\nu + N p \right)/2\}/\{\pi^{Np/2} \Gamma\left(\nu/2\right)\}$.
Then the $p \times p$ matrix $A = \sum_{j=1}^N \left(X_j - \bar{X}\right) \left(X_j - \bar{X}\right)'$ has the density $$g\left(A\right) = c\left(n,p;\nu\right)|\Lambda|^{-n/2} |A|^{(n-p-1)/2}\{\nu + tr\left(\Lambda^{-1} A \right)\}^{-(\nu+n p)/2},$$ for $A>0, \Lambda >0, n \geq p$ where $n=N-1$ and $$c\left(n,p;\nu\right) = \frac{\nu^{\nu/2}\Gamma\{\left(\nu+n p\right)/2\}}{\Gamma\left(\nu/2\right) \Gamma_p\left(n/2\right)}.$$