Let $\mathcal{S}_+^d$ denote the family of real $d \times d$ symmetric (strictly) positive definite matrices.
Define $\mathcal{P}_d$ to be those measures $\nu$ on $\mathcal{S}_+^d$ (assumed to have its usual $\sigma$-field) such that the following relation holds. $$ I_d = \int_{\mathcal{S}_+^d} X \, \mathrm{d}\nu(X). $$
Define $\mathcal{P}_d^{\rm iid}$ to be the laws of random positive semidefinite matrices given by $$ \Sigma_n := \frac{1}{n} \sum_{i=1}^n x_i \otimes x_i, $$ where $x_i \sim \mathbb{P}$ identically and independently. We assume that $\mathbb{P}$ is a probability measure on $\mathbb{R}^d$ with the usual $\sigma$-field. We further assume that $\mathbb{P}$ satisfies the following two properties: (i) assume $n \geq d$ and that $\Sigma_n$ is nonsingular almost surely; (ii) the distribution $\mathbb{P}$ satisfies the relation $$\mathbb{E}[\Sigma_n] = \int x \otimes x\, \mathrm{d}\mathbb{P}(x) = I_d$$
Thus, we clearly have the inclusion $\mathcal{P}^{\rm iid}_d \subset \mathcal{P}_d$.
Question: Is there a characterization (particularly in terms of eigenvalues and eigenvectors) of the measures in $\mathcal{P}^{\rm iid}_d$, as described above, which distinguishes this class from the more general class $\mathcal{P}_d$?
