Let $\{x_i\}_{i=1}^n \subset \mathbb{R}^d$ be iid random vectors drawn from probability measure $P$.
Define the random $d \times d$ real positive semidefinite matrix, $$ S_n = \frac{1}{n} \sum_{i=1}^n x_i x_i^T. $$ By the spectral theorem, we may also write $$ S_n = U_n D_n U_n^T \quad \mbox{where} \quad U_n \in \mathbb{R}^{d \times r_n}, D_n \in \mathbb{R}^{r_n \times r_n}, r_n = \mathrm{rank}(S_n). $$ Above, $U_n^T U_n = I_{r_n}$ and $D_n$ is diagonal, with nonincreasing positive entries.
Question: From $P$, we obtain a law $Q$ on $U_n$. What is the relationship between $P$ and $Q$?
I am especially interested in the following very special cases:
- Centered Gaussian with covariance $K$. When $P = \mathsf{N}_d(0, K)$, what is the corresponding $Q$?
- Symmetric Bernoulli. When $P = \mathsf{Unif}(\{-1, 1\}^d)$, what is $Q$?
I realize the questions above may not have known answers. The only case I am familiar is the classical situation when $P = \mathsf{N}_d(0, I_d)$, in which case $Q$ is the uniform measure on rank-$r$ projections where $r = \min\{d, n\}$. However, I am interested in references/results that address these questions.
