linear algebra – Limit spectrum of composition of projections onto random Gaussian vectors


Let $n > p$, let $X in mathbb{R}^{n times p}$ whose columns $X_1, ldots, X_p$ are zero-meaned Gaussian,of covariance $(rho^{|i – j|})_{i, j in (p)}$ ($rho in (0, 1)$).

Are there (asymptotics or not) known results on the eigenvalue distribution of:

$$left(mathrm{Id} – tfrac{1}{||{X_1}||^2} X_1 X_1^top right)
ldots left(mathrm{Id} – tfrac{1}{||{X_p}||^2} X_p X_p^top right) enspace ?$$

From a geometrical point of view, this is the matrix of the application which projects sequentially onto the orthogonal of the span of $X_p$, then onto that of $X_{p-1}$, etc, so all eigenvalues are in the unit disk, 0 is an eigenvalue, 1 also is an eigenvalue since $n > p$.
I would expect the spectrum to “move towards” 1 as $rho$ increases, but are there any quantitative results on that?