# 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?