Expectation value of inverse covariance matrix when sampling from unit sphere Let $X \sim \operatorname{Unif}S_{d-1}$, so $X\in\mathbb{R}^d$ and is distributed uniformly on the unit sphere.
Then let $X_1, \dots, X_n \sim X$ iid and define the matrix $\mathbf{X}\in\mathbb{R}^{n\times d}$ by $\mathbf{X}_{ij} = (X_i)_j$
Is there a name for the distribution of $(\mathbf{X}^\top \mathbf{X})^+$?
The + denotes Moore-Penrose Pseudo-Inverse. This would be to $\operatorname{Unif}S_{d-1}$ as the (generalised) Wishart is to the normal distribution.
 A: First note that the vector $x$ distributed uniformly on the $d$-dimensional hypersphere can be constructed from a vector $y$ with i.i.d. normal elements $y_1,y_2,\ldots y_d$, via
$$x=\left(\sum_{i=1}^d y_i^2\right)^{-1/2} y.$$
Consider a set of $n$ vectors $y^{(1)},y^{(2)}\ldots y^{(n)}$ and construct the $n\times d$ dimensional matrix $Y$ with elements $Y_{ij}=y^{(i)}_j$. The matrix product $Y^TY$ has a Wishart distribution, and $(Y^TY)^+$ has the inverse Wishart distribution, with expectation value
$$\mathbb{E}(Y^TY)^+=\frac{I}{n-d-1},\;\;n>d+1.$$
Also construct the $n\times n$ diagonal matrix $D$ with elements $D_{ij}=\delta_{ij}\left(\sum_{k=1}^d (y_k^{(i)})^2\right)^{-1/2}$. The matrix $X$ in the OP is related to the matrix $Y$ by $X=DY$.
We seek the expectation value of the scalar
$$R\equiv \frac{1}{d}\,{\rm tr}\,(X^TX)^+=\frac{1}{d}{\rm tr}\,(Y^T D^2Y)^+.$$
For $d\gg 1$ the matrix $D^2$ selfaverages to $1/d$ times the identity, hence we estimate
$$r_{n,d}\equiv\mathbb{E}(R)\approx\frac{d}{n-d-1},\;\;n>d+1\gg 1.$$
I have performed some numerical checks, averaging over 500 realizations.




d,n
d/(n-d-1)
numerics




100,105
25
25.1


100,110
11.11
11.0


100,120
5.26
5.15


50,100
1.02
1.00


50,200
0.336
0.332




The difference between the numerical value and the estimate $d/(n-d-1)$ is within the statistical uncertainty.
