Given a $M\times P$ matrix $\mathbf{X}$ with independent rows and the $m$-th row has the form:
$$ \mathbf{x} = \begin{bmatrix} f_1(x_{m,1},\dots,x_{m,R}) & f_2(x_{m,1},\dots,x_{m,R}) & \cdots f_P(x_{m,1},\dots,x_{m,R}) \end{bmatrix} . $$
$x_{m,r}$ are Gaussian i.i.d. random variables and $f_p$ are known (differentiable, smooth) functions. So each row is independent of each other and the columns are dependent and each column has a different distribution.
Is there anything I could say about spectral properties (eigenvalues, singular values, determinant, trace etc. of $\mathbf{X}$ or $(\mathbf{X}^T \mathbf{X})^{-1}$ or $(\mathbf{X}^T \mathbf{X})$?
Is there any similar known problem in random matrix theory that can help me with this problem?
Additional information:
The concrete case I am interested in is where $f_p$ is given as: $$f_p(x_{m,1},\dots,x_{m,R}) = \frac{1}{\sqrt{R}} \sum_{r=1}^R x_{m,r}^p $$ This means: 1) For $R=1$ the matrix is nothing but a Vandermonde matrix with random nodes 2) For $R>1$, each row is the normalized sum over $R$ rows of a random Vandermonde matrix 3) Each column contains sums of Gaussian random variables, raised to the power of $p$.