Is there a known expression for the eigenvalue distribution of a matrix of the form

$$\sum\limits_{i=1}^n k_ia_ia_i^T$$

where $a_i \in \mathcal{R}^m$, with $n > m$, $a_i \sim \mathcal{N}(0,\Sigma)$ and $k_i > 0$? For simplicity we can consider the case $\Sigma = \text{I}_d$. (Equivalently, the expression is $\sum\limits_{i=1}^n a_ia_i^T$ where the $a_i$'s are $\mathcal{N}(0,k_i\text{I}_d)$, or $A\,\text{diag}(\vec{k})\,A^T$.)

In fact, I'm only concerned with the distribution of the determinant of such a matrix. This type of matrix can be thought of as a generalization of the Wishart matrix (which has $k_i$ equal for all $i$), whose eigenvalue distribution is well known, for instance: Determinant of real Wishart matrix.

Most of the generalizations of this result that I have seen deal with when the $a_i$'s are noncentral, i.e. drawn from distributions with differing means but the same covariance. I'm wondering if this result also can be generalized somehow to my case, where they are still mean zero but have differing covariances.

(In fact, even an exact expression is not necessary; an asymptotic upper bound - i.e. where $n$ is very large, and the $k$'s are considered as fixed - on the determinant in terms of the $k_i$'s would also be of interest. Of course one can get such bounds via Hadamard's inequality or applying AM-GM to the trace and determinant, but these bounds are quite weak for matrices whose eigenvalues are not very uniform.)

[Note: I originally posted this on the Statistics StackExchange, but was advised by a commenter to try here instead (and delete the other).]


All you need is that the empirical measure of the $k_i$s converge. Whenever it does, just apply the original Pastur-Marcenko (1967) results to get that the empirical measure associated with your matrix converges to a computable limit $\mu$.

For the determinant $D_n$, if the $k_i$'s are bounded away from $0$ uniformly in $n$ and $n/m\to \alpha>1$, you will have that $n^{-1}\log D_n$ converges to $\int \log x \mu(dx)$, while the fluctuations of $\log D_n$ around its mean are of order $1$ by concentration of measure results; probably one can also prove a CLT.


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