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Is there a known expression for the eigenvalue distribution of a matrix of the form $$\sum_{i=1}^n k_ia_ia_i^T$$ where $a_i \in \mathcal{R}^m$, with $n > m$, $a_i \sim \mathcal{N}(0,\Sigma)$ and $...

I am concerned with the asymptotical behavior of integrals like this for large $n$ $$\frac{1}{n!}\intop_{\Omega}\prod_{1\leq i<j\leq n}(x_{j}-x_{i})^{2}\,\prod_{j=1}^{n}e^{-x_{j}^{2}}dx_{j},$$ ...