I'm trying to obtain the mean and variance of a function given by $f=\sum_{i=1}^{N} a \lambda_i$, where $a$ is a constant and $\lambda_i$ are the ordered eigenvalues of a Wishart matrix given by $\mathbf{W}=\mathbf{H}\mathbf{H}^H$, where the elements of $H$ are complex Gaussian distributed with zero-mean and unitary variance.

I know that when $N$ is large, the distribution of the eigenvalues of $\mathbf{W}$ is a Marchenko-Pastur distribution $p(\lambda)$ (I have validated this already). Therefore, following distribution on the inverse Wishart matrix eigenvalues summation , I'm considering $f$ as a Gaussian RV.

Although I can obtain the mean of $f$ with success, I'm not able to obtain its variance. I'm considering eq. (17) of https://arxiv.org/abs/cond-mat/9310010, but the numerical integration gives me a singularity.

Can someone help me with this? Thank you very much.