I consider the following random circulant matrix
$$ H_w = F^* \mathrm{diag}(w_1, \dots w_p)F, \, w_i\overset{iid}{\sim}\Gamma(1,1), $$ where $F$ is the matrix for discrete Fourier transform, $F^*$ its complex conjugate. Let $h_{w}$ be its square principal submatrix of order $r$, $r < p$ which is also leading (that is $h_w$ is an upper-left block of $H_w$ of size r).
I want to approximate the following expectation $$ \mathbb{E}_w[(h_w + \mu\Sigma)^{-1}h_w], \, \mu > 0, $$ where $\Sigma$ is diagonal with strictly positive elements, $\mu$ is a small but fixed parameter (one can consider limit $\mu \rightarrow 0$). I want to get the first-order correction to the above mean in $\mu$. Of course, this question can be generalized to arbitrary random matrices, however, I believe that there is a higher chance to get a closed answer for the special structure above. I would be also happy even with any concentration of measure result on the above product.
Note: Naive application of Neumann series (assuming that $h_w$ will dominate $\mu\Sigma$) gives first order correction $\mu h_w^{-1}\Sigma$. Expectation of such term in $w$ seems divergent, as $\int_{0}^{+\infty} x^{-1}e^{-x}\, dx = +\infty$. Surely, it is a problem of expansion but not of the expectation itself.
$h_w$ is invertible almost surely, it is quite easy to show using Courant-Fisher characterization of eigenvalues that smallest eigenvalue of $h_w$ is not smaller than $\min(w_1, \dots, w_p) \sim \mathrm{Exp}(p)$.
p.s. I have seen that there is a lot of literature on random matrices with iid components but I have never seen anything like this.
