I have a deterministic circulant matrix $R$ and a random diagonal matrix $X$ where all elements are IID and positive. I need to determine the expected inverse of $R+X$, that is: Evaluate, in closed, form $E[(R+X)^{-1}]$.

I am mostly interested in an inverse chi-square distribution for each element of $X$ with 2 degrees of freedom (so $E[X]$ does not exist). However, I would appreciate any help you can provide for any distribution.

By simple playing around in matlab, the solution appears to be of the form $(R+\lambda I)^{-1}$, where $\lambda$ depends on the distribution of $X$. This holds for all distributions I have tried (Gauss, chi-2, inverse-chi-2, a couple of discrete ones...)

One simplification that I am particularly interested is: Let the dimensions of $R$ and $X$ tend to infinity, and let $R$ have an eigenvalue spectrum $R(\omega)=1, |\omega|<\delta 2\pi$ and $R(\omega)=0, |\omega|\geq\delta 2\pi$ for some $0<\delta <1.$