Let $R$ be a semi-definite $N\times N$ circulant Toeplitz matrix and let $N\to \infty$. Let $D$ be an $N\times N$ diagonal matrix where the elements on the main diagonal are independent chi-square distributed random variables with two degrees of freedom. What is the expectation $Z=\mathbb{E}(R+D)^{-1}$?
The Circulant matrix is formed from a Fourer transform $R(\omega)$. So as $N$ grows, the eigenvalues converge to $R(\omega)$.
From simple experiments in Matlab, the following seems to hold:
The solution is of the form $(R+\lambda I)^{-1}$ for some $\lambda$. If true, the problem reduces into finding $\lambda$.
Due to 1) it holds that $R(\omega)+\frac{1}{Z(\omega)}=constant$, where $Z(\omega)$ is the induced Fourier transform of $Z$. Thus, since $R(\omega)$ is known, the problem reduces into finding a single value of $Z(\omega)$ or some other quantity such as $\int Z(\omega)d\omega$ etc.
