Let R be an $(N+1)\times (N+1)$ Toeplitz matrix. I would be mostly interested in the case of $N\to \infty$. Let $x$ be a complex Gaussian random $N\times 1$ vector with mean zero and covariance matrix $I$, i.e., $x\sim \mathcal{CN}(0,I)$. Let $X$ be a diagonal matrix with $x$ along its main diagonal. Let $\lambda$ be an arbitrary but fixed positive number. Let $\tilde{R}$ be the $N\times N$ sub matrix of $R$ constructed by removing the last row and the last column from $R$. Let $r$ be an $N\times 1$ vector that equals the first column of $R$, but with the first element removed.
Question: How to compute $$E_x\left[1-r^*X^*(X\tilde{R}X^*+\lambda I)^{-1}Xr\right].$$ This can be interpreted as the MMSE of an estimation of a correlated process observed in Gaussian noise through Gaussian training symbols. The diagonal of $R$ is assumed here to be $I$.