Let $\{Y_t\}_{t\in \mathbb{Z}_{>=0}}$ be a centered, stationary random process, with autocovariance $R(t)=Cov(Y_0Y_t)$. The spectral density $S$ is defined as the Fourier transform of $R$, $S(\omega):=\sum_t R(t)e^{i t\omega}$.
Let $C_{\infty}$ denote the infinite covariance matrix of the process and let $C_N$ denote the covariance matrix of the truncated process $\{Y_t\}_{t< N}$. For a fixed $N$, define the empirical spectral measure as $\mu_N=N^{-1}\sum_i \delta_{\lambda^N_i}$ where $\lambda_i^N$ are the eigenvalues of $C_N$. It is well known that for large $N$, the eigenvalues of $C_N$ are approximately discrete Fourier basis functions, with the corresponding eigenvalues being DFT of the autocovariance function. Thus it is natural to expect that in the limit the eigenvalues of $C_N$ approach the continuous Fourier transform, i.e. the spectral density. I would like a reference showing convergence of $\mu_N$ to the normalized spectral measure. More precisely, I would expect that something like $\mu_N([-\pi,t])\to {\frac 1 {2\pi}}|\{x:S(x)\leq t\}|$ holds under suitable conditions.