Szego limit theorems for Toeplitz matrix

Question

Let matrix $A$ to be a auto-correlation matrix for a stationary signal $x(t)$, hence matrix $A$ is symmetric Toeplitz matrix such that \begin{equation} A:=\begin{bmatrix}\phi(0) & \phi(-1) & \cdots & \phi(1-N) \\ \phi(1) & \phi(0) & \cdots & \phi(2-N)\\ \vdots & \vdots & \ddots & \vdots\\ \phi(N-1) & \phi(N-2) & \cdots & \phi(0) \end{bmatrix} \end{equation} It is easy to proof that \begin{equation} \lim_{N\rightarrow\infty}\frac{\lambda_0 + \lambda_1 + \cdots + \lambda_{N-1}}{N} = \phi(0) = \int^{f_N}_{-f_N}S(f)df, \tag1 \end{equation} where $f_N$ is the Nyquist frequency; $\lambda_i$ are eigenvalues of matrix $A$, $S(f)$ and $\phi(t)$ are Fourier Transform pairs such that \begin{equation} \phi(t) = \int^{f_N}_{-f_N} S(f)e^{i2\pi f\frac{t}{f_N}}df. \tag2 \end{equation} Now, I would like to prove that \begin{equation} \lim_{N\rightarrow\infty}\frac{\log(\lambda_0) + \log(\lambda_1) + \cdots + \log(\lambda_{N-1})}{N} = \frac{1}{2f_N}\int^{f_N}_{-f_N}\log[2f_N S(f)]df, \tag3 \end{equation} Does anyone know how to prove it? Many thanks!

Answer 1

The answer is fairly easy, because \begin{equation} \lim_{N\rightarrow\infty}\frac{\lambda_0 + \lambda_1 + \cdots + \lambda_{N-1}}{N} = \phi(0) = \int^{f_N}_{-f_N}S(f)df = \frac{1}{2f_N} \int^{f_N}_{-f_N}[2f_N S(f)]df, \end{equation} which is the Arithmetic mean of continuous function $2f_N S(f)$, as well as $\{\lambda_i\}_{0}^\infty$. Hence, the Geometric mean of $\{\lambda_i\}_{0}^\infty$ can be written as \begin{equation} \lim_{N\rightarrow\infty}\left(\prod_{i=1}^N \lambda_i\right)^{\frac{1}{N}} = e^{\frac{1}{2f_N}\int^{f_N}_{-f_N}\log[2f_N S(f)]df}. \end{equation} Finally, we have \begin{equation} \begin{aligned} \lim_{N\rightarrow\infty}\frac{\log(\lambda_0) + \cdots + \log(\lambda_{N-1})}{N} &= \lim_{N\rightarrow\infty}\log\left(\prod_{i=1}^N \lambda_i\right)^{\frac{1}{N}}\\ &=\frac{1}{2f_N}\int^{f_N}_{-f_N}\log[2f_N S(f)]df. \end{aligned} \end{equation}