Section 1. On "Distance Rate"
Definition 1. The Riemannian distance of two positive-definite symmetric matrices $P_1$ and $P_2$ is \[d\left( {{P_1},{P_2}} \right) = {\left\| {\log \left( {{P_1}^{ - 1}{P_2}} \right)} \right\|_F} = {\left( {\sum\limits_{i = 1}^n {{{\ln }^2}{\lambda _i}} } \right)^{\frac{1}{2}}}\] where $\lambda_i,~i=1,\ldots,n$ are the eigenvalues of $P_1^{-1}P_2$.
Lemma 1. If a function $H\left( z \right) = \sum\nolimits_{n = 0}^\infty {{h_n}{z^{ - n}}} $ is minimum-phase and $h_0 \ne 0$, then \begin{equation} \nonumber \ln h_0^2 = \frac{1}{{2\pi }}\int_{ - \pi }^\pi {\ln {{\left| {H\left( {{e^{j\omega }}} \right)} \right|}^2}d\omega }. \end{equation}
For a minimum-phase system, we have shown that \[\left[ {\begin{array}{*{20}{c}} {{y_0}}\\ {{y_1}}\\ \vdots \\ {{y_{n - 1}}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{h_0}}&0& \cdots &0\\ {{h_1}}&{{h_0}}& \cdots &0\\ \vdots & \vdots &{}& \vdots \\ {{h_{n - 1}}}&{{h_{n - 2}}}& \cdots &{{h_0}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{u_0}}\\ {{u_1}}\\ \vdots \\ {{u_{n - 1}}} \end{array}} \right] \buildrel \Delta \over = A_n\left[ {\begin{array}{*{20}{c}} {{u_0}}\\ {{u_1}}\\ \vdots \\ {{u_{n - 1}}} \end{array}} \right].\]
Suppose that $h_0 \ne 0$ which implies that $A_n$ is nonsingular and let the system input be Gaussian white with zero mean and $\sigma_u^2$ as covariance. Let ${\tilde u_k} = {[ {\begin{array}[b]{*{20}{c}} {{u_0}}&{{u_1}}& \cdots &{{u_{k-1}}} \end{array}}]^T}$ and ${\tilde y_k} = {[ {\begin{array}[b]{*{20}{c}} {{y_0}}&{{y_1}}& \cdots &{{y_{k-1}}} \end{array}}]^T}$, it's easy to see that \begin{equation} \label{def_Uk_Yk} \begin{aligned} {U_k} &= E\left\{ {{{\tilde u}_k}{{\tilde u}_k}^T} \right\} = \sigma_u^2 I_k\\ {Y_k} &= E\left\{ {{{\tilde y}_k}{{\tilde y}_k}^T} \right\} = \sigma_u^2 A_k A_k^T \end{aligned} \end{equation} which implies that ${U_k}^{ - 1}{Y_k} = A_k A_k^T$.
Now we define the distance rate as
\begin{equation} \label{def_dist_rate}
\begin{aligned}
\bar d\left( {U,Y} \right) &= \mathop {\lim \sup }\limits_{k \to \infty } \frac{{d\left( {{U_k},{Y_k}} \right)}}{k^{1/2}} \\
&= \mathop {\lim \sup }\limits_{k \to \infty }{\left( {\frac{1}{k}\sum\limits_{i = 1}^k {{{\ln }^2}{\lambda _i}\left( {U_k^{ - 1}{Y_k}} \right)} } \right)^{\frac{1}{2}}}.
\end{aligned}
\end{equation}
Directly substitute (\ref{def_Uk_Yk}) into (\ref{def_dist_rate}), we have \begin{equation}\nonumber \begin{aligned} \frac{{d\left( {{U_k},{Y_k}} \right)}}{{{k^{1/2}}}} &= {\left( {\frac{1}{k}\sum\limits_{i = 1}^k {{{\ln }^2}{\lambda _i}\left( {{A_k}A_k^T} \right)} } \right)^{\frac{1}{2}}} \\ &\ge \frac{1}{k}\sum\limits_{i = 1}^k {\ln {\lambda _i}\left( {{A_k}A_k^T} \right)} \\ &= \frac{1}{k}\ln \left| {{A_k}A_k^T} \right| \\ &= \frac{1}{k}\ln {\left| {{h_0}} \right|^{2k}} \\ &= \ln {\left| {{h_0}} \right|^2} \end{aligned} \end{equation} or, more clearly, $\bar d\left( {U,Y} \right) \ge \ln {\left| {{h_0}} \right|^2}$. Furthermore, by lemma 1 and Papoulis's theorem, we know that \[\bar d\left( {U,Y} \right) \ge \frac{1}{{2\pi }}\int_{ - \pi }^\pi {\ln {{\left| {H\left( {{e^{j\omega }}} \right)} \right|}^2}d\omega } = \bar h\left( y \right) - \bar h\left( u \right).\] On the other hand, the distance rate is bounded by the absolute value of the logarithm of the maximum singular value of $A_k$'s, i.e., \begin{equation} \begin{aligned} \bar d\left( {U,Y} \right) &= \mathop {\lim \sup }\limits_{k \to \infty } {\left( {\frac{1}{k}\sum\limits_{i = 1}^k {{{\ln }^2}{\lambda _i}\left( {{A_k}A_k^T} \right)} } \right)^{\frac{1}{2}}} \\ &\le \mathop {\lim \sup }\limits_{k \to \infty } {\left( {\frac{1}{k}\sum\limits_{i = 1}^k {\max\left\{ {{{\ln }^2}{\lambda _i}\left( {{A_k}A_k^T} \right)} \right\}} } \right)^{\frac{1}{2}}} \\ &\le \mathop {\sup }\limits_k \left( {\left| {\ln {\lambda _i}\left( {{A_k}A_k^T} \right)} \right|} \right). \end{aligned} \end{equation} Generally, we have \[\mathop {\lim \sup }\limits_{k \to \infty } \frac{1}{k}\ln \left| {U_k^{ - 1}{Y_k}} \right| \le \bar d\left( {U,Y} \right) \le \mathop {\sup }\limits_k \left( {\left| {\ln {\lambda _i}\left( {U_k^{ - 1}{Y_k}} \right)} \right|} \right).\] However, the upper bound $\mathop {\sup }\limits_k \left( {\left| {\ln {\lambda _i}\left( {U_k^{ - 1}{Y_k}} \right)} \right|} \right)$ may be conservative or even divergent.
Section 2. A Theorem
Let $f(x)$ be a real-valued function of the class $L$, \[{c_n} = \frac{1}{{2\pi }}\int_{ - \pi }^\pi {{e^{-inx}}f\left( x \right)dx} ,~~~n = 0, \pm 1, \pm 2, \ldots ,\] its Fourier coefficients. We consider the finite Toeplitz forms \begin{equation} \begin{aligned} {T_n}\left( f \right) &= \sum\limits_{i,j = 0,1,2, \ldots ,n} {{c_{j - i}}{u_i}{{\bar u}_j}} \\ &= \frac{1}{{2\pi }}\int_{ - \pi }^\pi {{{\left| {{u_0} + {u_1}{e^{ix}} + \cdots + {u_n}{e^{inx}}} \right|}^2}f\left( x \right)dx} ,\\ &~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~n = 0,1,2, \ldots. \end{aligned} \end{equation} The eigenvalues of the Hermitian form $T_n(f)$ are defined as the roots of the characteristic equation $\det T_n(f-\lambda)=0$; we denote them by $${\lambda _1^{\left( n \right)}},{\lambda _2^{\left( n \right)}},\ldots,{\lambda _n^{\left( n \right)}}.$$ As well known, these values are all real.
Theorem 1. [1] We denote by $m$ and $M$ the 'essential' lower bound and upper bound of $f(x)$, respectively, and assume that $m$ and $M$ are finite. If $G(\lambda)$ is any continuous function defined in the finite interval $m \le \lambda \le M$, we have \[\mathop {\lim }\limits_{n \to \infty } \frac{{G\left( {\lambda _1^{\left( n \right)}} \right) + G\left( {\lambda _2^{\left( n \right)}} \right) + \cdots + G\left( {\lambda _{n + 1}^{\left( n \right)}} \right)}}{{n + 1}} = \frac{1}{{2\pi }}\int_{ - \pi }^\pi {G\left[ {f\left( x \right)} \right]dx}.\]
Corollary. [2] \[\mathop {\lim }\limits_{n \to \infty } \frac{{\sum\nolimits_{i = 1}^n {G\left( {{\lambda _i}\left( {\Sigma \left( {e_1^k} \right)} \right)} \right)} }}{n} = \frac{1}{{2\pi }}\int_{ - \pi }^\pi {G\left[ {{{\hat F}_e}\left( \omega \right)} \right]d\omega } .\]
References
[1] Ulf Grenander and Gabor Szego. Toeplitz Forms and Their Applications. Univ. Calif. Press, 2001.
[2] N.C. Martins and M.A. Dahleh. Fundamental limitations of performance in the presence of finite capacity feedback. In American Control Conference, 2005. Proceedings of the 2005, pages 79–86 vol. 1, June 2005.