An AR process is generated by applying white Gaussian noise, with variance \(\sigma_{X}^{2}\), to a first-order filter with transfer function

\[H(z)=\frac{z}{z-a} .\]

This process has the autocorrelation matrix

\[\mathbf{R}_{Y}=\frac{\sigma_{X}^{2}}{1-a^{2}}\left[\begin{array}{cccc}1 & a & \cdots & a^{7} \\a & 1 & \cdots & a^{6} \\\vdots & \vdots & \ddots & \vdots \\a^{7} & a^{6} & \cdots & 1\end{array}\right]\]

whose inverse can be shown to be \[\mathbf{R}_{Y}^{-1}=\frac{1}{\sigma_{X}^{2}}\left[\begin{array}{ccccc}
1 & -a & \cdots & 0 & 0 \\
-a & 1+a^{2} & \cdots & 0 & 0 \\
0 & -a & \cdots & 0 & 0 \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
0 & 0 & \cdots & 1+a^{2} & -a \\
0 & 0 & \cdots & -a & 1 \end{array}\right]\]
For the signal above, calculate a closed-form solution for its minimum-variance estimate and comment on this solution when \(L\) approaches infinity.