## 4 Convergence Analysis of the Proposed Algorithms
 \tag{60}\]  
The matrix \(\left(\mathbf{I}-\sqrt{2/\pi}\mu\gamma^{-1/\alpha}\mathbf{R}\right)\) is symmetric, and its eigenvalue decomposition can be written as  
\[\mathbf{I}-\sqrt{2/\pi}\mu\gamma^{-1/\alpha}\mathbf{R}=\mathbf{U}\mathbf{V}\mathbf{U}^{T} \tag{61}\]with \(\mathbf{U}\) being the orthonormal matrix of eigenvectors and \(\mathbf{V}\) being a diagonal matrix of eigenvalues. Therefore, \(\left(\mathbf{I}-\sqrt{2/\pi}\mu\sigma_{e}^{-1}\mathbf{R}\right)=\mathbf{U}\mathbf{V}^{-1}\mathbf{U}^ {T}\). Let \(\uplambda_{\max}\) be the largest eigenvalue of the covariance matrix \(\mathbf{R}\) and \(\mu\) be small enough such that \(\left(1-\sqrt{2/\pi}\mu\gamma^{-1/\uplambda}\uplambda_{\max}\right)^{-1}>0\) under certain noise cases. Since \(\mathbf{V}^{-1}\) is a diagonal matrix, elements are all non-negative and less than or equal to \(\left(1-\sqrt{2/\pi}\mu\gamma^{-1/\uplambda}\uplambda_{\max}\right)^{-1}\), hence, \(\uplambda_{1}\triangleq\lim_{n\to\infty}\uplambda_{1}(n)\) and \(\uplambda_{2}\triangleq\lim_{n\to\infty}\uplambda_{2}(n)\) are further derived as  
\[\uplambda_{1} \triangleq\lim_{n\to\infty}\uplambda(n)=\lim_{n\to\infty}\text{ Tr}\left\{A_{4}\left(\mathbf{I}-\sqrt{2/\pi}\mu\sigma_{e}^{-1}\mathbf{R}\right)^{-1}\right\} \tag{62}\] \[=\lim_{n\to\infty}\rho^{2}E\left\{\text{Tr}\left[f\left(\mathbf{w}( n)\right)\left(\mathbf{I}-\sqrt{2/\pi}\mu\gamma^{1/\uplambda}\mathbf{R}\right)^{-1}f^{T} \left(\mathbf{w}(n)\right)\right]\right\}\] \[=\lim_{n\to\infty}\rho^{2}E\left\{\text{Tr}\left[f\left(\mathbf{w}( n)\right)\mathbf{U}\mathbf{V}^{-1}\mathbf{U}^{T}f^{T}\left(\mathbf{w}(n)\right)\right]\right\}\] \[=\lim_{n\to\infty}\rho^{2}E\left\{\text{Tr}\left[V^{-1}\mathbf{U}^{T }f\left(\mathbf{w}(n)\right)f^{T}\left(\mathbf{w}(n)\right)\mathbf{U}\right]\right\}\] \[\leq\lim_{n\to\infty}\frac{\rho^{2}}{1-\sqrt{2/\pi}\mu\gamma^{-1/ \uplambda}\uplambda_{\max}}E\left\{\text{Tr}\left[\mathbf{U}^{T}f\left(\mathbf{w}( n)\right)f^{T}\left(\mathbf{w}(n)\right)\mathbf{U}\right]\right\}\] \[=\frac{\rho^{2}}{1-\sqrt{2/\pi}\mu\gamma^{-1/\uplambda}\uplambda _{\max}}E\left\{\text{Tr}\left[f^{T}\left(\mathbf{w}(\infty)\right)\mathbf{U}\mathbf{U}^{ T}f\left(\mathbf{w}(\infty)\right)\right]\right\}\] \[\leq\frac{\rho^{2}}{1-\sqrt{2/\pi}\mu\gamma^{-1/\uplambda} \uplambda_{\max}}\sum_{\left\{f^{T}\left(\mathbf{w}(\infty)\right)f\left(\mathbf{w} (\infty)\right)\right\}}\leq\frac{N\rho^{2}}{1-\sqrt{2/\pi}\mu\gamma^{-1/ \uplambda}\uplambda_{\max}}\]  
\[\uplambda_{2} \triangleq\lim_{n\to\infty}\uplambda_{2}(n)=\lim_{n\to\infty} \text{Tr}\left\{A_{5}\left(\mathbf{I}-\sqrt{2/\pi}\mu\sigma_{e}^{-1}\mathbf{R}\right)^ {-1}\right\} \tag{63}\] \[=\lim_{n\to\infty}\rho E\left\{\text{Tr}\left[\mathbf{v}(n)f^{T} \left(\mathbf{w}(n)\right)+f\left(\mathbf{w}(n)\right)\mathbf{v}^{T}(n)\right]\right\}\] \[\uplambda_{1} \triangleq\lim_{n\to\infty}\uplambda(n)=\lim_{n\to\infty}\text{ Tr}\left\{A_{4}\left(\mathbf{I}-\sqrt{2/\pi}\mu\sigma_{e}^{-1}\mathbf{R}\right)^{-1}\right\} \tag{62}\]0 \[\uplambda_{1} \triangleq\lim_{n\to\infty}\uplambda(n)=\lim_{n\to\infty}\text{ Tr}\left\{A_{4}\left(\mathbf{I}-\sqrt{2/\pi}\mu\sigma_{e}^{-1}\mathbf{R}\right)^{-1}\right\} \tag{62}\]1  
Substituting Equations (62) and (63) into Equation (60), the excess MSE is finally derived as  
\[\uplambda_{1} \triangleq\lim_{n\to\infty}\uplambda(n)=\lim_{n\to\infty}\text{ Tr}\left\{A_{4}\left(\mathbf{I}-\sqrt{2/\pi}\mu\sigma_{e}^{-1}\mathbf{R}\right)^{-1}\right\} \tag{62}\]2  
where \(\uplambda_{1}^{\prime}\triangleq\uplambda_{1}/\rho^{2}\) and \(\uplambda_{2}^{\prime}\triangleq\uplambda_{2}/\rho^{2}\). According to Equation (62), one can find that \(\uplambda_{1}^{\prime}\) is bound as  
\[\uplambda_{1} \triangleq\lim_{n\to\infty}\uplambda(n)=\lim_{n\to\infty}\text{ Tr}\left\{A_{4}\left(\mathbf{I}-\sqrt{2/\pi}\mu\sigma_{e}^{-1}\mathbf{R}\right)^{-1}\right\} \tag{62}\]3  
The excess MSE of sparse SLMS in Equation (65) implies that choosing suitable \(\rho<\uplambda_{1}^{\prime}/\uplambda_{2}^{\prime}\) can lead to smaller excess MSE than standard SLMS algorithm.