## 4 Convergence Analysis of the Proposed Algorithms
Assuming \(\rho\) is sufficiently small so that for every \(i\in\Omega\), the excess MSE of sparse SLMS algorithms is_  
\[P_{ex}(\infty)=\frac{\mu\gamma^{1/\alpha}\varphi_{2}}{\sqrt{2\pi}\varphi_{1}} +\frac{\rho\gamma^{1/\alpha}\eta_{1}^{\prime}}{\sqrt{2\pi}\mu\varphi_{1}}\left( \rho-\frac{\eta_{2}^{\prime}}{\eta_{1}^{\prime}}\right) \tag{48}\]  
_where \(\varphi_{1}\), \(\varphi_{2}\), \(\eta_{1}^{\prime}\) and \(\eta_{2}^{\prime}\) are defined as:_  
\[\varphi_{1}=\text{Tr}[\left(\mathbf{I}-\sqrt{2/\pi}\mu\sigma_{e}^{-1}\mathbf{R} \right)^{-1}] \tag{49}\]  
\[\varphi_{2}=\text{Tr}[\mathbf{R}(\mathbf{I}-\sqrt{2/\pi}\mu\sigma_{e}^{-1}\mathbf{R})^{-1 }]\lhd \tag{50}\]  
\[\eta_{1}^{\prime}\triangleq\frac{N}{1-\sqrt{2/\pi}\mu\gamma^{-1/\alpha}\lambda _{\max}} \tag{51}\]  
\[\eta_{2}^{\prime}\triangleq 2E\left\{\text{Tr}\left[\mathbf{w}(\infty)f^{T}\left(\bm {w}(\infty)\right)\right]\right\}-2E\left\{\text{Tr}\left[\mathbf{w}f^{T}\left( \mathbf{w}(\infty)\right)\right]\right\} \tag{52}\]  
Proof: By using the above independent assumptions in Equations (49)-(52), the second moment \(\mathbf{C}(n+1)\) of the weight error vector \(\mathbf{v}(n+1)\) can be evaluated recursively as  
\[ \begin{array}{ll}\mathbf{C}(n+1)&=E\left\{\mathbf{v}(n+1)\mathbf{v}^{T}(n+1)\right\}\\ &=E\left\{\left[\mathbf{v}(n)+\mu\text{sgn}(e(n))\mathbf{x}(n)-\rho f\left(\mathbf{w}(n) \right)\right]\left[\mathbf{v}(n)+\mu\text{sgn}(e(n))\mathbf{x}(n)-\rho f\left(\mathbf{w}( n)\right)\right]^{T}\right\}\\ &=E\left\{\mathbf{v}(n)\mathbf{v}^{T}(n)\right\}+\mu^{2}E\left\{\mathbf{x}(n)\mathbf{x}^{T}(n) \right\}\\ &+\mu\left\{E\left[\text{sgn}\left(e(n)\right)\mathbf{v}(n)\mathbf{x}^{T}(n)\right]+E \left[\text{sgn}\left(e(n)\right)\mathbf{x}(n)\mathbf{v}^{T}(n)\right]\right\}\\ &-\underbrace{\mu\rho\left\{E\left[\text{sgn}\left(e(n)\right)\mathbf{x}(n)f^{T} \left(\mathbf{w}(n)\right)\right]+E\left[\text{sgn}\left(e(n)\right)f\left(\mathbf{w }(n)\right)\mathbf{x}^{T}(n)\right]\right\}}_{A_{2}(n)}\\ &-\underbrace{\rho\left\{E\left[\mathbf{v}(n)f^{T}\left(\mathbf{w}(n)\right)\right]+E \left[f\left(\mathbf{w}(n)\right)\mathbf{v}^{T}(n)\right]\right\}}_{A_{3}(n)}+\underbrace{ \rho^{2}E\left\{f\left(\mathbf{w}(n)\right)f^{T}\left(\mathbf{w}(n)\right)\right\}}_{A_{4} (n)}\\ &=\mathbf{C}(n)+\mu^{2}\mathbf{R}+\mathbf{A}_{1}(n)-\underbrace{\left(\mathbf{A}_{2}(n)+\mathbf{A}_ {3}(n)\right)}_{A_{5}(n)}+\mathbf{A}_{4}(n)\end{array}  \tag{53}\]where \(\mathbf{A}_{1}(n)\), \(\mathbf{A}_{2}(n)\), and \(\mathbf{A}_{5}(n)\) are further derived as  
\[ \begin{array}{ll}\mathbf{A}_{1}(n)&\approx\mu E\left\{\text{sgn}\left(e(n)\right) \mathbf{v}(n)\mathbf{x}^{T}(n)\left|\mathbf{v}(n)\right.\right\}+\mu E\left\{\text{sgn} \left(e(n)\right)\mathbf{x}(n)\mathbf{v}^{T}(n)\left|\mathbf{v}(n)\right.\right\}\\ &=-\sqrt{\pi/2}\mu\sigma_{e}^{-1}(n)\left\{E\left[\mathbf{v}(n)E\left[e(n)\mathbf{x}^{ T}(n)\left|\mathbf{v}(n)\right.\right]\right]+E\left[E\left[e(n)\mathbf{x}(n)\left|\mathbf{v}(n )\right.\right]\mathbf{v}^{T}(n)\right]\right\}\\ &=-\sqrt{\pi/2}\mu\sigma_{e}^{-1}(n)\left\{E\left[\mathbf{v}(n)\mathbf{v}^{T}(n) \right]\mathbf{R}+\mathbf{R}E\left[\mathbf{v}(n)\mathbf{v}^{T}(n)\right]\right\}\\ &=-\sqrt{\pi/2}\mu\sigma_{e}^{-1}(n)\left[\mathbf{C}(\mathbf{n})\mathbf{R}+\mathbf{R}\mathbf{C}( \mathbf{n})\right]\end{array}  \tag{54}\]  
\[