## 4 Convergence Analysis of the Proposed Algorithms
\begin{array}{ll}\mathbf{A}_{2}(n)&=\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\}\\ &\approx\mu\rho E\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)\left|\mathbf{v}(n)\right.\right]\right\}\\ &=-\sqrt{2/\pi}\mu\sigma_{e}^{-1}E\left\{E\left[\mathbf{x}(n)e(n)\left|\mathbf{v}(n) \right.\right]f^{T}\left(\mathbf{w}(n)\right)+f\left(\mathbf{w}(n)\right)E\left[e(n) \mathbf{x}^{T}(n)\left|\mathbf{v}(n)\right.\right]\right\}\\ &=-\sqrt{2/\pi}\mu\sigma_{e}^{-1}\left\{\mathbf{R}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]\mathbf{R} \right\}\\ \end{array}  \tag{55}\]  
\[ \begin{array}{ll}\mathbf{A}_{5}(n)&\triangleq\mathbf{A}_{2}(n)+\mathbf{A}_{3}(n)\\ &=-\sqrt{2/\pi}\mu\sigma_{e}^{-1}\left\{\mathbf{R}E\left[\mathbf{v}(n)f^{T}\left(\mathbf{ w}(n)\right)\right]+E\left[f\left(\mathbf{w}(n)\right)\mathbf{v}^{T}(n)\right]\mathbf{R} \right\}\\ &+\rho\left\{E\left[\mathbf{v}(n)f^{T}\left(\mathbf{w}(n)\right)\right]+E\left[f\left( \mathbf{w}(n)\right)\mathbf{v}^{T}(n)\right]\right\}\\ &=\rho\left(I-\sqrt{2/\pi}\mu\sigma_{e}^{-1}\mathbf{R}\right)E\left[\mathbf{v}(n)f^{T} \left(\mathbf{w}(n)\right)\right]+\rho E\left[f\left(\mathbf{w}(n)\right)\mathbf{v}^{T}(n) \right]\left(I-\sqrt{2/\pi}\mu\sigma_{e}^{-1}\mathbf{R}\right)\end{array}  \tag{56}\]  
By substituting Equations (54) and (56) into Equation (53), we obtain  
\[ \begin{array}{ll}\mathbf{C}(n+1)&=\mathbf{C}(n)+\mu^{2}\mathbf{R}+\mathbf{A}_{1}(n)-\mathbf{A}_ {5}(n)+\mathbf{A}_{4}(n)\\ &=\mathbf{C}(n)+\mu^{2}\mathbf{R}-\sqrt{2/\pi}\mu\sigma_{e}^{-1}(n)\left[\mathbf{C}(\mathbf{n} )\mathbf{R}+\mathbf{R}\mathbf{C}(\mathbf{n})\right]-\mathbf{A}_{5}(n)+\mathbf{A}_{4}(n)\end{array}  \tag{57}\]  
Letting \(n\to\infty\) and using Equation (47), Equation (57) is further rewritten as  
\[ \begin{array}{ll}\mathbf{C}(\infty)\mathbf{R}+\mathbf{R}\mathbf{C}(\infty)\approx\sqrt{2/ \pi}\mu\gamma^{1/\alpha}\mathbf{R}+\sqrt{2/\pi}\mu^{-1}\gamma^{1/\alpha}\lim_{n\to \infty}\left[\mathbf{A}_{4}(n)-\mathbf{A}_{5}(n)\right]\end{array}  \tag{58}\]  
Multiplying both sides of Equation (58) by \(\left(\mathbf{I}-\sqrt{2/\pi}\mu\sigma_{e}^{-1}\mathbf{R}\right)^{-1}\) from right, the following can be derived as  
\[ \begin{array}{ll}\left[\mathbf{C}(\infty)\mathbf{R}+\mathbf{R}\mathbf{C}(\infty)\right] \left(\mathbf{I}-\sqrt{2/\pi}\mu\sigma_{e}^{-1}\mathbf{R}\right)^{-1}&\approx\sqrt{2/ \pi}\mu\gamma^{1/\alpha}\mathbf{R}\left(\mathbf{I}-\sqrt{2/\pi}\mu\sigma_{e}^{-1}\mathbf{R }\right)^{-1}\\ &+\sqrt{2/\pi}\mu^{-1}\gamma^{1/\alpha}\lim_{n\to\infty}\mathbf{A}_{4}(n)\left( \mathbf{I}-\sqrt{2/\pi}\mu\sigma_{e}^{-1}\mathbf{R}\right)^{-1}\\ &-\sqrt{2/\pi}\mu^{-1}\gamma^{1/\alpha}\lim_{n\to\infty}\mathbf{A}_{5}(n)\left( \mathbf{I}-\sqrt{2/\pi}\mu\sigma_{e}^{-1}\mathbf{R}\right)^{-1}\end{array}  \tag{59}\]  
Taking the trace of the two sides of Equation (59), since \(\text{Tr}\left(\mathbf{C}(\infty)\mathbf{R}\right)=\text{Tr}\left(\mathbf{R}\mathbf{C}(\infty)\right)\), the excess MSE is derived as  
\[ \begin{array}{ll}P_{ex}(\infty)&=\text{Tr}\left[\mathbf{R}\mathbf{C}(\infty)\right] \\ &=\frac{\sqrt{2/\pi}\mu\gamma^{1/\alpha}}{2\varphi_{1}}\text{Tr}\underbrace{ \left[\mathbf{R}\left(\mathbf{I}-\sqrt{2/\pi}\mu\sigma_{e}^{-1}\mathbf{R}\right)^{-1} \right]}_{\varphi_{2}}\\ &+\frac{\sqrt{2/\pi}\mu^{-1}\gamma^{1/\alpha}}{2\varphi_{1}}\lim_{n\to\infty} \underbrace{\text{Tr}\left[\mathbf{A}_{4}(n)\left(\mathbf{I}-\sqrt{2/\pi}\mu\sigma_{e}^{ -1}\mathbf{R}\right)^{-1}\right]}_{n_{1}(n)}\\ &-\frac{\sqrt{2/\pi}\mu^{-1}\gamma^{1/\alpha}}{2\varphi_{1}}\lim_{n\to\infty} \underbrace{\text{Tr}\left[\mathbf{A}_{5}(n)\left(\mathbf{I}-\sqrt{2/\pi}\mu\sigma_{e}^{-1 }\mathbf{R}\right)^{-1}\right]}_{n_{2}(n)}\end{array}