## 4 Convergence Analysis of the Revised CIM-PNMCC
### Mean Square Convergence (MSC)  
As for the MSC of our CIM-PNMCC, we give the auto-covariance of \(\hat{\mathbf{w}}\left(n\right)\)  
\[\text{S}\left(n\right)=\text{E}\left[z\left(n\right)z^{T}\left(n\right)\right], \tag{38}\]  
where  
\[z\left(n\right)=\hat{\mathbf{w}}\left(n\right)-\text{E}\left[\hat{\mathbf{w}} \left(n\right)\right]. \tag{39}\]Then, considering Equations (39), (32) and (33), we get  
\[z\left(n+1\right) =\left[\mathbf{I}-\frac{\chi_{1}\mathbf{G}\left(n\right)f\left(e \left(n\right)\right)\mathbf{x}\left(n\right)\mathbf{x}^{T}\left(n\right)}{ \mathbf{x}^{T}\left(n\right)\mathbf{G}\left(n\right)\mathbf{x}\left(n\right)+ \varepsilon_{\text{CIM}}}\right]\mathbf{\hat{w}}\left(n\right)+\frac{\chi_{1} \mathbf{G}\left(n\right)f\left(e\left(n\right)\right)v\left(n\right)\mathbf{x} \left(n\right)}{\mathbf{x}^{T}\left(n\right)\mathbf{G}\left(n\right)\mathbf{x} \left(n\right)+\varepsilon_{\text{CIM}}}+\rho_{\text{CIM}}m\left(n\right) \tag{40}\] \[-\rho_{\text{CIM}}\text{E}\left[m\left(n\right)\right]+\chi_{1} \left[\frac{\mathbf{G}\left(n\right)\mathbf{R}}{N\sigma_{x}^{2}+\varepsilon_{ \text{CIM}}}-\frac{\mathbf{G}\left(n\right)f\left(e\left(n\right)\right) \mathbf{x}\left(n\right)\mathbf{x}^{T}\left(n\right)}{\mathbf{x}^{T}\left(n \right)\mathbf{G}\left(n\right)\mathbf{x}\left(n\right)+\varepsilon_{\text{ CIM}}}\right]\text{E}\left[\mathbf{\hat{w}}\left(n\right)\right]\] \[=\mathbf{\Phi}\left(n\right)z\left(n\right)+\chi_{1}\text{Y}\left( n\right)\text{E}\left[\mathbf{\hat{w}}\left(n\right)\right]+\frac{\chi_{1} \mathbf{G}\left(n\right)f\left(e\left(n\right)\right)v\left(n\right)\mathbf{x} \left(n\right)}{\mathbf{x}^{T}\left(n\right)\mathbf{G}\left(n\right)\mathbf{x }\left(n\right)+\varepsilon_{\text{CIM}}}+\rho_{\text{CIM}}\text{B}\left(n \right),\]  
where  
\[\mathbf{\Phi}\left(n\right) =\left[\mathbf{I}-\frac{\chi_{1}\mathbf{G}\left(n\right)f\left(e \left(n\right)\right)\mathbf{x}\left(n\right)\mathbf{x}^{T}\left(n\right)}{ \mathbf{x}^{T}\left(n\right)\mathbf{G}\left(n\right)\mathbf{x}\left(n\right)+ \varepsilon_{\text{CIM}}}\right], \tag{41}\] \[\text{Y}\left(n\right) =\left[\frac{\mathbf{G}\left(n\right)\mathbf{R}}{N\sigma_{x}^{2 }+\varepsilon_{\text{CIM}}}-\frac{\mathbf{G}\left(n\right)f\left(e\left(n \right)\right)\mathbf{x}\left(n\right)\mathbf{x}^{T}\left(n\right)}{\mathbf{x }^{T}\left(n\right)\mathbf{G}\left(n\right)\mathbf{x}\left(n\right)+ \varepsilon_{\text{CIM}}}\right],\] \[\text{B}\left(n\right) =m\left(n\right)-\text{E}\left[m\left(n\right)\right].\]  
Under the assumptions 1-5, we can say that \(\text{Y}\left(n\right)\) and \(\text{B}\left(n\right)\) are both zero means. In addition, we know that \(\mathbf{\hat{w}}\left(n\right)\), \(\mathbf{x}\left(n\right)\) and \(v\left(n\right)\) are independent with each other. Putting Equation (40) into Equation (38), we have  
\[\text{S}\left(n+1\right) =\text{E}\left[ \begin{array}{c}\left(\mathbf{\Phi}\left(n \right)z\left(n\right)+\chi_{1}\text{Y}\left(n\right)\text{E}\left[\mathbf{\hat {w}}\left(n\right)\right]+\frac{\chi_{1}\mathbf{G}\left(n\right)f\left(e \left(n\right)\right)v\left(n\right)\mathbf{x}\left(n\right)}{\mathbf{x}^{T} \left(n\right)\mathbf{G}\left(n\right)\mathbf{x}\left(n\right)+\varepsilon_{ \text{CIM}}}+\rho_{\text{CIM}}\text{B}\left(n\right)\right)\times\\ \left(\mathbf{\Phi}\left(n\right)z\left(n\right)+\chi_{1}\text{Y}\left(n\right) \text{E}\left[\mathbf{\hat{w}}\left(n\right)\right]+\frac{\chi_{1}\mathbf{G} \left(n\right)f\left(e\left(n\right)\right)v\left(n\right)\mathbf{x}\left(n \right)}{\mathbf{x}^{T}\left(n\right)\mathbf{G}\left(n\right)\mathbf{x}\left(n \right)+\varepsilon_{\text{CIM}}}+\rho_{\text{CIM}}\text{B}\left(n\right) \right)^{T}\end{array}