## 4 Convergence Analysis of the Revised CIM-PNMCC
### Mean Square Convergence (MSC)
\right] \tag{42}\] \[=\text{E}\left[\mathbf{\Phi}\left(n\right)z\left(n\right)z^{T} \left(n\right)\mathbf{\Phi}^{T}\left(n\right)\right]+\rho_{\text{CIM}}\text{E} \left[\mathbf{\Phi}\left(n\right)z\left(n\right)\text{B}^{T}\left(n\right)\right]\] \[+\chi_{1}^{2}\text{E}\left[\text{Y}\left(n\right)\varphi\left(n \right)\varphi^{T}\left(n\right)\text{Y}^{T}\left(n\right)\right]+\frac{\chi_{1} \mathbf{G}^{2}\left(n\right)\text{E}\left[f^{2}\left(e\left(n\right)\right) \right]\sigma_{x}^{2}\sigma_{v}^{2}}{\left(N\sigma_{x}^{2}+\varepsilon_{\text{ CIM}}\right)^{2}}+\rho_{\text{CIM}}^{2}\text{E}\left[\text{B}\left(n \right)\text{B}^{T}\left(n\right)\right]\] \[+\rho_{\text{CIM}}\text{E}\left[\text{B}\left(n\right)z^{T} \left(n\right)\mathbf{\Phi}^{T}\left(n\right)\right],\]  
where \(\varphi\left(n\right)=\text{E}\left[\mathbf{\hat{w}}\left(n\right)\right]\). Since the Gaussian variable with fourth-order moment is three times the variable square [52], and we also know that \(\text{S}\left(n\right)\) is symmetric [52], we obtain  
\[\text{E}\left[\mathbf{\Phi}\left(n\right)z\left(n\right)z^{T} \left(n\right)\mathbf{\Phi}^{T}\left(n\right)\right]=\left[1-\frac{2\chi_{1} \mathbf{G}\left(n\right)\text{E}\left[f\left(e\left(n\right)\right)\right] \sigma_{x}^{2}}{N\sigma_{x}^{2}+\varepsilon_{\text{CIM}}}+\frac{2\chi_{1}^{2} \mathbf{G}^{2}\left(n\right)\text{E}\left[f^{2}\left(e\left(n\right)\right) \right]\sigma_{x}^{4}}{\left(N\sigma_{x}^{2}+\varepsilon_{\text{CIM}}\right)^{2}} \right]S\left(n\right) \tag{43}\] \[+\frac{\chi_{1}^{2}\mathbf{G}^{2}\left(n\right)\text{E}\left[f^{2} \left(e\left(n\right)\right)\right]\sigma_{x}^{4}}{\left(N\sigma_{x}^{2}+ \varepsilon_{\text{CIM}}\right)^{2}}tr\left[\text{S}\left(n\right)\right]\] \[\text{E}\left[\text{Y}\left(n\right)\varphi\left(n\right)\varphi^{T} \left(n\right)\text{Y}^{T}\left(n\right)\right]=\frac{\mathbf{G}^{2}\left(n \right)\text{E}\left[f^{2}\left(e\left(n\right)\right)\right]\sigma_{x}^{4}}{ \left(N\sigma_{x}^{2}+\varepsilon_{\text{CIM}}\right)^{2}}\left[\varphi\left(n \right)\varphi^{T}\left(n\right)+tr\left[\varphi\left(n\right)\varphi^{T}\left(n \right)\right]I_{\text{N}}\right].\]  
Moreover, using Equations (31) and (39) and \(\text{B}\left(n\right)\), we have  
\[\text{E}\left[\mathbf{\Phi}\left(n\right)z\left(n\right)\text{B}^{T} \left(n\right)\right]=\text{E}\left[\text{B}\left(n\right)z^{T}\left(n\right) \mathbf{\Phi}^{T}\left(n\right)\right]=\left[1-\frac{\chi_{1}\mathbf{G} \left(n\right)\text{E}[f\left(e\left(n\right)\right)]\varphi_{x}^{2}}{N \sigma_{x}^{2}+\varepsilon_{\text{CIM}}}\right]\text{E}\left[\mathbf{\hat{w}} \left(n\right)\text{B}^{T}\left(n\right)\right]. \tag{44}\]According to Equations (42)-(44), we have  
\[ \begin{split} tr\left[\mathrm{S}\left(n+1\right)\right]& =\left[1-\frac{2\chi_{1}tr[\mathbf{G}\left(n\right)]\mathrm{E}[f \left(\left(e\left(n\right)\right)]\varphi_{x}^{2}}{N\sigma_{x}^{2}+\epsilon_{ \mathrm{CM}}}+\left(N+2\right)\frac{\chi_{1}^{2}tr\left[\mathbf{G}^{2}\left(n \right)\right]\mathrm{E}\left[f^{2}\left(e\left(n\right)\right)\right]\sigma_{ x}^{4}}{\left(N\sigma_{x}^{2}+\epsilon_{\mathrm{CM}}\right)^{2}}\right]tr\left[ \mathrm{S}\left(n\right)\right]\\ &+2\rho_{\mathrm{CM}}\left[1-\frac{\chi_{1}tr[\mathbf{G}\left(n \right)]\mathrm{E}[f\left(e\left(n\right)\right)]\varphi_{x}^{2}}{N\sigma_{x}^ {2}+\epsilon_{\mathrm{CM}}}\right]\mathrm{E}\left[\tilde{\mathbf{w}}\left(n \right)\mathrm{B}^{T}\left(n\right)\right]\\ &+\chi_{1}^{2}\left(N+1\right)\frac{tr[\mathbf{G}^{2}\left(n \right)]\mathrm{E}[f^{2}\left(e\left(n\right)\right)]\sigma_{x}^{4}}{\left(N \sigma_{x}^{2}+\epsilon_{\mathrm{CM}}\right)^{2}}\varphi\left(n\right)\varphi ^{T}\left(n\right)\\ &+\frac{N\chi_{1}^{2}\mathrm{G}^{2}\left(n\right)\mathrm{E} \left[f^{2}\left(e\left(n\right)\right)\right]\sigma_{x}^{2}\varphi_{x}^{2}}{ \left(N\sigma_{x}^{2}+\epsilon_{\mathrm{CM}}\right)^{2}}+\rho_{\mathrm{CM}}^{2 }\mathrm{E}\left[\mathrm{B}\left(n\right)\mathrm{B}^{T}\left(n\right)\right]. \end{split}