## Appendix B Model with Non-Gaussian Linear Scalar Filter
### Moment Equations and Stationary Solutions for Non-GLSF Model
\begin{split}\frac{dE[z_{x_{2}}(t,l)z_{x_{3}}(t,l)]}{dt}=(\alpha_{x}^{l }+\lambda_{x_{2}}^{l}(\gamma_{1}^{l})^{2})E[z_{x_{3}}(t,l)]-(\beta_{x_{1}}^{l} \lambda_{x_{1}}^{l}-3(\gamma_{1}^{l})^{2}\lambda_{x_{3}}^{l})E[z_{x_{2}}(t,l)z_ {x_{3}}(t,l)]\\ \\ -2\beta_{x_{1}}^{l}\lambda_{x_{2}}^{l}E[z_{x_{3}}(t,l)]-2\beta_{x_ {1}}^{l}E[z_{x_{1}}(t,l)z_{x_{3}}(t,l)]\\ \\ -3\beta_{x_{1}}^{l}\lambda_{x_{3}}^{l}E[z_{x_{3}}(t,l)z_{x_{4}}(t,l)]+2(\gamma_{x_{1}}^{l})^{2}\lambda_{x_{1}}^{l}E[z_{x_{2}}(t,l)]+\\ +4(\gamma_{x_{1}}^{l})^{2}\lambda_{x_{2}}^{l}E[z_{x_{2}}^{2}(t,l )]+(\gamma_{x_{1}}^{l})^{2}E[z_{x_{1}}(t,l)]+6(\gamma_{x_{1}}^{l})^{2}\lambda_ {x_{3}}^{l}E[z_{x_{2}}(t,l)z_{x_{3}}(t,l)],\\ \frac{dE[z_{x_{1}}(t,l)z_{x_{4}}(t,l)]}{dt}=(\alpha_{x}^{l}+ \lambda_{x_{2}}^{l}(\gamma_{1}^{l})^{2})E[z_{x_{4}}(t,l)]-(\beta_{x_{1}}^{l} \lambda_{x_{1}}^{l}-3(\gamma_{1}^{l})^{2}\lambda_{x_{3}}^{l})E[z_{x_{2}}(t,l) z_{x_{4}}(t,l)]\\ \\ -2\beta_{x_{1}}^{l}\lambda_{x_{2}}^{l}E[z_{x_{3}}(t,l)z_{x_{4}}(t,l)]-3\beta_{x_{1}}^{l}E[z_{x_{1}}(t,l)z_{x_{4}}(t,l)]\\ -3\beta_{x_{1}}^{l}\lambda_{x_{3}}^{l}E[z_{x_{4}}^{2}(t,l)]+3( \gamma_{x_{1}}^{l})^{2}\lambda_{x_{1}}^{l}E[z_{x_{3}}(t,l)]\\ +6(\gamma_{x_{1}}^{l})^{2}\lambda_{x_{2}}^{l}E[z_{x_{2}}(t,l)z_{x _{3}}(t,l)]+3(\gamma_{x_{1}}^{l})^{2}E[z_{x_{1}}(t,l)z_{x_{2}}(t,l)]+9(\gamma_ {x_{1}}^{l})^{2}\lambda_{x_{3}}^{l}E[z_{x_{3}}^{2}(t,l)],\\ \frac{dE[z_{x_{2}}^{2}(t,l)]}{dt}=-2\beta_{x_{1}}^{l}E[z_{x_{2}}^{ 2}(t,l)]+\gamma_{x_{1}}^{l},\\ \frac{dE[z_{x_{3}}^{2}(t,l)]}{dt}=-4\beta_{x_{1}}^{l}E[z_{x_{3}}^{ 2}(t,l)]+2(\gamma_{x_{1}}^{l})^{2}E[z_{x_{3}}(t,l)]+4(\gamma_{x_{1}}^{l})^{2} E[z_{x_{2}}^{2}(t,l)],\\ \frac{dE[z_{x_{2}}(t,l)z_{x_{3}}(t,l)]}{dt}=-\beta_{x_{1}}^{l}E[z_ {x_{2}}(t,l)z_{x_{3}}(t,l)]-2\beta_{x_{1}}^{l}E[z_{x_{2}}(t,l)z_{x_{3}}(t,l)] \\ +(\gamma_{x_{1}}^{l})^{2}E[z_{x_{2}}(t,l)]+2(\gamma_{x_{1}}^{l})^{ 2}E[z_{x_{2}}(t,l)],\\ \frac{dE[z_{x_{2}}(t,l)z_{x_{4}}(t,l)]}{dt}=-\beta_{x_{1}}^{l}E[z_ {x_{2}}(t,l)z_{x_{4}}(t,l)]-3\beta_{x_{1}}^{l}E[z_{x_{2}}(t,l)z_{x_{4}}(t,l)] \\ +3(\gamma_{x_{1}}^{l})^{2}E[z_{x_{2}}^{2}(t,l)]+3(\gamma_{x_{1}}^{ l})^{2}E[z_{x_{3}}(t,l)],\\ \frac{dE[z_{x_{4}}^{2}(t,l)]}{dt}=-6\beta_{x_{1}}^{l}E[z_{x_{4}}^{ 2}(t,l)]+6(\gamma_{x_{1}}^{l})^{2}E[z_{x_{2}}(t,l)z_{x_{4}}(t,l)]+9(\gamma_{x _{1}}^{l})^{2}E[z_{x_{3}}^{2}(t,l)],\\ \frac{dE[z_{x_{3}}(t,l)z_{x_{4}}(t,l)]}{dt}=-2\beta_{x_{1}}^{l}E[z_ {x_{3}}(t,l)z_{x_{4}}(t,l)]-3\beta_{x_{1}}^{l}E[z_{x_{3}}(t,l)z_{x_{4}}(t,l)] \\ +(\gamma_{x_{1}}^{l})^{2}E[z_{x_{4}}(t,l)]+3(\gamma_{x_{1}}^{l})^{ 2}E[z_{x_{2}}(t,l)z_{x_{3}}(t,l)]\\ +6(\gamma_{x_{1}}^{l})^{2}E[z_{x_{2}}(t,l)z_{x_{3}}(t,l)].\end{split}  \tag{123}\]  
_Appendix B.2. Partially Stationary Solutions for Non-GLSF Model_  
Equating the derivatives in Equations (A18)-(A20) to zero, we obtain  
\[E[z_{x_{2}}(t,l)]=E[z_{x_{4}}(t,l)]=0,\quad E[z_{x_{3}}(t,l)]=\frac{\gamma_{x_ {1}}^{2}}{2\beta_{x_{1}}} \tag{124}\]  
Hence, from conditions (A21) and equalities (A31), we find the nonstationary solution for the first moment of the process \(z_{x_{1}}(t,l)\)  
\[E[z_{x_{1}}(t,l)]=\alpha_{x}^{l}t+\alpha_{\
u}^{l}, \tag{125}\]where \(\alpha_{0}^{l}\) is an integration constant.