B.2 Proof of Proposition 3
By the definition of Z^{(H)}=Z^{(H)}_{X,F} ,

\lim_{H\rightarrow\infty}\operatorname{vec}[Z^{(H)}]=\lim_{H\rightarrow\infty}\operatorname{vec}[\gamma g(F)Z^{(H-1)}S+X]=\gamma[S\otimes g(F)]\left(\lim_{H\rightarrow\infty}\operatorname{vec}[Z^{(H-1)}]\right)+\operatorname{vec}[X].

Since we have shown that the limit exists in Proposition 1 , we can define Z_{X,F}=\lim_{H\rightarrow\infty}Z^{(H)} . Then the above equation can be rewritten as

\operatorname{vec}[Z_{X,F}]=\gamma[S\otimes g(F)]\operatorname{vec}[Z_{X,F}]+\operatorname{vec}[X],

which is equivalent to

\operatorname{vec}[Z_{X,F}]-\gamma[S\otimes g(F)]\operatorname{vec}[Z_{X,F}]-\operatorname{vec}[X]=0.

Define a function

\varphi(\operatorname{vec}[F],\operatorname{vec}[Z])=\operatorname{vec}[Z]-\gamma[S\otimes g(F)]\operatorname{vec}[Z]-\operatorname{vec}[X],

where F and Z are independent variables of this function \varphi . For general F and Z , \varphi(\operatorname{vec}[F],\operatorname{vec}[Z]) is allowed to be nonzero. In contrast, we have \varphi(\operatorname{vec}[F],\operatorname{vec}[Z_{X,F}])=0 where F and Z_{X,F} are dependent variables because of the definition of Z_{X,F}=\lim_{H\rightarrow\infty}Z^{(H)}_{X,F} .

With this definition, we have

J(\bar{F},\bar{Z})=\left.\frac{\partial\varphi(\operatorname{vec}[F],\operatorname{vec}[Z])}{\partial\operatorname{vec}[Z]}\right|_{(F,Z)=(\bar{F},\bar{Z})}=I-\gamma[S\otimes g(\bar{F})].

Here, \varphi(\operatorname{vec}[F],\operatorname{vec}[Z_{X,F}])=0 and the Jacobian J(F,Z_{X,F})=I-\gamma[S\otimes g(F)] is invertible as shown in Appendix B.1 . Therefore, applying the implicit function theorem to the function \varphi yields the following: there exists an open set containing F and a function \psi defined on the open set such that \psi(F)=\operatorname{vec}[Z_{X,F}] , \varphi(\operatorname{vec}[\bar{F}],\psi(\bar{F}))=0 for all \bar{F} in the open set, and \frac{\partial\psi(\bar{F})}{\partial\operatorname{vec}[\bar{F}]}=-J(\bar{F},\bar{Z})^{-1}\frac{\partial\varphi(\operatorname{vec}[\bar{F}],\operatorname{vec}[\bar{Z}])}{\partial\operatorname{vec}[\bar{F}]} for all \bar{F} in the open set. From the above equivalent definition of Z_{X,F} (i.e., \varphi(\operatorname{vec}[F],\operatorname{vec}[Z_{X,F}])=0 ) ,
this implies that

\frac{\partial\operatorname{vec}[Z_{X,F}]}{\partial\operatorname{vec}[F]}=-J(F,Z_{X,F})^{-1}\left.\frac{\partial\varphi(\operatorname{vec}[F],\operatorname{vec}[Z])}{\partial\operatorname{vec}[F]}\right|_{Z=Z_{X,F}}

Using the definition of \varphi

\frac{\partial\varphi(\operatorname{vec}[F],\operatorname{vec}[Z])}{\partial\operatorname{vec}[F]} =\frac{\partial}{\partial\operatorname{vec}[F]}\operatorname{vec}[Z]-\gamma\operatorname{vec}[g(F)ZS]-\operatorname{vec}[X] =\frac{\partial}{\partial\operatorname{vec}[F]}\operatorname{vec}[Z]-\gamma[SZ^{\top}\otimes I_{m}]\operatorname{vec}[g(F)]-\operatorname{vec}[X] =-\gamma[SZ^{\top}\otimes I_{m}]\frac{\partial\operatorname{vec}[g(F)]}{\partial\operatorname{vec}[F]}

Combining,

\frac{\partial\operatorname{vec}[Z_{X,F}]}{\partial\operatorname{vec}[F]}=-J(F,Z_{X,F})^{-1}\left.\frac{\partial\varphi(\operatorname{vec}[F],\operatorname{vec}[Z])}{\partial\operatorname{vec}[F]}\right|_{Z=Z_{X,F}}=\gamma(I-\gamma[S\otimes g(F)])^{-1}[SZ_{X,F}^{\top}\otimes I_{m}]\frac{\partial\operatorname{vec}[g(F)]}{\partial\operatorname{vec}[F]},

where Z_{X,F}=\lim_{H\rightarrow\infty}Z^{(H)}_{X,F} and
Proposition 1 has shown that

\lim_{H\rightarrow\infty}\operatorname{vec}[Z^{(H)}_{X,F}]=\left(I-\gamma[S\otimes g(F)]\right)^{-1}\operatorname{vec}[X].

∎