any $ \bP \in \sP_\mu $, let $ \lambda = (\lambda_k , k \in -\N) $ denote the marginal distributions of $ \eta_k $ on $ G $ , i.e., $$\lambda_k(\cdot) = \bP (\eta_k \in \cdot) , 
\qquad k \in -\N. 
\label{marginal}$$ Then the equation implies that holds. Conversely, the following holds.

\[ent law\] Let $ \mu $ be a given noise law. Let $ \lambda = (\lambda_k , k \in -\N) $ be an entrance law for the noise law $ \mu $: The family $ (\lambda_k , k \in -\N) $ satisfies the consistency condition . Then there exists a unique element $ \bP \in \sP_\mu $ such that holds.

This is obvious by Kolmogorov’s extention theorem, so we omit the proof.

By Lemma \[ent law\], we see that there exists a solution $ \bP^*_\mu $ with marginal distributions given by . Let $ \bE^*_\mu $ denote the expectation with respect to $ \bP^*_\mu $. Let $ \phi $ be a bounded measurable function on $ G \times G^{{\left\{ k-n+1,\ldots,0 \right\}}} $. Noting that the independence of $ \eta_{k-n} $ and $ \sigma (\xi_{k-n+1},\ldots,\xi_0) $, we have $$\begin{aligned}
&& \bE^*_\mu {\left[  \phi {\left( \eta_k; \xi_{k-n+1}, \ldots, \xi_0 \right)}  