\right]} 
\\
&& = \bE^*_\mu {\left[  \phi {\left( \eta_{k-n} \cdot \xi_{k-n+1} \cdots \xi_k; \xi_{k-n+1}, \ldots, \xi_0 \right)}  \right]} 
\\
&& = \bE^*_\mu {\left[  \phi {\left( \eta_{k-n}; \xi_{k-n+1}, \ldots, \xi_0 \right)}  \right]} . \end{aligned}$$ This implies .

Homogeneous space structure of the set of extremal points {#sec: hom sp}
=========================================================

Let $ \mu = (\mu_k , k \in -\N) $ be given. For each $ g \in G $, we define a continuous map $ T_g $ on $ \sP_\mu $ by $$T_g(\bP)(\cdot) = \bP {\left(  g \cdot \eta \in \cdot  \right)} , 
\quad \bP \in \sP_\mu$$ where we write $$g \cdot \eta = {\left(  g \cdot \eta_k , k \in -\N  \right)} , 
\quad g \in G , \, \eta \in G^{-\N}.$$ Then the following is obvious.

\[Tg action\] The family of maps $ (T_g : \, g \in G ) $ defines a group action of $ G $ on $ \sP_\mu $ with $ \ext $ an invariant subset, i.e., $$T_g(\sP_\mu) \subset \sP_\mu 
\quad \mbox{and} \quad 
T_g(\ext) \subset \ext 
\quad 
\mbox{for} \ g \in G 
\label{G action 1}$$ and $$T_g T_{h^{-1}} = T_{gh^{-1}} \quad \mbox{for $ g,h \in G $}. 
\label{G action 2}$$

The following proposition shows that 