of $(d-k)$-tuples, i.e. some orientations are missing. For $k=d-2$ e.g. all facets tend to be parallel (with any orientation). Additionally when the assertion of Theorem \[tecko\] is valid for $\mu_a^{(d-k)}$ then it holds also for expectations ${\mathbb E} G_j(\mu^{(d-k)}_a) $ with $j=d-k+1,\dots ,d.$

The behavior of Metropolis-Hastings chain for simulation of a realization of $\mu_a^{d-k}$ with fixed parameter $a$ depends on $\nu_{d-k}$ in the model with finitely many orientations. For $|\nu_{d-k}|$ large typically it converges quickly to a realization with some missing orientations. However when $\nu_{d-k}$ is close to zero it does not converge quickly at all. This property does not take place when the orientation distribution is absolutely continuous w.r.t. spherical Lebesgue measure.

$U$-statistics of the order smaller than the submodel
=====================================================

In this section we continue to study the model from previous subsection, see (\[ymod\]). Asymptotic moments (when $a\rightarrow\infty $) of functionals $G_{d-k}({\mathbf x})$ in the submodel $\mu_{a}^{(d)},$ where $k=1,\ldots,d-1;\; d \geq 3,$ will be investigated. Here the vanishing property (\[a\]) is not expected, see the following Table where crosses mean that the expected value is non-zero.

   $a\rightarrow\infty$                                                                     
  ---------------------- ------------------- --------------------- --- -------------------- -------------------
         Submodel         $\mathbb{E}G_{d}$   $\mathbb{E}G_{d-1}$   …   $\mathbb{E} G_{2}$   $\mathbb{E}G_{1}$
     $\mu^{(2)}_{a}$             $0$                  $0$           …          $0$               $\times$
     $\mu^{(3)}_{a}$             