planar segment process satisfying certain conditions on absolute regularity coefficients.

In facet processes having densities we restrict ourselves to the model with finitely many orientations corresponding to canonical vectors. Submodels of the facet process are investigated where a detailed analysis of the correlation function yields different asymptotics. When the order of the submodel is not greater than the order of the observed $U$-statistic then asymptoticaly the mean value of the $U$-statistic vanishes. This leads to a degeneracy in the sense that some orientations are missing. On the other hand when the order of the submodel is greater than the order of the observed $U$-statistic then the limit of correlation function is finite and nonzero and under selected standardization $U$-statistic tends almost surely to its non-zero expectation. By changing the standardization, however, we achieve a finite non-zero asymptotic variance. Even if these results are obtained in a special situation with facets of a fixed shape and size related to the window size, it is important that they allow us to understand the ongoing problems for a possible further investigation of the complex model.

The background of point processes having a density {#sec:1}
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Consider a bounded Borel set $B\subset {\mathbb R}^d$ 