$$\label{assd}\int_B |T_1F^{(i)}|^3{\text{\rm d}}\lambda <\infty,\;i=1,\dots ,l,$$ there exists a constant $c$ such that $$\label{mltv}d_3((\hat{F}_a^{(1)},\dots ,\hat{F}_a^{(l)}),Z)\leq ca^{-\frac{1}{2}},\; a\geq 1.$$

For the Poisson facet processes $\eta_a,\,a\geq 1$ on $Y$ (\[bck\]) with intensity measure $a\lambda $ (\[inti\]) and the $U$-statistics $G_j(\eta), j=1,\dots ,d,$ in (\[hrm\]) we obtain that $$T_1G_j(x)=\frac{1}{(j-1)!}\int_Y\dots\int_Y{\mathbb H}^{d-j}(\cap_{i=1}^{j-1}y_i\cap x)\lambda^{j-1} ({\text{\rm d}}(y_1,\dots ,y_{j-1})).$$ The finiteness of the intensity measure $\lambda $ in (\[inti\]) and the boundedness of the facets guarantee that all integrals (\[assd\]) and (\[asco\]) are finite. Thus for the random vector $(G_j(\eta_a),j=1,\dots ,d)$ both the central limit theorem when $a\rightarrow\infty $ and the Berry-Esseen type inequality (\[mltv\]) hold.

The non-Poisson case
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Let facet processes $\mu_a,\,a\geq 1,$ have densities $$p_a({\mathbf x})=c_{\nu ,a}\exp (\nu\cdot G({\mathbf x}))$$ w.r.t. Poisson processes $\eta_a$ with intensities $\lambda_a=a\lambda ,$ respectively. Here $G({\mathbf x})$ is given in (\[hrm\]) and $\nu_j\leq 0,\; j=2,\dots ,d.$ We investigate $U$-statistics and submodels of the same order $d-k,$ from formula (\[effa\]) we have $$\label{znovu}{\mathbb E} G_{d-k}(\mu_a^{(d-k)})=$$$$=\frac{a^{d-k}}{(d-k)!}\int_{Y^{d-k}}{\mathbb H}^k(\cap_{i=1}^{d-k}y_i)\rho_{d-k}(y_1,\dots ,y_{d-k};\mu_a^{(d-k)})\lambda^{d-k}({\text{\rm d}}(y_1,\dots ,y_{d-k})),$$$ k=0,\dots ,d-2.$

\[ll1\]For the processes $\mu_a^{(d-k)}$ the correlation function $\rho_{d-k},\;k=0,\dots ,$ $d-2,$ has form $$\label{rods}
\rho_{d-k}(y_1,\dots ,y_{d-k};\mu_a^{(d-k)})=\frac{A(a)}{B(a)},$$ where $A(a)=$ $$\sum_{n=0}^\infty \frac{a^n}{n!}\int_Y\dots \int_Y\exp\left(\frac{\nu_{d-k}}{(d-k)!}\sum_{(x_1,\dots ,x_{d-k})\subset\{u_1,\dots ,u_n,y_1,\dots ,y_{d-k}\}}{\mathbb H}^k(\cap_{i=1}^{d-k}x_i)\right)$$$$\times\lambda^n ({\text{\rm d}}(u_1,\dots ,u_n)),$$$$B(a)=\sum_{n=0}^\infty \frac{a^n}{n!}\int_Y\dots \int_Y\exp\left(\frac{\nu_{d-k}}{(d-k)!}\sum_{(x_1,\dots ,x_{d-k})\subset\{u_1,\dots ,u_n\}}{\mathbb H}^k(\cap_{i=1}^{d-k}x_i)\right)$$$$\times\lambda^n ({\text{\rm d}}(u_1,\dots ,u_n)).$$

[**Proof:**]{} Using formulas (\[coref\]),(\[odhad\]) we obtain 