F={\mathbb E}F(\mu ),$ whenever the expectations exist. We write $T_nF$ for $T^\eta_nF.$ For the functionals of a Poisson process Theorem 1.1 in [@RefJ] says that given $F,\tilde{F}\in L^2(P_\eta )$ it holds $$\label{skaso}{\mathbb E}[F(\eta ){\tilde F}(\eta )]={\mathbb E}F(\eta ){\mathbb E}{\tilde F}(\eta )+\sum_{n=1}^\infty\frac{1}{n!}\langle T_nF,T_n{\tilde F}\rangle_n,$$ where $\langle .,.\rangle_n$ is the scalar product in $L_2(\lambda^n).$ We will use symbol $[n]=\{1,\dots ,n\},$ $n\in {\mathbb N}.$

For $p \in L_2(P_\eta ),\;n\in {\mathbb N},$ it holds $$\label{tnp}T_np(y_1,\dots , y_n)=\sum_{J\subset [n]}(-1)^{n-|J|}\rho_{|J|}(\{y_j,\:j\in J\};\mu )$$ for $\lambda^n$-almost all $(y_1,\dots , y_n),$ where $|J|$ is the cardinality of $J.$

U-statistics
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A $U$-statistic of order $k\in {\mathbb N}$ of a finite point process $\mu $ is a functional defined by $$\label{ust}F(\mu )=\sum_{(x_1,\dots ,x_k)\in\mu^k_{\neq }} f(x_1,\dots ,x_k),$$ where $f:B^k\rightarrow {\mathbb R}$ is a function symmetric w.r.t. to the permutations of its variables, $f\in L_1(\lambda^k ).$ Here $\mu^k_{\neq }$ is the set of $k$-tuples of different points of $\mu .$ We say that $F$ is driven by $f.$ In this Section basic results on $U$-statistics for point processes having densities, obtained in [@RefB] are reviewed for later use. By the Slivnyak-Mecke theorem [@RefS] we have $${\mathbb E}F(\eta )=\int_{B^k}f(x_1,\dots ,x_k)\lambda^k ({\text{\rm d}}(x_1,\dots ,x_k)).$$ For a $U$-statistic $F\in L_2(P_\eta )$ of order $k$ and 