$$\label{divi}{\mathbb E}\left[\prod_{i=1}^mF_i(\mu )\right]=\sum_{\sigma\in\prod_{k_1\dots k_m}}\int_{B^{|\sigma |}}(\otimes_{i=1}^mf_i)_{\sigma }(x_1,\dots ,x_{|\sigma |})\times$$$$\times \rho_{|\sigma |}(x_1,\dots ,x_{|\sigma |};\mu )\lambda({\text{\rm d}}(x_1,\dots ,x_{|\sigma |})).$$

Similar formulas to (\[divi\]), but under a stronger assumption of a product form of the driving function $f$ of $F,$ were derived in [@RefD] using the Georgii-Nguyen-Zessin formula [@RefA]. As an application consider processes $\mu_a$ with densities $p_a$ w.r.t. $\eta_a,\,a\geq 1.$ Let formula (\[divi\]) be applied with $ \lambda_a.$ The term at the highest power of $a$ which comes from $a\lambda $ is called the leading term. The rate of the leading term is $\sum_{i=1}^{m}k_{i}$. There remains the dependence on $a$ hidden in $\rho ,$ which will be investigated later.

\[tht\] For a $U$-statistic $$F(\mu_a)=\sum_{(x_1,\dots ,x_{k})\in\mu^{k}_{a\neq }} f(x_1,\dots ,x_{k})$$ of order $k$ on a bounded set $B$ we have the following expression for the leading term of centered moments, the rate being $mk:$ ${\mathbb E}[(F(\mu_a)-{\mathbb E}F(\mu_a))^m]=$ $$\sum_{l=0}^m\binom{m}{l} (-1)^{m-l}\int_{B^{lk}} f^{\otimes l}(x_1,\dots ,x_{lk})\rho_{lk}(x_1,\dots ,x_{lk};\mu_a)\lambda^{lk}({\text{\rm d}}(x_1,\dots ,x_{lk}))$$$$\times\left(\int_{B^k}f(x_1,\dots ,x_{k})\rho_{k}(x_1,\dots ,x_{k};\mu_a)\lambda^{k}({\text{\rm d}}(x_1,\dots ,x_{k}))\right)^{m-l}.$$

[**Proof:**]{} We have $${\mathbb E}[(F(\mu_a)-{\mathbb E} F(\mu_a))^m]=\sum_{l=0}^m\binom{m}{l} (-1)^{m-l}{\mathbb E}F(\mu_a)^l({\mathbb E} F(\mu_a))^{m-l}.$$ From (\[effa\]) it is $({\mathbb E} F(\mu_a))^{m-l}=$$$=a^{k(m-l)}\left(\int_{B^k}f(x_1,\dots ,x_{k})\rho_{k}(x_1,\dots ,x_{k};\mu_a)\lambda^{k}({\text{\rm d}}(x_1,\dots ,x_{k}))\right)^{m-l}$$ and from (\[divi\]) applied to ${\mathbb E}[F(\mu_a)^l]$ we take the term with highest power of $a.$ It corresponds to $$a^{lk}\int_{B^{lk}} f^{\otimes l}(x_1,\dots ,x_{lk})\rho_{lk}(x_1,\dots ,x_{lk};\mu_a)\lambda^{lk}({\text{\rm 