density $p\in L_2(P_\eta )$ it holds $$\label{effa}{\mathbb E}F(\mu )=\int_{B^k}f(x_1,\dots ,x_k)\rho_k(x_1,\dots ,x_k;\mu)\lambda^k ({\text{\rm d}}(x_1,\dots ,x_k)).$$ In integrals like (\[effa\]) it does not mind that $\rho_k$ is defined only for distinct arguments $x_j.$ We are interested in higher-order and mixed moments in the following.

We can use a short expression of formulas for moments using diagrams and partitions, see [@RefT], [@RefL]. Let $\tilde{\prod}_k$ be the set of all partitions $\{J_i\}$ of $[k],$ where $J_i$ are disjoint blocks and $\cup J_i=[k].$ For $k=k_1+\dots +k_m$ and blocks $$J_i=\{ j: k_1+\dots +k_{i-1}< j\leq k_1+\dots +k_i\},\; i=1,\dots ,m,$$ consider the partition $\pi =\{J_i,\;1\leq i\leq m\}$ and let $\prod_{k_1,\dots ,k_m}\subset \tilde{\prod}_k$ be the set of all partitions $\sigma\in\tilde{\prod}_k$ such that $|J\cap J'|\leq 1$ for all $J\in\pi $ and all $J'\in\sigma .$ Here $|J|$ is the cardinality of a block $J\in\sigma .$ For a partition $\sigma\in\prod_{k_1\dots k_m}$ we define the function $(\otimes_{j=1}^mf_j)_\sigma:B^{|\sigma |}\rightarrow {\mathbb R}$ by replacing all variables of the tensor product $\otimes_{j=1}^mf_j$ that belong to the same block of $\sigma $ by a new common variable, $|\sigma |$ is the number of blocks in $\sigma .$

\[theo2\]Let $m\in\mathbb{N},\;\prod_{i=1}^mF_i\in L_2(P_{\eta}),$ $p\in L_2(P_\eta),$ where $F_i$ are $U$-statistics of orders $k_i$ driven by nonnegative functions $f_i,$ respectively, $i=1,\dots,m$. Then 