I would like your expertise or recommandations for a problem I am trying to solve. Let us take $n$ $i.i.d$ samples $X_1, \cdots, X_n \sim \alpha$. Let us consider a symmetric kernel $h$ of order m. I define the following U-statistics of the empirical distribution $\alpha$ as: \begin{equation} U_{n}=\left(\begin{array}{c}{n} \\ {m}\end{array}\right)^{-1} \sum_{c} h\left(X_{i_{1}}, \ldots, X_{i_{m}}\right) \end{equation}
where $\sum_{c}$ denotes summation over the $\left(\begin{array}{c}{n} \\ {m}\end{array}\right)$ combinations of distinct elements ${(i_{1}, \ldots, i_{m})}$ from $\{1, \ldots, n\}$. We see that $U_{n}$ is an unbiased estimator of the parameter of:
\begin{align} \theta(F)&=E_{F}\left[h\left(X_{1}, \ldots, X_{m}\right)\right] \\ &=\int \cdots \int h\left(x_{1}, \ldots, x_{m}\right) d \alpha\left(x_{1}\right) \cdots d \alpha \left(x_{m}\right) \end{align}
I am interested in the asymptotic minimax rate of the complete U-statistic $U_n$ to $\theta(F)$ as a function of $n$. I found TCL results, deviation bound results but I have not found minimax rate for U-statistic. Would you have some references for this problem ?
Thank you very much