Suppose a we have a $U$-statistics $U_{1,N}$ (with kernel of order $n$) and another $U_{2,N}$ (with kernel of order $m$) both are non-degenerate and $n$ and $m$ not necessarily equal such that \begin{align} a_N\left(U_{1,N}-E\left(U_{1,N}\right)\right)&\Rightarrow N(0,\sigma_1^2),\\ b_N\left(U_{2,N}-E\left(U_{2,N}\right)\right)&\Rightarrow N(0,\sigma_2^2),\quad\text{as $N\to\infty$} \end{align} My questions are the following
If $a_N < b_N,\quad \forall N\geq 1$ and if we consider the joint distribution of $c_N\left(U_{1,N}-E\left(U_{1,N}\right), U_{2,N}-E\left(U_{2,N}\right)\right)$, with some scaling sequence $c_N$ , then what is the relationship between $\{c_N\}$ and $\{a_N\},\{b_N\}$?
What can we say about the above joint distribution if only $U_{1,N}$ is the degenerate $U$-statistics?
Any kind of help such as reference paper, ideas are helpful.
My primary attack of this problem using Cramer-Wold device. but that doesn't help since rates are different. (original data in i.i.d. setup.)
I consider $U$-statistics without the $\binom Nm$ or $\binom Nn$ normalization. When both $U$-statistics are non-degenerated, the appropriated normalization is $a_N=N^{-m+1/2}$ and $b_N=N^{-n+1/2}$. If $m\neq n$ and we take the same normalization for $U_{1,n}$ and $U_{2,n}$, then the normalization will kill one of these terms. If $m=n$, then we take $c_N=N^{-m+1/2}$ and get the convergence of $c_N\left(U_{1,N}-E\left(U_{1,N}\right), U_{2,N}-E\left(U_{2,N}\right)\right)$ to a Gaussian vector whose covariance matrix can be expressed in terms of $\operatorname{Cov}(h_a(X_1),h_b(X_1))$ where $a,b=1,2$ and $h_1,h_2$ are the functions associated to the linear part in the Hoeffding's decomposition.
If one of the statistics is degenerated, then it also depends on the order of degeneracy. If the appropriate normalization is the same for both, I do not know whether there exists a general answer.