Permutation CLT for a two-sample U-statistic under mixed-assignment centering

Assume AC. Let $X_1,\dots ,X_m$ be IID $P$ and independently let $Y_1,\dots ,Y_n$ be IID $Q$, with $N=m+n$, $m\mathrm{/}N\to p\in (0,1)$, and $\theta (P,Q;h)=0$. Fix $r$ and a measurable kernel $h$ that is symmetric in each block and antisymmetric under block exchange. Assume for every assignment of independent P- and Q-distributed arguments to the $2r$ positions that $E[h]=0$ and $0<E[h^2]<\mathrm{\infty }$. Let ${\sigma }_p^2=E_{\bar{P}}[g_{\bar{P}}^2]>0$ for $\bar{P}=pP+(1-p)Q$, where $g_{\bar{P}}$ is its first projection. Then the conditional permutation CDF of $\sqrt{m}{U}_{m,n}$ satisfies

${\sup }_{t\in \mathbb{R}}\mid {\widehat{R}}_{m,n}^{\sqrt{m}U}(t)-\mathrm{\Phi }\text{\hspace{0.17em}⁣}\left(\frac{t}{{\tau }_p}\right)\mid {\to }_{P}0,{\tau }_p^2=\frac{r^2}{1-p}{\sigma }_p^2\mathrm{.}$

This is the all-assignment-centered specialization of the general weak-null theorem. Its extra centering assumptions are sufficient but are not needed by the locally proved permutation coupling.