A studentized rank U-statistic under a weak null

The kernel is antisymmetric. For $X\sim P$, its conditional mean against $Q$ is $g_X(0)=1\mathrm{/}2$ and $g_X(2)=-1\mathrm{/}2$, so $\theta (P,Q;h)=\frac{1}{2}(1\mathrm{/}2)+\frac{1}{2}(-1\mathrm{/}2)=0$. For the reverse P/Q assignment, the mean is $-\theta =0$ by antisymmetry. For P/P and Q/Q assignments, the two arguments are IID with the same law, so swapping them leaves their joint law unchanged while negating $h$; each such mean is therefore zero. Also, ${\sigma }_X^2=E_P[g_X^2]=1\mathrm{/}4$. For $Y\sim Q$, the conditional means are $g_Y(-1)=-1$, $g_Y(1)=0$, and $g_Y(3)=1$, so $E_Q{g}_Y=0$ and ${\sigma }_Y^2=1\mathrm{/}2$. Every P/Q or Q/P pair is distinct, so its second moment is $1$. Under P/P the second moment is the probability of unequal values, $1\mathrm{/}2$; under Q/Q it is $1-\{(1\mathrm{/}4{)}^2+(1\mathrm{/}2{)}^2+(1\mathrm{/}4{)}^2\}=5\mathrm{/}8$. All four assignment-wise second moments are positive and finite.

For $p=1\mathrm{/}2$, the pooled law $\bar{P}=(P+Q)\mathrm{/}2$ assigns masses $1\mathrm{/}8,1\mathrm{/}4,1\mathrm{/}4,1\mathrm{/}4,1\mathrm{/}8$ to $-1,0,1,2,3$. The pooled projection $g_{\bar{P}}(z)=E_{\bar{P}}h(z,Y)$ has values $7\mathrm{/}8,1\mathrm{/}2,0,-1\mathrm{/}2,-7\mathrm{/}8$ at those points. Its mean is zero and ${\sigma }_{\bar{P}}^2=E_{\bar{P}}[g_{\bar{P}}^2]=\frac{49}{256}+\frac{1}{8}=\frac{81}{256}>0.$ For general $p_N\to 1\mathrm{/}2$, the same finite support probabilities and projection values vary continuously with $p_N$, so the pooled projection variance tends to this positive value.

For $x$ in the finite support of $P$, set ${\widehat{g}}_{X,i}=n^{-1}{\sum }_{j}h(X_i,Y_j)-U_{m,n}$; for $y$ in the finite support of $Q$, set ${\widehat{g}}_{Y,j}=m^{-1}{\sum }_{i}h(X_i,Y_j)-U_{m,n}$. Define ${\widehat{\sigma }}_m^2$ and ${\widehat{\sigma }}_n^2$ as the sample variances of these estimated projection values. They are nonnegative measurable statistics. In either sampling regime, each empirical frequency on the finite support converges to its row probability by [F8], uniformly over that finite support. Since $h$ is bounded, the finite sums expressing $U_{m,n}$ and each empirical section mean then converge to their corresponding product-law means. Under the true samples these limits are $\theta =0$, $E_{Q}h(x,Y)$, and $E_{P}h(X,y)$; under two IID ${\bar{P}}_N$ samples the limits are $0$ and the pooled section integrals, because the pooled parameter is zero by antisymmetry. Subtracting $U_{m,n}$ therefore makes the estimated projections converge uniformly on each support to their centered section-integral values. The sample averages of their squares converge to the corresponding projection second moments, again by finite support and [F8]; their sample means are zero by construction. Thus the variance estimates converge to ${\sigma }_X^2,{\sigma }_Y^2$ under the true laws and to ${\sigma }_{{\bar{P}}_N}^2\to 81\mathrm{/}256$ under the pooled-mixture rows.

For each support point $z\in \{-1,0,1,2,3\}$, the empirical frequency of $z$ in an IID ${\bar{P}}_N$ group converges to ${\bar{P}}_N(\{z\})$ by the bounded Bernoulli Chebyshev estimate [F8]. Applying [F9] to this single-group statistic and taking a finite union over points and both groups proves that the two permuted groups' empirical frequencies converge jointly to the corresponding pooled frequencies. Their conditional failure probabilities given the pooled sample tend to zero in probability, by Markov's inequality applied to the unconditional failure probabilities. The two variance estimates, though depending on both groups, are finite polynomial expressions in those frequencies times $m\mathrm{/}(m-1)$ or $n\mathrm{/}(n-1)$; the kernel is bounded and the finite row probabilities converge. Therefore both estimates converge to ${\sigma }_{{\bar{P}}_N}^2$ in conditional permutation probability, in probability over the data. This directly verifies the joint-estimator premise of [F5].

Under AC [A1], steps 1.1, 1.2, 2.1, 3.1 verify assignment-wise centering, finite positive mixed second moments, positive direct and pooled projection variances, original-sample consistency and conditional permutation consistency of the joint estimates. Thus [F5] applies: both the true-sampling pivot and its conditional permutation distribution converge to the standard normal despite $P\mathrm{\ne }Q$.