Studentized permutation validity under mixed-assignment centering

Assume AC and the sampling and kernel conditions of Permutation CLT for a two-sample U-statistic under mixed-assignment centering. Let the direct first-projection variances ${\sigma }_X^2,{\sigma }_Y^2$ and pooled variance ${\sigma }_p^2$ be positive and finite, and put ${\sigma }_N^2=E_{{\bar{P}}_N}{g}_N^2\to {\sigma }_p^2$. Use the studentized statistic in Studentized two-sample U-statistic and permutation distribution. Suppose both variance estimates are consistent for their direct projection variances under the original samples and are permutation-consistent for ${\sigma }_N^2$ in the conditional sense of Studentized permutation validity under a weak two-sample parameter null. A sufficient condition for the latter is that each estimate depends on its own group alone and is consistent for ${\sigma }_N^2$ on an IID ${\bar{P}}_N$ sample of that size.

Then the conditional permutation CDF of $S_{m,n}=U_{m,n}\mathrm{/}{V}_{m,n}^{\mathrm{S}\mathrm{E}}$ converges uniformly in probability to $\mathrm{\Phi }$, and under the true sampling law $S_{m,n}\Rightarrow N(0,1)$.

This specialization retains the original all-assignment centered branch. Joint two-group variance estimates satisfy the theorem when their permutation consistency is established directly; the separate-group transfer lemma does not by itself prove consistency of arbitrary joint estimates.