Are asymptotic confidence sequences and anytime $p$-values uniformly valid for a nontrivial class of distributions $\mathcal{P}$? We give a positive answer to this question by deriving distribution-uniform anytime-valid inference procedures. Historically, anytime-valid methods -- including confidence sequences, anytime $p$-values, and sequential hypothesis tests that enable inference at stopping times -- have been justified nonasymptotically. Nevertheless, asymptotic procedures such as those based on the central limit theorem occupy an important part of statistical toolbox due to their simplicity, universality, and weak assumptions. While recent work has derived asymptotic analogues of anytime-valid methods with the aforementioned benefits, these were not shown to be $\mathcal{P}$-uniform, meaning that their asymptotics are not uniformly valid in a class of distributions $\mathcal{P}$. Indeed, the anytime-valid inference literature currently has no central limit theory to draw from that is both uniform in $\mathcal{P}$ and in the sample size $n$. This paper fills that gap by deriving a novel $\mathcal{P}$-uniform strong Gaussian approximation theorem. We apply some of these results to obtain an anytime-valid test of conditional independence without the Model-X assumption, as well as a $\mathcal{P}$-uniform law of the iterated logarithm.

翻译：渐近置信序列和任意时间 $p$ 值是否对一类非平凡分布 $\mathcal{P}$ 具有均匀有效性？我们通过推导分布均匀的任意时间有效推断程序，对此问题给出肯定答案。历史上，任意时间有效方法（包括置信序列、任意时间 $p$ 值以及支持在停止时间进行推断的序贯假设检验）一直通过非渐近方式得到证明。尽管如此，基于中心极限定理等渐近程序因其简洁性、普适性和弱假设条件，在统计工具库中占据重要地位。虽然近期研究已推导出具有上述优点的任意时间有效方法的渐近版本，但尚未证明这些方法对 $\mathcal{P}$ 具有均匀性，即其渐近性质在分布类 $\mathcal{P}$ 中并非均匀有效。事实上，当前任意时间有效推断文献中缺乏既对 $\mathcal{P}$ 均匀、又对样本量 $n$ 均匀的中心极限理论。本文通过推导一个新的 $\mathcal{P}$-均匀强高斯逼近定理填补了这一空白。我们应用部分结果，在无需Model-X假设的情况下获得了条件独立性的任意时间有效检验，同时建立了 $\mathcal{P}$-均匀重对数律。