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, enabling $\mathcal{P}$-uniform anytime-valid inference for the first time. Along the way, our Gaussian approximation also yields 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的分布$\mathcal{P}$一致有效？我们通过推导分布一致的任意时间有效推断程序对这一问题给出肯定回答。历史上，任意时间有效方法（包括置信序列、任意时间$p$值以及能在停止时刻进行推断的序贯假设检验）一直是基于非渐近方法论证的。尽管如此，基于中心极限定理等渐近方法因其简洁性、普适性和弱假设条件在统计工具箱中占据重要地位。尽管近期工作已推导出具备上述优点的渐近形式的任意时间有效方法，但这些方法尚未被证明是$\mathcal{P}$一致的，即其渐近性在分布类$\mathcal{P}$中并非一致有效。事实上，当前任意时间有效推断文献中尚缺乏同时满足$\mathcal{P}$一致性和样本量$n$一致性的中心极限理论可供借鉴。本文通过推导一个新颖的$\mathcal{P}$一致强高斯逼近定理填补了这一空白，首次实现了$\mathcal{P}$一致的任意时间有效推断。在此过程中，我们的高斯逼近还导出了$\mathcal{P}$一致的重对数律。