This paper develops a theory of distribution- and time-uniform asymptotics, culminating in the first large-sample anytime-valid inference procedures that are shown to be uniformly valid in a rich class of distributions. Historically, anytime-valid methods -- including confidence sequences, anytime $p$-values, and sequential hypothesis tests -- have been justified nonasymptotically. By contrast, large-sample inference procedures such as those based on the central limit theorem occupy an important part of statistical toolbox due to their simplicity, universality, and the weak assumptions they make. While recent work has derived asymptotic analogues of anytime-valid methods, they were not distribution-uniform (also called \emph{honest}), meaning that their type-I errors may not be uniformly upper-bounded by the desired level in the limit. The theory and methods we outline resolve this tension, and they do so without imposing assumptions that are any stronger than the distribution-uniform fixed-$n$ (non-anytime-valid) counterparts or distribution-pointwise anytime-valid special cases. It is shown that certain ``Robbins-Siegmund'' probability distributions play roles in anytime-valid asymptotics analogous to those played by Gaussian distributions in standard asymptotics. As an application, we derive the first anytime-valid test of conditional independence without the Model-X assumption.

翻译：本文发展了分布与时间一致性的渐近理论，最终构建出首个被证明在丰富分布类中具有一致有效性的大样本任意时间有效推断方法。历史上，任意时间有效方法——包括置信序列、任意时间$p$值与序贯假设检验——均通过非渐近论证获得理论支撑。相比之下，基于中心极限定理的大样本推断方法因其简洁性、普适性及所需假设较弱，始终占据统计学工具箱的重要地位。尽管近期研究已推导出任意时间有效方法的渐近类比形式，但这些方法不具备分布均匀性（亦称\emph{诚实性}），意味着其第一类错误在极限情况下可能无法被期望水平一致上界控制。本文阐述的理论与方法解决了这一矛盾，且未施加比分布均匀的固定样本量（非任意时间有效）对应方法或分布逐点任意时间有效特例更强的假设。研究证明，某些“Robbins-Siegmund”概率分布在任意时间有效渐近理论中扮演的角色，类似于高斯分布在标准渐近理论中的角色。作为应用，我们首次在不依赖Model-X假设的前提下推导出条件独立性检验的任意时间有效方法。