Confidence intervals based on the central limit theorem (CLT) are a cornerstone of classical statistics. Despite being only asymptotically valid, they are ubiquitous because they permit statistical inference under weak assumptions and can often be applied to problems even when nonasymptotic inference is impossible. This paper introduces time-uniform analogues of such asymptotic confidence intervals, adding to the literature on confidence sequences (CS) -- sequences of confidence intervals that are uniformly valid over time -- which provide valid inference at arbitrary stopping times and incur no penalties for "peeking" at the data, unlike classical confidence intervals which require the sample size to be fixed in advance. Existing CSs in the literature are nonasymptotic, enjoying finite-sample guarantees but not the aforementioned broad applicability of asymptotic confidence intervals. This work provides a definition for "asymptotic CSs" and a general recipe for deriving them. Asymptotic CSs forgo nonasymptotic validity for CLT-like versatility and (asymptotic) time-uniform guarantees. While the CLT approximates the distribution of a sample average by that of a Gaussian for a fixed sample size, we use strong invariance principles (stemming from the seminal 1960s work of Strassen) to uniformly approximate the entire sample average process by an implicit Gaussian process. As an illustration, we derive asymptotic CSs for the average treatment effect in observational studies (for which nonasymptotic bounds are essentially impossible to derive even in the fixed-time regime) as well as randomized experiments, enabling causal inference in sequential environments.

翻译：基于中心极限定理（CLT）的置信区间是经典统计学的基石。尽管仅具有渐近有效性，但由于其能在较弱的假设下进行统计推断，且即使无法进行非渐近推断时也常可应用于问题，因此被广泛使用。本文引入了此类渐近置信区间的时间一致类似物，丰富了关于置信序列（CS）的文献——即随时间一致有效的置信区间序列——这些序列允许在任意停止时间进行有效推断，且不像经典置信区间要求预先固定样本量那样，对数据“窥视”不产生惩罚。现有文献中的CS是非渐近的，具有有限样本保证，但不具备前述渐近置信区间的广泛适用性。本文为“渐近CS”提供了定义和推导通用方法。渐近CS放弃非渐近有效性，以换取类似CLT的通用性和（渐近）时间一致保证。CLT通过高斯分布近似固定样本量下的样本均值分布，而我们利用强不变原理（源于斯特拉森1960年代的开创性工作）将整个样本均值过程一致地近似为隐式高斯过程。作为示例，我们推导了观察性研究（即使在固定时间框架下也几乎无法推导非渐近界）以及随机实验中平均处理效应的渐近CS，从而实现了序贯环境中的因果推断。