Time-uniform central limit theory and asymptotic confidence sequences

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 very 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. To elaborate, our methods take the form of confidence sequences (CS) -- sequences of confidence intervals that are uniformly valid over time. CSs provide valid inference at arbitrary stopping times, incurring 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, and hence do not enjoy the aforementioned broad applicability of asymptotic confidence intervals. Our work bridges the gap by giving a definition for "asymptotic CSs", and deriving a universal asymptotic CS that requires only weak CLT-like assumptions. While the CLT approximates the distribution of a sample average by that of a Gaussian at a fixed sample size, we use strong invariance principles (stemming from the seminal 1960s work of Strassen and improvements by Koml\'os, Major, and Tusn\'ady) to uniformly approximate the entire sample average process by an implicit Gaussian process. We demonstrate their utility by deriving nonparametric asymptotic CSs for the average treatment effect based on doubly robust estimators in observational studies, for which no nonasymptotic methods can exist even in the fixed-time regime. This enables causal inference that can be continuously monitored and adaptively stopped.