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Confidence intervals based on the central limit theorem (CLT) are acornerstone of classical statistics. Despite being only asymptotically valid,they are ubiquitous because they permit statistical inference under very weakassumptions, and can often be applied to problems even when nonasymptoticinference is impossible. This paper introduces time-uniform analogues of suchasymptotic confidence intervals. To elaborate, our methods take the form ofconfidence sequences (CS) -- sequences of confidence intervals that areuniformly valid over time. CSs provide valid inference at arbitrary stoppingtimes, incurring no penalties for "peeking" at the data, unlike classicalconfidence intervals which require the sample size to be fixed in advance.Existing CSs in the literature are nonasymptotic, and hence do not enjoy theaforementioned broad applicability of asymptotic confidence intervals. Our workbridges the gap by giving a definition for "asymptotic CSs", and deriving auniversal asymptotic CS that requires only weak CLT-like assumptions. While theCLT approximates the distribution of a sample average by that of a Gaussian ata fixed sample size, we use strong invariance principles (stemming from theseminal 1960s work of Strassen and improvements by Koml\'os, Major, andTusn\'ady) to uniformly approximate the entire sample average process by animplicit Gaussian process. As an illustration of our theory, we deriveasymptotic CSs for the average treatment effect using efficient estimators inobservational studies (for which no nonasymptotic bounds can exist even in thefixed-time regime) as well as randomized experiments, enabling causal inferencethat can be continuously monitored and adaptively stopped.

Time-uniform central limit theory and asymptotic confidence sequences