# Time-uniform central limit theory and asymptotic confidence sequences arXiv: https://arxiv.org/abs/2103.06476 ยท HTML: https://mlsanity.com/paper/2103.06476/ ## Key facts - Cited by 29 papers in the map (data through 14 Oct 2026). - Builds on 9 of its 74 references that are in the map. - First posted on arXiv 11 Mar 2021. ## Builds on (most cited first, 9 of 9) - [Time-uniform, nonparametric, nonasymptotic confidence sequences](https://mlsanity.com/paper/1810.08240/) (2018) - [Estimating means of bounded random variables by betting](https://mlsanity.com/paper/2010.09686/) (2020) - [There are natural scores: Full comment on Shafer, "Testing by betting: A strategy for statistical and scientific communication"](https://mlsanity.com/paper/2102.05569/) (2021) - [Admissible anytime-valid sequential inference must rely on nonnegative martingales](https://mlsanity.com/paper/2009.03167/) (2020) - [Universal Inference](https://mlsanity.com/paper/1912.11436/) (2019) - [Sequential estimation of quantiles with applications to A/B-testing and best-arm identification](https://mlsanity.com/paper/1906.09712/) (2019) - [Confidence sequences for sampling without replacement](https://mlsanity.com/paper/2006.04347/) (2020) - [Sharp instruments for classifying compliers and generalizing causal effects](https://mlsanity.com/paper/1801.03635/) (2018) - [A New Framework for Online Testing of Heterogeneous Treatment Effect](https://mlsanity.com/paper/2002.03277/) (2020) ## Built on by (most cited first, 29 of 29) - [Semiparametric Efficient Inference in Adaptive Experiments](https://mlsanity.com/paper/2311.18274/) (2023) - [Prediction-Powered E-Values](https://mlsanity.com/paper/2502.04294/) (2025) - [An Experimental Design for Anytime-Valid Causal Inference on Multi-Armed Bandits](https://mlsanity.com/paper/2311.05794/) (2023) - [Peeking with PEAK: Sequential, Nonparametric Composite Hypothesis Tests for Means of Multiple Data Streams](https://mlsanity.com/paper/2402.06122/) (2024) - [Comparing Sequential Forecasters](https://mlsanity.com/paper/2110.00115/) (2021) - [Anytime-valid, Bayes-assisted, Prediction-Powered Inference](https://mlsanity.com/paper/2505.18000/) (2025) - [Time-Uniform Confidence Spheres for Means of Random Vectors](https://mlsanity.com/paper/2311.08168/) (2023) - [Anytime-Valid Inference for Double/Debiased Machine Learning of Causal Parameters](https://mlsanity.com/paper/2408.09598/) (2024) - [Certified Interventional Fidelity: Anytime-Valid, Adaptive Evaluation of Causal Claims in Mechanistic Interpretability](https://mlsanity.com/paper/2607.08349/) (2026) - [Asymptotic Time-Uniform Inference for Parameters in Averaged Stochastic Approximation](https://mlsanity.com/paper/2410.15057/) (2024) - [Simulation-Based Inference for Adaptive Experiments](https://mlsanity.com/paper/2506.02881/) (2025) - [Anytime-valid t-tests and confidence sequences for Gaussian means with unknown variance](https://mlsanity.com/paper/2310.03722/) (2023) - [Efficient Adaptive Experimentation with Noncompliance](https://mlsanity.com/paper/2505.17468/) (2025) - [Demistifying Inference after Adaptive Experiments](https://mlsanity.com/paper/2405.01281/) (2024) - [Early Stopping Based on Repeated Significance](https://mlsanity.com/paper/2408.00908/) (2024) - [A new and flexible class of sharp asymptotic time-uniform confidence sequences](https://mlsanity.com/paper/2502.10380/) (2025) - [Sharp Gaussian approximations for Decentralized Federated Learning](https://mlsanity.com/paper/2505.08125/) (2025) - [Design-Based Bandits Under Network Interference: Trade-Off Between Regret and Statistical Inference](https://mlsanity.com/paper/2510.07646/) (2025) - [Multi-Armed Sequential Hypothesis Testing by Betting](https://mlsanity.com/paper/2603.17925/) (2026) - [AV-AIVAT: 74x Cheaper Agent Evaluation with Certified Anytime-Valid Stopping in Imperfect-Information Games](https://mlsanity.com/paper/2608.06362/) (2026) - [Steady Continuous Monitoring is (Just Barely) Impossible for Tests of Unbounded Length](https://mlsanity.com/paper/2408.02821/) (2024) - [Online Data Collection for Efficient Semiparametric Inference](https://mlsanity.com/paper/2411.03195/) (2024) - [On the Precise Asymptotics of Universal Inference](https://mlsanity.com/paper/2503.14717/) (2025) - [GAAVI: Global Asymptotic Anytime Valid Inference for the Conditional Mean Function](https://mlsanity.com/paper/2602.08096/) (2026) - [Fixed-Horizon Self-Normalized Inference for Adaptive Experiments via Martingale AIPW/DML with Logged Propensities](https://mlsanity.com/paper/2602.15559/) (2026) - [Sharp asymptotic theory for Q-learning with LDTZ learning rate and its generalization](https://mlsanity.com/paper/2604.04218/) (2026) - [Asymptotically Log-Optimal Bayes-Assisted Confidence Sequences for Bounded Means](https://mlsanity.com/paper/2605.07964/) (2026) - [Adaptive Experimentation for Censored Survival Outcomes](https://mlsanity.com/paper/2605.18459/) (2026) - [Bandit Simulation for Average Reward Inference](https://mlsanity.com/paper/2606.00913/) (2026) ## Abstract (arXiv) 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.