# Optimal sequential tests yield log-optimal e-processes

arXiv: https://arxiv.org/abs/2605.12720 · HTML: https://mlsanity.com/paper/2605.12720/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 32 references that are in the map.
- First posted on arXiv 12 May 2026.

## Builds on (most cited first, 14 of 14)

- [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)
- [Game-theoretic statistics and safe anytime-valid inference](https://mlsanity.com/paper/2210.01948/) (2022)
- [Safe Testing](https://mlsanity.com/paper/1906.07801/) (2019)
- [Hypothesis testing with e-values](https://mlsanity.com/paper/2410.23614/) (2024)
- [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)
- [Universal Inference](https://mlsanity.com/paper/1912.11436/) (2019)
- [Testing exchangeability: fork-convexity, supermartingales, and e-processes](https://mlsanity.com/paper/2102.00630/) (2021)
- [E-detectors: a nonparametric framework for sequential change detection](https://mlsanity.com/paper/2203.03532/) (2022)
- [Sequential Predictive Two-Sample and Independence Testing](https://mlsanity.com/paper/2305.00143/) (2023)
- [The numeraire e-variable and reverse information projection](https://mlsanity.com/paper/2402.18810/) (2024)
- [A composite generalization of Ville's martingale theorem using e-processes](https://mlsanity.com/paper/2203.04485/) (2022)
- [Asymptotically optimal sequential change detection for bounded means](https://mlsanity.com/paper/2602.05272/) (2026)
- [Generic E-Variables for Exact Sequential k-Sample Tests that allow for Optional Stopping](https://mlsanity.com/paper/2106.02693/) (2021)

## Abstract (arXiv)

It has been recently shown that e-processes are sufficient for sequential testing in the following sense: every level-α sequential test can be obtained by thresholding an e-process at 1/α. However, in the above result, neither does the test have to be asymptotically optimal (in terms of stopping times) nor does the e-process have to be asymptotically log-optimal. It has separately been shown that asymptotically log-optimal e-processes yield asymptotically optimal sequential tests. In this paper, we prove the converse, arguably completing the story: it is possible to aggregate asymptotically optimal sequential tests into asymptotically log-optimal e-processes. This is accomplished by using a new class of WAIT e-processes: those that are Weighted Aggregates of Indicators of stopping Times that begin at zero, are nondecreasing and increase to infinity under the alternative at the optimal rate. Importantly, the paper discusses several nuances in the varied definitions of asymptotic (log-)optimality.