# A complete characterization of sequential testability and change detectability in i.i.d. models

arXiv: https://arxiv.org/abs/2609.05752 · HTML: https://mlsanity.com/paper/2609.05752/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 47 references that are in the map.
- First posted on arXiv 4 Sep 2026.

## Builds on (most cited first, 19 of 19)

- [Time-uniform, nonparametric, nonasymptotic confidence sequences](https://mlsanity.com/paper/1810.08240/) (2018)
- [Game-theoretic statistics and safe anytime-valid inference](https://mlsanity.com/paper/2210.01948/) (2022)
- [Safe Testing](https://mlsanity.com/paper/1906.07801/) (2019)
- [Nonparametric Two-Sample Testing by Betting](https://mlsanity.com/paper/2112.09162/) (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)
- [Sequential Kernelized Independence Testing](https://mlsanity.com/paper/2212.07383/) (2022)
- [Model-X Sequential Testing for Conditional Independence via Testing by Betting](https://mlsanity.com/paper/2210.00354/) (2022)
- [A composite generalization of Ville's martingale theorem using e-processes](https://mlsanity.com/paper/2203.04485/) (2022)
- [Reducing sequential change detection to sequential estimation](https://mlsanity.com/paper/2309.09111/) (2023)
- [Deep anytime-valid hypothesis testing](https://mlsanity.com/paper/2310.19384/) (2023)
- [On Stopping Times of Power-one Sequential Tests: Tight Lower and Upper Bounds](https://mlsanity.com/paper/2504.19952/) (2025)
- [Asymptotically optimal sequential change detection for bounded means](https://mlsanity.com/paper/2602.05272/) (2026)
- [A Rank-Based Sequential Test of Independence](https://mlsanity.com/paper/2305.13818/) (2023)
- [On the existence of powerful p-values and e-values for composite hypotheses](https://mlsanity.com/paper/2305.16539/) (2023)
- [The optimal betting wealth growth rate](https://mlsanity.com/paper/2604.25280/) (2026)
- [Strong duality for the GROW criterion](https://mlsanity.com/paper/2606.24768/) (2026)
- [Non-partitioned e-detectors for nonparametric sequential change detection](https://mlsanity.com/paper/2607.28322/) (2026)
- [Universality of e-detectors for ARL control](https://mlsanity.com/paper/2608.12660/) (2026)

## Built on by (most cited first, 1 of 1)

- [Change detection with conformal martingales: new optimal constructions, and suboptimality of existing methods](https://mlsanity.com/paper/2609.27179/) (2026)

## Abstract (arXiv)

We give a necessary and sufficient condition for the existence of power-one sequential tests in an i.i.d. composite testing problem. A level-(α) test with power one against every alternative exists if and only if the alternatives are separated from the null by a countable family of finite-block events. We provide other equivalent conditions using randomized fixed-sample tests, bounded finite-block scores, e-processes, reduced-filtration test supermartingales, and a countable cover whose finite-block weak-* closed convex hulls are positively separated in total variation. As a bonus, the constructive proof yields tests have pointwise expected sample size (OQ(log(1/α))). Exactly the same conditions also characterize i.i.d. change detectability under optional-horizon average-run-length control: for every (η>0), they are equivalent to an alarm family ((Tγ)γ≥1) satisfying (ProbP^∞(Tγ≤σ)≤ EP^∞σ/γ) for every null law and every stopping time (σ). In fact, when these conditions hold, we can construct a single e-detector such that every null-law average run length lies between (γ) and ((1+η)γ+1), and having robust Lorden delay (OQ(logγ)).