# On Stopping Times of Power-one Sequential Tests: Tight Lower and Upper Bounds

arXiv: https://arxiv.org/abs/2504.19952 · HTML: https://mlsanity.com/paper/2504.19952/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 38 references that are in the map.
- First posted on arXiv 28 Apr 2025.

## Builds on (most cited first, 13 of 13)

- [On the Complexity of Best Arm Identification in Multi-Armed Bandit Models](https://mlsanity.com/paper/1407.4443/) (2014)
- [Time-uniform, nonparametric, nonasymptotic confidence sequences](https://mlsanity.com/paper/1810.08240/) (2018)
- [Explore First, Exploit Next: The True Shape of Regret in Bandit Problems](https://mlsanity.com/paper/1602.07182/) (2016)
- [Hypothesis testing with e-values](https://mlsanity.com/paper/2410.23614/) (2024)
- [Tight Concentrations and Confidence Sequences from the Regret of Universal Portfolio](https://mlsanity.com/paper/2110.14099/) (2021)
- [Nonparametric Two-Sample Testing by Betting](https://mlsanity.com/paper/2112.09162/) (2021)
- [Testing exchangeability: fork-convexity, supermartingales, and e-processes](https://mlsanity.com/paper/2102.00630/) (2021)
- [Auditing Fairness by Betting](https://mlsanity.com/paper/2305.17570/) (2023)
- [Optimal Best-Arm Identification Methods for Tail-Risk Measures](https://mlsanity.com/paper/2008.07606/) (2020)
- [On the Optimal Sample Complexity for Best Arm Identification](https://mlsanity.com/paper/1511.03774/) (2015)
- [The numeraire e-variable and reverse information projection](https://mlsanity.com/paper/2402.18810/) (2024)
- [Instance-Log-Optimality of Portfolio-Based E-Processes and their Sequential Hypothesis Tests](https://mlsanity.com/paper/2504.02818/) (2025)
- [Optimistic Interior Point Methods for Sequential Hypothesis Testing by Betting](https://mlsanity.com/paper/2502.07774/) (2025)

## Built on by (most cited first, 7 of 7)

- [Asymptotically Optimal Sequential Testing with Markovian Data](https://mlsanity.com/paper/2602.17587/) (2026)
- [Optimal Anytime-Valid Tests for Composite Nulls](https://mlsanity.com/paper/2512.20039/) (2025)
- [Multi-Armed Sequential Hypothesis Testing by Betting](https://mlsanity.com/paper/2603.17925/) (2026)
- [The optimal betting wealth growth rate](https://mlsanity.com/paper/2604.25280/) (2026)
- [A complete characterization of sequential testability and change detectability in i.i.d. models](https://mlsanity.com/paper/2609.05752/) (2026)
- [GAAVI: Global Asymptotic Anytime Valid Inference for the Conditional Mean Function](https://mlsanity.com/paper/2602.08096/) (2026)
- [Confidence Horizons](https://mlsanity.com/paper/2608.03889/) (2026)

## Abstract (arXiv)

We present two general lower bounds for stopping times of sequential tests between arbitrary composite nulls mathcal P and alternatives mathcal Q. The first lower bound is for the ``Wald setting'' where the type-1 error level α approaches zero for a fixed alternative Q ∈ mathcal Q, and equals log(1/α) divided by a certain infimum KL divergence between mathcal P and Q, termed operatornameKLinf. The second lower bound applies to the ``Farrell setting'', where α is fixed and operatornameKLinf approaches 0 along a sequence of alternatives such that the required expected sample size along that sequence is of order at least operatornameKL-1inf log log operatornameKL-1inf. Our main contribution is the generality of these bounds, which hold in non-parametric, composite settings, without requiring a dominating reference measure, substantially generalizing the known parametric results. We also provide sufficient conditions for matching upper bounds and show that these are met in several nontrivial non-parametric cases.