# Power one sequential tests exist for weakly compact mathscr P against mathscr Pc

arXiv: https://arxiv.org/abs/2604.03218 · HTML: https://mlsanity.com/paper/2604.03218/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 19 references that are in the map.
- First posted on arXiv 3 Apr 2026.

## Builds on (most cited first, 4 of 4)

- [Game-theoretic statistics and safe anytime-valid inference](https://mlsanity.com/paper/2210.01948/) (2022)
- [Safe Testing](https://mlsanity.com/paper/1906.07801/) (2019)
- [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)

## Built on by (most cited first, 2 of 2)

- [The optimal betting wealth growth rate](https://mlsanity.com/paper/2604.25280/) (2026)
- [Non-partitioned e-detectors for nonparametric sequential change detection](https://mlsanity.com/paper/2607.28322/) (2026)

## Abstract (arXiv)

We study power-one sequential testing for an i.i.d. law on a Polish sample space. Given a nonempty composite null class Pcalsubseteqmathcal M1(X), we ask when there exists a level-α stopping rule that rejects almost surely under every alternative in a prescribed class QcalsubseteqPcalc. Our main sufficient condition is local weak lower semicontinuity and positivity of the information projection functional ( Phi_Pcal(Q):=infP∈PcalKL(Q|P). ) In particular, if Pcal is weakly compact, then for every α∈(0,1) there is a single level-α sequential test with power one against the entire complement Pcalc. The proof combines Csisz'ar's nonasymptotic Sanov bound for weakly closed convex empirical-measure sets with a Lindel"of countable-subcover argument. We also show that weak lower semicontinuity is sufficient but not necessary by giving examples where discontinuous finite-sample events separate alternatives that weak neighborhoods cannot detect. Finally, we construct an e-process that is asymptotically relatively growth-rate optimal under weak compactness. We verify the weak-lower-semicontinuity condition for weakly compact nulls, f-divergence balls, several integral probability metric balls, Wasserstein balls on proper spaces, and a number of non-weakly-compact semiparametric examples.