A complete characterization of sequential testability and change detectability in i.i.d. models
Abstract
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).
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BibTeX
@misc{ramdas2026complete,
title = {{A complete characterization of sequential testability and change detectability in i.i.d. models}},
author = {Aaditya Ramdas},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05752},
eprint = {2609.05752},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05752},
}
RIS
TY - UNPB
TI - A complete characterization of sequential testability and change detectability in i.i.d. models
AU - Aaditya Ramdas
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.05752
UR - https://arxiv.org/abs/2609.05752
AB - 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 O_Q(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 (T_γ)_γ≥1 satisfying Prob_P^∞(T_γ≤σ)≤ E_P^∞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 O_Q(logγ).
ER -