{
  "abstract": "In safe hypothesis testing with test supermartingals, Ville's inequality provides anytime-valid type-I error guarantees for every significance level $α\\in(0,1]$, if one rejects the null hypothesis whenever the wealth process first exceeds $\\frac{1}α$. Due to an inherent asymmetry, the type-II error does not have such guarantees: a heavy concentration of the probability on the lower tail of the log-increments can lead to one catastrophic bet that undoes any amount of accumulated evidence. This paper studies how different hypotheses on those lower-tail probabilities lead to different bounds on the type-II error of the sequential test. They all reduce to one master inequality, which bounds the type-II error at level $α$, at a fixed horizon and sequentially, in terms of a one-sided Legendre transform of the (inverse-)moment generating function of the e-variables, evaluated at one number: the amount by which the lower bound of the accumulated e-powers exceeds $\\log\\frac{1}α$. And, the step is lossless, in the sense, that it extracts exactly a constrained information projection. Every bound presented here is a corollary, obtained by a certain majorant of the above function. The hypotheses are: a finite negative moment; an exponentially small crash probability with a moment on the winning side; a wealth floor with a conditional variance, and its Bernstein variant, which interpolates between a Gaussian regime set by the variance and an exponential one set by the scale; a sub-Gaussian or bounded-tilt lower tail; bounded log-increments; and i.i.d. increments, where the majorant is the truth. We also provide an empirical-Bernstein variant. Each hypothesis may either be read as a condition on the e-variables one has, or as the price of betting with an approximation to the likelihood ratio rather than the ratio itself, which satisfies the weakest condition for free.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.27766",
  "author_total": 1,
  "authors": [
    {
      "name": "Patrick Forré"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3trbvffqxtj6k.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.27766",
  "fulltext_url": null,
  "id": "3trbvffqxtj6k",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.27766",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "2613|Probability and Statistical Research"
    ]
  ],
  "openalex": "W7214153646",
  "pdf_url": "https://arxiv.org/pdf/2609.27766",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3trbvffqxtj6k.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.27766",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.27766"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Type-II Error Bounds for Test Supermartingales from Lower-Tail Hypotheses",
  "title_text": "Type-II Error Bounds for Test Supermartingales from Lower-Tail Hypotheses",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:06Z",
  "url": "https://research.draconitic.com/paper/3trbvffqxtj6k",
  "venue": "arXiv",
  "versions": []
}