{
  "abstract": "In safe hypothesis testing with test supermartingales, 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 $1/α$. Due to an inherent asymmetry, the type-II error behaves differently. We prove two things about the latter, for a simple null and alternative. First, the mean growth rate $\\mathbb{E}_{P_1}[\\log E]$, the e-power, that Kelly betting and growth-rate-optimal e-variables maximise, bounds nothing on its own. For every level $c>0$, every $α$ and horizon $t$ we construct e-variables of conditional e-power exactly $c$ whose probability of not rejecting by $t$ is arbitrarily close to one. It forces eventual rejection, but no finite-horizon guarantee follows. Second, the quantity that does control the type-II error is the Chernoff-Stein exponent of an e-variable, $Λ(E)=\\sup_{s\\ge0}\\{-\\log \\mathbb{E}_{P_1}[E^{-s}]\\}$, whose range is exactly determined: $\\sup_E Λ(E)=\\mathrm{KL}(P_0\\|P_1)$, the classical Chernoff-Stein exponent, and so the ceiling of its own per-e-variable form. One conditional application of Hoelder's inequality per step gives it, for every test supermartingale on an arbitrary filtered space, with no independence or product structure; the i.i.d. case adds that it is matched, and attained by nothing. The e-power has its own ceiling, $\\mathrm{KL}(P_1\\|P_0)$, and that one is attained, $P_0$-a.s. uniquely, by the likelihood ratio $R$. The two optima are the same divergence in opposite arguments, at opposite ends of the flattened family $R^β/\\mathbb{E}_{P_0}[R^β]$: the ceiling as $β\\downarrow0$, $R$ at $β=1$. Which $β$ is best is settled by the horizon, exactly: $R$ is optimal at $t=\\log(1/α)/\\mathrm{KL}(P_1\\|P_0)$ alone, beaten by sharpening $(β>1)$ below it and by flattening above.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.27765",
  "author_total": 1,
  "authors": [
    {
      "name": "Patrick Forré"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e63r031vq3a1k.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.27765",
  "fulltext_url": null,
  "id": "e63r031vq3a1k",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.27765",
  "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": "W7214175227",
  "pdf_url": "https://arxiv.org/pdf/2609.27765",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e63r031vq3a1k.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.27765",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.27765"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "The Type-II Error of Test Supermartingales: e-Power versus the Chernoff-Stein Exponent",
  "title_text": "The Type-II Error of Test Supermartingales: e-Power versus the Chernoff-Stein Exponent",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:06Z",
  "url": "https://research.draconitic.com/paper/e63r031vq3a1k",
  "venue": "arXiv",
  "versions": []
}