{
  "abstract": "Let $M_n$ be a nonnegative supermartingale with $M_0=1$, and let $T_b$ be the first time $M_n$ crosses $b>1$. Ville's inequality gives $P(T_b<\\infty)\\leq 1/b$. We show that, at every finite time horizon $N$, $$ 1=bP(T_b\\leq N)+V_{b,N}+D_{b,N}+R_{b,N}, $$ where $V_{b,N}$ is the expected overshoot at crossing, $D_{b,N}$ is the cumulative predictable decrease before crossing, and $R_{b,N}$ is the expected value of paths that up to the time horizon $N$ remained below the boundary $b$. The identity follows from telescoping and can be made horizon free without additional integrability assumption. It gives a formula for the slack in Ville's inequality and thus the conservatism of the threshold test $M_n\\geq b$. For likelihood ratio martingales, the same terms also describe power: $R_{b,N}$ is the probability, under the alternative, of not crossing by $N$, whereas $bP(T_b\\leq N)+V_{b,N}$ is the power. The identity also allows us to verify horizon-free guarantees from a finite computation: $P(T_b<\\infty)$ lies between $P(T_b\\leq N)$ and $P(T_b\\leq N)+R_{b,N}/b$, so a proposed threshold test $M_n\\geq b_α$ can be shown to be of level $α\\in(0,1)$ for every time horizon based on computations at the single horizon $N$. We illustrate how Ville's identity yields sharper thresholds for Bernoulli and exponential likelihood-ratio tests, and for a conformal test of exchangeability, demonstrating Monte Carlo calibration under a composite, nonparametric null. Lastly, we also establish a continuous-time analogue of Ville's identity.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.04620",
  "author_total": 6,
  "authors": [
    {
      "name": "Victor H. de la Pena"
    },
    {
      "name": "Michael J. Klass"
    },
    {
      "name": "Victor K. de la Pena"
    },
    {
      "name": "Alexander Ly"
    },
    {
      "name": "Logan Neishloss"
    },
    {
      "name": "Fangyuan Lin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6jpfjp0v5tn4z.bib",
  "cited_by": 1,
  "doi": "10.48550/arxiv.2607.04620",
  "fulltext_url": null,
  "id": "6jpfjp0v5tn4z",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.04620",
  "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",
      "math.PR"
    ],
    [
      "openalex",
      "2613|Probability and Statistical Research"
    ]
  ],
  "openalex": "W7167628487",
  "pdf_url": "https://arxiv.org/pdf/2607.04620",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6jpfjp0v5tn4z.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:28Z",
      "native_id": "2607.04620",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.04620"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "The Exact Ville Identity with Applications to E-Values",
  "title_text": "The Exact Ville Identity with Applications to E-Values",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/6jpfjp0v5tn4z",
  "venue": "arXiv",
  "versions": []
}