{"paper":{"title":"Asymptotic e-processes","license":"http://creativecommons.org/licenses/by/4.0/","headline":"A doubly indexed process approximating an e-process satisfies an asymptotic Ville inequality that uniformly bounds its excursions up to a time horizon growing with approximation quality.","cross_cats":["stat.ME","stat.TH"],"primary_cat":"math.ST","authors_text":"Mattes Mollenhauer, Pierre-Fran\\c{c}ois Massiani, Sebastian Schulze","submitted_at":"2026-04-21T11:34:24Z","abstract_excerpt":"We investigate the concept of an asymptotic e-process, which is a doubly-indexed stochastic process $(E_{m,n})_{m,n\\in\\mathbb{N}}$ that possesses, asymptotically for an approximation index $m\\to\\infty$, the properties of an e-process along a monitoring time index $n$. This constitutes the first in-depth study of this recently introduced concept, which is relevant in asymptotic sequential anytime-valid inference. Our theory is motivated by practical applications in sequential hypothesis testing, in which e-variables and e-processes can only be constructed approximately from observations due to "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We derive an asymptotic version of Ville's inequality, which bounds excursion probabilities of (E_{m,n})_{m,n∈ℕ} over some threshold uniformly over n up to a time horizon r_m that is determined by the quality of process approximation over m.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The approximation quality of the doubly indexed process to an e-process as m→∞ is sufficient to determine a growing time horizon r_m over which the uniform bound holds, as stated in the abstract's description of the limiting behavior.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Asymptotic e-processes approximate e-processes for large m, enabling an asymptotic Ville's inequality that bounds uniform excursion probabilities up to a time horizon r_m determined by approximation quality.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A doubly indexed process approximating an e-process satisfies an asymptotic Ville inequality that uniformly bounds its excursions up to a time horizon growing with approximation quality.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"a0c3954425e721992af4b5cb69ebf01e28da0f94faf70212fee8759adad34d9b"},"source":{"id":"2604.19353","kind":"arxiv","version":2},"verdict":{"id":"53ad64bb-6489-425a-a83f-1c07372d799f","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T01:34:44.177933Z","strongest_claim":"We derive an asymptotic version of Ville's inequality, which bounds excursion probabilities of (E_{m,n})_{m,n∈ℕ} over some threshold uniformly over n up to a time horizon r_m that is determined by the quality of process approximation over m.","one_line_summary":"Asymptotic e-processes approximate e-processes for large m, enabling an asymptotic Ville's inequality that bounds uniform excursion probabilities up to a time horizon r_m determined by approximation quality.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The approximation quality of the doubly indexed process to an e-process as m→∞ is sufficient to determine a growing time horizon r_m over which the uniform bound holds, as stated in the abstract's description of the limiting behavior.","pith_extraction_headline":"A doubly indexed process approximating an e-process satisfies an asymptotic Ville inequality that uniformly bounds its excursions up to a time horizon growing with approximation quality."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.19353/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_compliance","ran_at":"2026-05-20T03:04:23.787679Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"d29b5909a2e28753a251a729ac5210149a380e4041e23f26abad9d113919e273"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}